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0 such that !1 T D !2 T D 0 mod 2. If !1 /!2 is irrational, the motion never returns to its initial value. On the other hand it densely fills T 2 , in the sense that it comes arbitrarily close to any point of T 2 . We say in that case that the motion is quasi-periodic. The definition extends to T d , d > 2: a linear motion ˛ ! ˛ C ! t on T d is quasi-periodic if the components of ! are rationally independent, that is if ! D !1 1 C C!d d D 0 for 2 Zd if and only if D 0 (a b is the standard scalar product between the two vectors a, b). More generally we say that a motion on a manifold is quasi-periodic if, in suitable coordinates, it can be described as a linear quasi-periodic motion. The vector ! is usually called the frequency or rotation vector. Renormalization group By renormalization group one denotes the set of techniques and concepts used to study problems where there are some scale invariance properties. The basic mechanism consists in considering equations depending on some parameters and defining some transformations on the equations, including a suitable rescaling, such that after the transformation the equations can be expressed, up to irrelevant corrections, in the same form as before but with new values for the parameters. Torus The 1-torus T is defined as T D R/2Z, that is the set of real numbers defined modulo 2 (this means that x is identified with y if x y is a multiple of 2). So it is the natural domain of an angle. One defines the d-torus T d as a product of d 1-tori, that is T d D T T . For instance one can imagine T 2 as a square with the opposite sides glued together. Tree A graph is a collection of points, called nodes, and of lines which connect the nodes. A walk on the graph is a sequence of lines such that any two successive lines in the sequence share a node; a walk is nontrivial if it
Diagrammatic Methods in Classical Perturbation Theory
contains at least one line. A tree is a planar graph with no closed loops, that is, such that there is no nontrivial walk connecting any node to itself. An oriented tree is a tree with a special node such that all lines of the tree are oriented toward that node. If we add a further oriented line connecting the special node to another point, called the root, we obtain a rooted tree (see Fig. 1 in Sect. “Trees and Graphical Representation”). Definition of the Subject Recursive equations naturally arise whenever a dynamical system is considered in the regime of perturbation theory; for an introductory article on perturbation theory see Perturbation Theory. A classical example is provided by Celestial Mechanics, where perturbation series, known as Lindstedt series, are widely used; see Gallavotti [21] and Perturbation Theory in Celestial Mechanics. A typical problem in Celestial Mechanics is to study formal solutions of given ordinary differential equations in the form of expansions in a suitable small parameter, the perturbation parameter. In the case of quasi-periodic solutions, the study of the series, in particular of its convergence, is made difficult by the presence of the small divisors – which will be defined later on. Under some non-resonance condition on the frequency vector, one can show that the series are well-defined to any order. The first proof of such a property was given by Poincaré [53], even if the convergence of the series remained an open problem up to the advent of KAM theory – an account can be found in Gallavotti [17] and in Arnold et al. [2]; see also Kolmogorov–Arnold–Moser (KAM) Theory. KAM is an acronym standing for Kolmogorov [47], Arnold [1] and Moser [51], who proved in the middle of last century the persistence of most of invariant tori for quasi-integrable systems. Kolmogorov and Arnold proofs apply to analytic Hamiltonian systems, while Moser’s approach deals also with the differentiable case; the smoothness condition on the Hamiltonian function was thereafter improved by Pöschel [54]. In the analytic case, the persisting tori turn out to be analytic in the perturbation parameter, as explicitly showed by Moser [52]. In particular, this means that the perturbation series not only are well-defined, but also converge. However, a systematic analysis with diagrammatic techniques started only recently after the pioneering, fundamental works by Eliasson [16] and Gallavotti [18], and were subsequently extended to many other problems with small divisors, including dynamical systems with infinitely many degrees of freedom, such as nonlinear partial differential equations, and non-Hamiltonian systems.
Some of these extensions will be discussed in Sect. “Generalizations”. From a technical point of view, the diagrammatic techniques used in classical perturbation theory are strongly reminiscent of the Feynman diagrams used in quantum field theory: this was first pointed out by Gallavotti [18]. Also the multiscale analysis used to control the small divisors is typical of renormalization group techniques, which have been successfully used in problems of quantum field theory, statistical mechanics and classical mechanics; see Gallavotti [20] and Gentile & Mastropietro [38] for some reviews. Note that there exist other renormalization group approaches to the study of dynamical systems, and of KAMlike problems in particular, different from that outlined in this article. By confining ourselves to the framework of problems of KAM-type, we can mention the paper by Bricmont et al. [11], which also stressed the similarity of the technique with quantum field theory, and the so called dynamical renormalization group method – see MacKay [50] – which recently produced rigorous proofs of persistence of quasi-periodic solutions; see for instance Koch [46] and Khanin et al. [45]. Introduction Consider the ordinary differential equation on Rd Du D G(u) C "F(u) ;
(1)
where D is a pseudo-differential operator and G, F are real analytic functions. Assume that (1) admits a solution u(0) (t) for " D 0, that is Du(0) D G(u(0) ). The problem we are interested in is to investigate whether there exists a solution of (1) which reduces to u(0) as " ! 0. For simplicity assume G D 0 in the following. The first attempt one can try is to look for solutions in the form of power series in ", u(t) D
1 X
" k u(k) (t) ;
(2)
kD0
which, inserted into (1), when equating the left and right hand sides order by order, gives the list of recursive equations Du(0) D 0, Du(1) D F(u(0) ), Du(2) D @u F(u(0) )u(1) , and so on. In general to order k 1 one has Du(k) D
k1 X 1 s @u F(u(0) ) s! sD0
X
u(k 1 ) : : : u(k s ) ;
(3)
k 1 CCk s Dk1 k i 1
where @su F, the sth derivative of F, is a tensor with s C 1 indices (s must be contracted with the vectors
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Diagrammatic Methods in Classical Perturbation Theory
u(k 1 ) ; : : : ; u(k s ) ), and the term with s D 0 in the sum has to be interpreted as F(u(0) ) and appears only for k D 1. For instance for F(u) D u3 the first orders give Du(1) D u(0)3 ; Du(2) D 3u(0)2 u(1) ; Du(3) D 3u(0)2 u(2) C 3u(0) u(1)2 ;
(4)
Du(4) D 3u(0)2 u(3) C 6u(0) u(1) u(2) C u(1)3 ; as is easy to check. If the operator D can be inverted then the recursions (3) provide an algorithm to compute the functions u(k) (t). In that case we say that (2) defines a formal power series: by this we mean that the functions u(k) (t) are well-defined for all k 0. Of course, even if this can be obtained, there is still the issue of the convergence of the series that must be dealt with. Examples In this section we consider a few paradigmatic examples of dynamical systems which can be described by equations of the form (1). A Class of Quasi-integrable Hamiltonian Systems Consider the Hamiltonian system described by the Hamiltonian function H (˛; A) D 12 A2 C " f (˛) ;
Td
(5)
Rd
where (˛; A) 2 are angle-action variables, with T D R/2Z, f is a real analytic function, 2-periodic in each of its arguments, and A2 D A A, if (here and henceforth) denotes the standard scalar product in Rd , that is a b D a1 b1 C C ad bd . Assume also for simplicity that f is a trigonometric polynomial of degree N. The corresponding Hamilton equations are (we shorten @x D @/@x )
(k) ˛¨ (k) D " f (˛) :D
k1 X 1 sC1 @˛ f (! t) s! sD0
X
˛ (k 1 ) : : : ˛ (k s ) : (7)
k 1 CCk s Dk1 k i 1
We look for a quasi-periodic solution of (6), that is a solution of the form ˛(t) D ! t C h(! t), with h(! t) D O("). We call h the conjugation function, as it “conjugates” (that is, maps) the perturbed solution ˛(t) to the unperturbed solution ! t. In terms of the function h (6) becomes h¨ D "@˛ f (! t C h) ;
(8)
where @˛ denotes derivative with respect to the argument. Then (8) can be more conveniently written in Fourier space, where the operator D acts as a multiplication operator. If we write h(! t) D
X
e i!t h ;
2Z d
h D
1 X
" k h(k) ;
which can be written as an equation involving only the angle variables: (6)
which is of the form (1) with u D ˛, G D 0, F D @˛ f , and D D d2 /dt 2 . For " D 0, ˛ (0) (t) D ˛0 C! t is a solution of (6) for any choice of ˛0 2 T d and ! 2 Rd . Take for simplicity ˛0 D
(9)
kD1
and insert (9) into (8) we obtain (k) (! )2 h(k) D "@˛ f (˛) :D
k1 X
X
X
sD0 k 1 CCk s Dk1 0 C1 CC s D i 2Z n k i 1
(i0 )sC1 f0 h(k11 ) : : : h(ks s ) :
˛˙ D @A H (˛; A) D A ; A˙ D @˛ H (˛; A) D "@˛ f (˛) ;
˛¨ D "@˛ f (˛) ;
0: we shall see that this choice makes sense. We say that for " D 0 the Hamiltonian function (5) describes a system of d rotators. We call ! the frequency vector, and we say that ! is irrational if its components are rationally independent, that is if ! D 0 for 2 Zd if and only if D 0. For irrational ! the solution ˛ (0) (t) describes a quasi-periodic motion with frequency vector !, and it densely fills T d . Then (3) becomes
1 s! (10)
These equations are well-defined to all orders provided ["@˛ f (˛)](k) D 0 for all such that ! D 0. If ! is an irrational vector we need ["@˛ f (˛)](k) 0 D 0 for the equations to be well-defined. In that case the coefficients h0(k) are left undetermined, and we can fix them arbitrarily to vanish (which is a convenient choice). We shall see that under some condition on ! a quasiperiodic solution ˛(t) exists, and densely fills a d-dimensional manifold. The analysis carried out above for ˛0 D 0 can be repeated unchanged for all values of ˛0 2 T d : ˛0
Diagrammatic Methods in Classical Perturbation Theory
represents the initial phase of the solution, and by varying ˛0 we cover all the manifold. Such a manifold can be parametrized in terms of ˛0 , so it represents an invariant torus for the perturbed system. A Simplified Model with No Small Divisors Consider the same equation as (8) with D D d2 /dt 2 replaced by 1, that is h D "@˛ f (! t C h) :
(11)
Of course in this case we no longer have a differential equation; still, we can look again for quasi-periodic solutions h(! t) D O(") with frequency vector !. In such a case in Fourier space we have (k) h(k) D "@˛ f (˛) :D
k1 X
X
X
sD0 k 1 CCk s Dk1 0 C1 CC s D i 2Z n k i 1
1 (i0 )sC1 s!
f0 h(k11 ) : : : h(ks s ) :
(12)
For instance if d D 1 and f (˛) D cos ˛ the equation, which is known as the Kepler equation, can be explicitly solved by the Lagrange inversion theorem [55], and gives
h(k)
D
8 ˆ < ˆ :
jj k;
i(1) kC(k)/2 ; 2 ((k )/2)!((k C )/2)!
C k even ;
0;
otherwise :
k
(13) We shall show in Sect. “Small Divisors” that a different derivation can be provided by using the forthcoming diagrammatic techniques. The Standard Map Consider the finite difference equation D˛ D " sin ˛ ;
(14)
on T , where now D is defined by D˛( ) :D 2˛( ) ˛( By writing ˛ D Dh D " sin(
C !) ˛(
k1
h(k) D
X 1 2 4 sin (!/2) sD0
X
X
k 1 CCk s Dk1 0 C1 CC s D i 2Z k i 1
(i0 )sC1 f0 h(k11 ) : : : h(ks s ) ;
1 s!
(18)
where 0 D ˙1 and f˙1 D 1/2. Note that (17) is a discrete dynamical system. However, when passing to Fourier space, (18) acquires the same form as for the continuous dynamical systems previously considered, simply with a different kernel for D. In particular if we replace D with 1 we recover the Kepler equation. The number ! is called the rotation number. We say that ! is irrational if the vector (2; !) is irrational according to the previous definition. Trees and Graphical Representation Take ! to be irrational. We study the recursive equations ( (k) h(k) D g(! ) "@˛ f (˛) ; ¤ 0 ; (19) (k) D0; "@˛ f (˛) 0 D 0 ; where the form of g depends on the particular model we are investigating. Hence one has either g(! ) D (! )2 or g(! ) D 1 or g(! ) D (2 sin(!/2))2 according to models described in Sect. “Examples”. For ¤ 0 we have equations which express the co0 efficients h(k) , 2 Zd , in terms of the coefficients h(k ) , 2 Zd , with k 0 < k, provided the equations for D 0 are satisfied for all k 1. Recursive equations, such as (19), naturally lead to a graphical representation in terms of trees. Trees
!) :
(15)
C h( ), (14) becomes C h) ;
In other words, by writing x D C h( ) and y D ! C h( ) h( !), with ( ; !) solving (17) for " D 0, that is ( 0 ; ! 0 ) D ( C !; !), we obtain a closed-form equation for h, which is exactly (16). In Fourier space the operator D acts as D : e i ! 4 sin2 (!/2)e i , so that, by expanding h according to (9), we can write (16) as
(16)
which is the functional equation that must be solved by the conjugation function of the standard map ( x 0 D x C y C " sin x ; (17) y 0 D y C " sin x :
A connected graph G is a collection of points (nodes) and lines connecting all of them. Denote with N(G ) and L(G ) the set of nodes and the set of lines, respectively. A path between two nodes is the minimal subset of L(G ) connecting the two nodes. A graph is planar if it can be drawn in a plane without graph lines crossing. A tree is a planar graph G containing no closed loops. Consider a tree G with a single special node v0 : this introduces a natural partial ordering on the set of lines and nodes, and one can imagine that each line carries an arrow
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Diagrammatic Methods in Classical Perturbation Theory
pointing toward the node v0 . We add an extra oriented line `0 exiting the special node v0 ; the added line will be called the root line and the point it enters (which is not a node) will be called the root of the tree. In this way we obtain a tree defined by N( ) D N(G ) and L( ) D L(G ) [ `0 . A labeled tree is a rooted tree together with a label function defined on the sets L( ) and N( ). We call equivalent two rooted trees which can be transformed into each other by continuously deforming the lines in the plane in such a way that the lines do not cross each other. We can extend the notion of equivalence also to labeled trees, by considering equivalent two labeled trees if they can be transformed into each other in such a way that the labels also match. In the following we shall deal mostly with nonequivalent labeled trees: for simplicity, where no confusion can arise, we call them just trees. Given two nodes v; w 2 N( ), we say that w v if v is on the path connecting w to the root line. We can identify a line ` through the node v it exits by writing ` D `v . We call internal nodes the nodes such that there is at least one line entering them, and end-points the nodes which have no entering line. We denote with L( ), V ( ) and E( ) the set of lines, internal nodes and end-points, respectively. Of course N( ) D V( ) [ E( ). The number of unlabeled trees with k nodes (and hence with k lines) is bounded by 22k , which is a bound on the number of random walks with 2k steps [38]. For each node v denote by S(v) the set of the lines entering v and set sv D jS(v)j. Hence sv D 0 if v is an endnode, and sv 1 if v is an internal node. One has X X sv D sv D k 1 ; (20) v2N( )
v2V ( )
this can be easily checked by induction on the order of the tree. An example of unlabeled tree is represented in Fig. 1. For further details on graphs and trees we refer to the literature; cf. for instance Harary [43]. Labels and Diagrammatic Rules We associate with each node v 2 N( ) a mode label v 2 Zd , and with each line ` 2 L( ) a momentum label ` 2 Zd , with the constraint X X w D v C ` ; (21) `v D w2N( ) wv
`2S(v)
which represents a conservation rule for each node. Call T k; the set of all trees with k nodes and momentum associated with the root line. We call k and the order and the momentum of , respectively.
Diagrammatic Methods in Classical Perturbation Theory, Figure 1 An unlabeled tree with 17 nodes
Diagrammatic Methods in Classical Perturbation Theory, Figure 2 Graph element
Diagrammatic Methods in Classical Perturbation Theory, Figure 3 Graphical representation of the recursive equations
We want to show that trees naturally arise when studying (19). Let h(k) be represented with the graph element in Fig. 2 as a line with label exiting from a ball with label (k). Then we can represent (19) graphically as depicted in Fig. 3. Simply represent each factor h(ki i ) on the right hand side as a graph element according to Fig. 2. The lines of all such graph elements enter the same node v0 . This is a graphical expedient to recall the conservation rule: the momentum of the root line is the sum of the mode label 0 of the node v0 plus the sum of the momenta of the lines entering v0 . The first few orders k 4 are as depicted in Fig. 4. For each node the conservation rule (21) holds: for instance for k D 2 one has D 1 C 2 , for k D 3 one has D 1 C `1 and `1 D 2 C 3 in the first tree and D 1 C 2 C 3 in
Diagrammatic Methods in Classical Perturbation Theory
Diagrammatic Methods in Classical Perturbation Theory, Figure 4 Trees of lower orders
the second tree, and so on. Moreover one has to sum over all possible choices of the labels v , v 2 N( ), which sum up to . Given any tree 2 T k; we associate with each node v 2 N( ) a node factor F v and with each line ` 2 L( ) a propagator g` , by setting Fv :D
1 (iv )s v C1 fv ; sv !
g` :D g(! ` ) ;
and define the value of the tree as Y Y Fv g` : Val( ) :D v2N( )
(22)
(23)
`2L( )
The propagators g` are scalars, whereas each F v is a tensor with sv C 1 indices, which can be associated with the sv C 1 lines entering or exiting v. In (23) the indices of the tensors F v must be contracted: this means that if a node v is connected to a node v0 by a line ` then the indices of F v and Fv 0 associated with ` are equal to each other, and eventually one has to sum over all the indices except that associated with the root line. For instance the value of the tree in Fig. 4 contributing to h(2) is given by Val( ) D (i1 )2 f1 (i2 ) f2 g(! )g(! 2 ) ;
with 1 C 2 D , while the value of the last tree in Fig. 4 contributing to h(4) is given by Val( ) D
(i1 )4 f1 (i2 ) f2 (i3 ) f3 (i4 ) f4 3! g(! )g(! 2 )g(! 3 )g(! 4 ) ;
with 1 C 2 C 3 C 4 D . It is straightforward to prove that one can write X h(k) D Val( ) ; ¤0; k 1:
(24)
2T k;
This follows from the fact that the recursive equations (19) can be graphically represented through Fig. 3: one iterates the graphical representation of Fig. 3 until only graph elements of order k D 1 appear, and if is of order 1 (cf. Fig. 4) then Val( ) D (i) f g(! ). Each line ` 2 L( ) can be seen as the root line of the tree consisting of all nodes and lines preceding `. The choice h0(k) D 0 for all k 1 implies that no line can have zero momentum: in other words we have ` ¤ 0 for all ` 2 L( ). Therefore in order to prove that (9) with h(k) given by (24) solves formally, that is order by order, the Eqs. (19),
131
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Diagrammatic Methods in Classical Perturbation Theory
we have only to check that ["@˛ f (! t C h(! t))](k) 0 D 0 for all k 1. If we define g` D 1 for ` D 0, then also the second relation in (19) can be graphically represented as in Fig. 3 by setting D 0 and requiring h0(k) D 0, which yields that the sum of the values of all trees on the right hand side must vanish. Note that this is not an equation to solve, but just an identity that has to be checked to hold at all orders. For instance for k D 2 (the case k D 1 is trivial) the identity ["@˛ f (! t C h(! t))](2) 0 D 0 reads (cf. the second line in Fig. 4) X (i1 )2 f1 (i2 ) f2 g(! 2 ) D 0 ; 1 C2 D0
which is found to be satisfied because the propagators are even in their arguments. Such a cancellation can be graphically interpreted as follows. Consider the tree with mode labels 1 and 2 , with 1 C 2 D 0: its value is (i1 )2 f1 (i2 ) f2 g(! 2 ). One can detach the root line from the node with mode label 1 and attach it to the node with mode label 2 , and reverse the arrow of the other line so that it points toward the new root line. In this way we obtain a new tree (cf. Fig. 5): the value of the new tree is (i1 ) f1 (i2 )2 f2 g(! 1 ), where g(! 1 ) D g(! 2 ) D g(! 2 ), so that the values of the two trees contain a common factor (i1 ) f1 (i2 ) f2 g(! 2 ) times an extra factor which is (i1 ) for the first tree and (i2 ) for the second tree. Hence the sum of the two values gives zero. The cancellation mechanism described above can be generalized to all orders. Given a tree one considers all trees which can be obtained by detaching the root line and attaching to the other nodes of the tree, and by reversing the arrows of the lines (when needed) to make them point toward the root line. Then one sums together the values of all the trees so obtained: such values contain a common factor times a factor iv , if v is the node which the root line exits (the only nontrivial part of the proof is to check that the combinatorial factors match each other: we refer to Gentile & Mastropietro [37] for details). Hence the sum gives zero, as the sum of all the mode labels vanishes. For instance for k D 3 the cancellation operates by considering the three trees in Fig. 5: such trees can be considered to be obtained from each other by shifting the root line and consistently reversing the arrows of the lines.
In such a case the combinatorial factors of the node factors are different, because in the second tree the node factor associated with the node with mode label 2 contains a factor 1/2: on the other hand if 1 ¤ 3 there are two nonequivalent trees with that shape (with the labels 1 and 3 exchanged between themselves), whereas if 1 D 3 there is only one such tree, but then the first and third trees are equivalent, so that only one of them must be counted. Thus, by using that 1 C 2 C 3 D 0 – which implies g(! (2 C 3 )) D g(! 1 ) and g(! (1 C 2 )) D g(! 3 ) – in all cases we find that the sum of the values of the trees gives a common factor (i1 ) f1 (i2 )2 f2 (i3 ) f3 g(! 3 )g(! 1 ) times a factor 1 or 1/2 times i(1 C2 C3 ), and hence vanishes: once more the property that g is even is crucial. Small Divisors We want to study the convergence properties of the series h(! t) D
X
e i!t h ;
h D
2Z d
" k h(k) ;
(25)
kD1
which has been shown to be well–defined as a formal power series for the models considered in Sect. “Examples”. Recall that the number of unlabeled trees of order k is bounded by 22k . To sum over the labels we can confine ourselves to the mode labels, as the momenta are uniquely determined by the mode labels. If f is a trigonometric polynomial of degree N, that is f D 0 for all such that jj :D j1 j C C jd j > N, we have that h(k) D 0 for all jj > kN (which can be easily proved by induction), and moreover we can bound the sum over the mode labels of any tree of order k by (2N C 1)d k . Finally we can bound Y Y jv js vC1 N s vC1 N 2k ; (26) v2N( )
v2N( )
because of (20). For the model (11), where g` D 1 in (22), we can bound ˇ ˇ ˇ X ˇˇ ˇ ˇ (k) ˇ ˇh(k) ˇ 22k (2N C 1)d k N 2k ˚ k ; ˇh ˇ 2Z d
(27) ˚ D max j f j ; jjN
which shows that the series (25) converges for " small enough, more precisely for j"j < "0 , with "0 :D C0 (4N 2 ˚(2N C 1)d )1 ;
Diagrammatic Methods in Classical Perturbation Theory, Figure 5 Trees to be considered together to prove that ["@f(˛)]20 D 0
1 X
(28)
where C0 D 1. Hence the function h(! t) in that case is analytic in ". For d D 1 and f (˛) D cos ˛, we can eas-
Diagrammatic Methods in Classical Perturbation Theory
Diagrammatic Methods in Classical Perturbation Theory, Figure 6 Trees to be considered together to prove that ["@f(˛)]30 D 0
ily provide an exact expression for the coefficients h(k) : all the computational difficulties reduce to a combinatorial check, which can be found in Gentile & van Erp [42], and the formula (13) is recovered. However for the models where g` ¤ 1, the situation is much more involved: the propagators can be arbitrarily close to zero for large enough. This is the so-called small divisor problem. The series (25) is formally well–defined, assuming only an irrationality condition on !. But to prove the convergence of the series, we need a stronger condition. For instance one can require the standard Diophantine condition j! j >
jj
8 ¤ 0 ;
(29)
for suitable positive constants and . For fixed > d 1, the sets of vectors which satisfy (29) for some constant
> 0 has full Lebesgue measure in Rd [17]. We can also impose a weaker condition, known as the Bryuno condition, which can be expressed by requiring B(!) :D
1 X 1 1 log 2(n1)/ , hence E(n; k) > 1. If m D 1 call 1 and 2 the momenta of the lines `0 and `1 , respectively. By construction T cannot be a self-energy cluster, hence 1 ¤ 2 , so that, by the Diophantine condition (29), 2nC2 j! 1 j C j! 2 j j! (1 2 )j
; > j1 2 j because n`0 D n and n`1 n. Thus, one has X N kT jv j j1 2 j > 2(n2)/ ;
hence T must contain “many nodes”. In particular, one finds also in this case Nn ( ) D 1 C Nn ( 1 ) 1 C E(n; k1 ) 1 C E(n; k) E(n; k T ) E(n; k), where we have used that E(n; k T ) 1 by (37). The argument above shows that small divisors can accumulate only by allowing self-energy clusters. That accumulation really occurs is shown by the example in Fig. 10, where a tree of order k containing a chain of p self-energy clusters is depicted. Assume for simplicity that k/3 is an integer: then if p D k/3 the subtree 1 with root line ` is of order k/3. If the line ` entering the rightmost self-energy cluster T p has momentum , also the lines exiting the p self-energy clusters have the same momentum . Suppose that jj N k/3 and j! j /jj (this is certainly possible for some ). Then the value of the tree grows like a1k (k!) a 2 , for some constants a1 and a2 : a bound of this kind prevents the convergence of the perturbation series (25). If no self-energy clusters could occur (so that Rn ( ) D 0) the Siegel–Bryuno lemma would allow us to bound in (32) 1 Y
22nNn ( ) D
nD0
1 Y
22nNn ( )
nD0 1 X exp C1 k n2n/ C2k ;
(38)
nD0
for suitable constants C1 and C2 . In that case convergence of the series for j"j < "0 would follow, with "0 defined as in (26) with C0 D 2 /C2 . However, there are selfenergy clusters and they produce factorials, as the example in Fig. 10 shows, so that we have to deal with them.
(36) Resummation (37)
v2N(T)
Let us come back to the Eq. (10). If we expand g(! )["@˛ f (˛)](k) in trees according to the diagrammatic rules described in Sect. “Trees and Graphical Representation”, we can distinguish between contributions in which the root line exits a self-energy cluster T, that we can write as X g(! ) VT (! )h(kk T ) ; (39) T : k T 1). For the last s components, for k D 1 it reads @ˇ f0 (ˇ0 ) D 0, hence it fixes ˇ0 to be a stationary point for f0 (ˇ), while for higher values of k it fixes the corrections of higher order of these values (to do this we need the non-degeneracy condition). Thus, we are free to choose only ˛¯ 0 as a free parameter, since the last s components of ˛0 have to be fixed.
137
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Diagrammatic Methods in Classical Perturbation Theory
Clusters and self-energy clusters are defined as in Sect. “Multiscale Analysis”. Note that only the first r components ¯ of the momenta intervene in the definition of the scales – again because ! D !¯ ¯ . In particular, in the definition of self-energy clusters, in (33) we must replace v with ¯ v . Thus, already to first order the value of a self-energy cluster can be non-zero: for k T D 1, that is for T containing only a node v with mode label (¯v ; ˜v ) D (0; ˜ v ), the matrix VT (x; ") is of the form 0 VT (x) D 0 (b˜ v ) i; j D e
0 ; b˜ v
i ˜ v ˇ0
(50)
(i˜v;i )(i˜v; j ) f(0;˜v ) ;
with i; j D r C 1; : : : ; d. If we sum over ˜ v 2 Zs and multiply times ", we obtain
0 0 ; 0 "B (51) X ˜ D e i ˇ0 (i˜ i )(i˜ j ) f(0;˜) D @ˇ i @ˇ j f0 (ˇ0 ) :
M0 :D B i; j
˜ 2Z s
The s s block B is non-zero: in fact, the non-degeneracy condition yields that it is invertible. To higher orders one finds that the matrix M[n] (x; "), with x D ! D !¯ ¯ , is a self-adjoint matrix and M[n] (x; ") D (M[n] (x; "))T , as in the case of maximal tori. Moreover the corresponding eigenvalues [n] i (x; ") 2 x 2 ) for i D 1; : : : ; r and [n] (x; (x; ") D O(" satisfy [n] i i ") D O(") for i D r C 1; : : : ; d; this property is not trivial because of the off-diagonal blocks (which in general do not vanish at orders k 2), and to prove it one has to use the self-adjointness of the matrix M[n] (x; "). More 2 precisely one has [n] i (x; ") D "a i C O(" ) for i > r, where a rC1 ; : : : ; ad are the s eigenvalues of the matrix B in (51). From this point on the discussion proceeds in a very different way according to the sign of " (recall that we are assuming that a i > 0 for all i > r). 2 For " < 0 one has [n] i (x; ") D a i " C O(" ) < 0 for i > r, so that we can bound the last s eigenvalues of x 2 M[n] (x; ") with x2 , and the first r with x 2 /2 by the same argument as in Sect. “Resummation”. Hence we obtain easily the convergence of the series (43); of course, analyticity at the origin is prevented because of the condition " < 0. We say in that case that the lower-dimensional tori are hyperbolic. We refer to Gallavotti & Gentile [22] and Gallavotti et al. [23] for details. The case of elliptic lower-dimensional tori – that is " > 0 when a i > 0 for all i > r – is more difficult. Essentially the idea is as follows (we only sketch the strat-
egy: the details can be found in Gentile & Gallavotti [36]). One has to define the scales recursively, by using a variant, first introduced in Gentile [27], of the resummation technique described in Sect. “Resummation”. We say that is on scale 0 if j!¯ ¯ j and on scale [ 1] otherwise: for on scale 0 we write (42) with M D M0 , as given in (51). This defines the propagators of the lines ` on scale n D 0 as 1 g` D g [0] (! ` ) D (!¯ ¯ ` )2 M0 :
(52)
Denote by i the eigenvalues of M 0 : given on scale [ 1] wepsay that is on scale 1 if 21 min iD1;:::;d pj(!¯ ¯ )2 i j, and on scale [ 2] if min iD1;:::;d j(!¯ ¯ )2 i j < 21 . For on scale 1 we write (42) with M replaced by M[0] (! ¯ ; "), which is given by M 0 plus the sum of the values of all self-energy clusters T on scale n T D 0. Then the propagators of the lines ` on scale n` D 1 is defined as 1 g` D g [1] (! ` ) D (!¯ ¯ ` )2 M[0] (!¯ ¯ ` ; ") : (53) [n] Call [n] i (x; ") the eigenvalues of M (x; "): given on scale [ 2]qwe say that is on scale 2 if 22
¯ ¯ ; ")j, and on scale [ 3] j(!¯ ¯ )2 [0] i (! q ¯ ¯ ; ")j < 22 . For if min iD1;:::;d j(!¯ ¯ )2 [0] i (! on scale 2 we write (42) with M replaced by M[1] (!¯ ¯ ; "), which is given by M[0] (!¯ ¯ ; ") plus the sum of the values of all self-energy clusters T on scale n T D 1. Thus, the propagators of the lines ` on scale n` D 2 will be defined as min iD1;:::;d
1 ; (54) g` D g [2] (! ` ) D (!¯ ¯ ` )2 M[1] (!¯ ¯ ` ; ") and so on. The propagators are self-adjoint matrices, hence their norms can be bounded through the corresponding eigenvalues. In order to proceed as in Sects. “Multiscale Analysis” and “Resummation” we need some Diophantine conditions on these eigenvalues. We can assume for some 0 > ˇ ˇ q ˇ ˇ ˇj!¯ ¯ j j[n] (!¯ ¯ ; ")jˇ >
i ˇ ˇ j¯j0
8¯ ¤ 0 ; (55)
for all i D 1; : : : ; d and n 0. These are known as the first Melnikov conditions. Unfortunately, things do not proceed so plainly. In order to prove a bound like (35), possibly with a different 0 replacing , we need to compare the propagators of the
Diagrammatic Methods in Classical Perturbation Theory
lines entering and exiting clusters T which are not selfenergy clusters. This requires replacing (36) with 2nC2 ˇ ˇ q q ˇ ˇ ˇ!¯ (¯1 ¯2 ) ˙ j[n] (!¯ ¯ 1 ; ")j ˙ j[n] (!¯ ¯ 2 ; ")jˇ i j ˇ ˇ
; > j¯1 ¯ 2 j0 (56) for all i; j D 1; : : : ; d and choices of the signs ˙, and hence introduces further Diophantine conditions, known as the second Melnikov conditions. The conditions in (56) turn out to be too many, because for all n 0 and all ¯ 2 Zr such that ¯ D ¯ 1 ¯ 2 there are infinitely many conditions to be considered, one per pair (¯1 ; ¯2 ). However we can impose both the condi¯ ¯ ; "), tions (55) and (56) not for the eigenvalues [n] i (! ¯ (") independent of and then but for some quantities [n] i use the smoothness of the eigenvalues in x to control ¯ ¯ ; ") in terms of (!¯ ¯ )2 [n] (!¯ ¯ )2 [n] i (! i ("). Even¯ we have to tually, beside the Diophantine condition on !, impose the Melnikov conditions ˇ ˇ q ˇ ˇ ˇj!¯ ¯ j j[n] (")jˇ > 0 ; i ˇ ˇ j¯j ˇ ˇ (57) q q ˇ ˇ ˇj!¯ ¯ j ˙ j[n] (")j ˙ j[n] (")jˇ > 0 ; i j ˇ ˇ j¯j for all ¯ ¤ 0 and all n 0. Each condition in (57) leads us to eliminate a small interval of values of ". For the values of " which are left define h(! t) according to (43) and (46), with the new definition of the propagators. If " is small enough, say j"j < "0 , then the series (43) converges. Denote by E [0; "0 ] the set of values of " for which the conditions (57) are satisfied. One can prove that E is a Cantor set, that is a perfect, nowhere dense set. Moreover E has large relative Lebesgue measure, in the sense that lim
"!0
meas(E \ [0; "]) D1; "
(58)
provided 0 in (57) is large enough with respect to . The property (58) yields that, notwithstanding that we are eliminating infinitely many intervals, the measure of the union of all these intervals is small. If a i < 0 for all i > r we reason in the same way, simply exchanging the role of positive and negative ". On the contrary if a i D 0 for some i > r, the problem becomes much more difficult. For instance if s D 1 and arC1 D 0, then in general perturbation theory in " is not possible, not even at a formal level. However, under some conditions,
one can still construct fractional series in ", and prove that the series can be resummed [24]. Other Ordinary Differential Equations The formalism described above extends to other models, such as skew-product systems [29] and systems with strong damping in the presence of a quasi-periodic forcing term [32]. As an example of skew-product system one can con sider the linear differential equation x˙ D A C " f (! t) x on SL(2; R), where 2 R, " is a small real parameter, ! 2 Rn is an irrational vector, and A; f 2 sl(2; R), with A is a constant matrix and f an analytic function periodic in its arguments. Trees for skew-products were considered by Iserles and Nørsett [44], but they used expansions in time, hence not suited for the study of global properties, such as quasi-periodicity. Quasi-periodically forced one-dimensional systems with strong damping are described by the ordinary differential equations x¨ C x˙ C g(x) D f (! t), where x 2 R, " D 1/ is a small real parameter, ! 2 Rn is irrational, and f ,g are analytic functions (g is the “force”), with f periodic in its arguments. We refer to the bibliography for details and results on the existence of quasi-periodic solutions. Bryuno Vectors The diagrammatic methods can be used to prove the any unperturbed maximal torus with frequency vector which is a Bryuno vector persists under perturbation for " small enough [31]. One could speculate whether the Bryuno condition (30) is optimal. In general the problem is open. However, in the case of the standard map – see Sect. “The Standard Map”, – one can prove [6,15] that, by considering the radius of convergence "0 of the perturbation series as a function of !, say "0 D 0 (!), then there exists a universal constant C such that jlog 0 (!) C 2B(!)j C :
(59)
In particular this yields that the invariant curve with rotation number ! persists under perturbation if and only if ! is a Bryuno number. The proof of (59) requires the study of a more refined cancellation than that discussed in Sect. “Resummation”. We refer to Berretti & Gentile [6] and Gentile [28] for details. Extensions to Bryuno vectors for lower-dimensional tori can also be found in Gentile [31]. For the models considered in Sect. “Other Ordinary Differential Equations” we refer to Gentile [29] and Gentile et al. [33].
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Partial Differential Equations Existence of quasi-periodic solutions in systems described by one-dimensional nonlinear partial differential equations (finite-dimensional tori in infinite-dimensional systems) was first studied by Kuksin [48], Craig and Wayne [14] and Bourgain [9]. In these systems, even the case of periodic solutions yields small divisors, and hence requires a multiscale analysis. The study of persistence of periodic solutions for nonlinear Schrödinger equations and nonlinear wave equations, with the techniques discussed here, can be found in Gentile & Mastropietro [39], Gentile et al. [40], and Gentile & Procesi [41]. The models are still described by (1), with G(u) D 0, but now D is given by D D @2t C in the case of the wave equation and by D D i@ t C in the case of the Schrödinger equation, where is the Laplacian and 2 R. In dimension 1, one has D @2x . If we look for periodic solutions with frequency ! it can be convenient to pass to Fourier space, where the operator D acts as D : e i!ntCi mx ! ! 2 n2 C m2 C e i!ntCi mx ; (60) for the wave equation; a similar expression holds for the Schrödinger equation. Therefore the kernel of D can be arbitrarily close to zero for n and m large enough. Then one can consider, say, (1) for x 2 [0; ] and F(u) D u3 , with Dirichlet boundary conditions u(0) D u() D 0, and study the existence of periodic solutions with frequency ! close to some of the unperturbed frequencies. We refer to the cited bibliography for results and proofs.
Conclusions and Future Directions The diagrammatic techniques described above have been applied also in cases where no small divisors appear; cf. Berretti & Gentile [4] and Gentile et al. [34]. Of course, such problems are much easier from a technical point of view, and can be considered as propaedeutic examples to become familiar with the tree formalism. Also the study of lower-dimensional tori becomes easy for r D 1 (periodic ¯ j j!j ¯ for all ¯ ¤ 0, so solutions): in that case one has j!¯ ¯ 2k , that the product of the propagators is bounded by j!j and one can proceed as in Sect. “Small Divisors” to obtain analyticity of the solutions. In the case of hyperbolic lower-dimensional tori, if ! is a two-dimensional Diophantine vector of constant type (that is, with D 1) the conjugation function h can be proved to be Borel summable [13]. Analogous considerations hold for the one-dimensional systems in the presence
of friction and of a quasiperiodic forcing term described in Sect. “Other Ordinary Differential Equations”; in that case one has Borel summability also for one-dimensional !, that is for periodic forcing [33]. It would be interesting to investigate whether Borel summability could be obtained for higher values of . Recently existence of finite-dimensional tori in the nonlinear Schrödinger equation in higher dimensions was proved by Bourgain [10]. It would be interesting to investigate how far the diagrammatic techniques extend to deal with such higher dimensional generalizations. The main problem is that (the analogues of) the second Melnikov conditions in (57) cannot be imposed. In certain cases the tree formalism was extended to non-analytic systems, such as some quasi-integrable systems of the form (5) with f in a class of Cp functions for some finite p [7,8]. However, up to exceptional cases, the method described here seems to be intrinsically suited in cases in which the vector fields are analytic. The reason is that in order to exploit the expansion (3), we need that F be infinitely many times differentiable and we need a bound on the derivatives. It is a remarkable property that the perturbation series can be given a meaning also in cases where the solutions are not analytic in ". An advantage of the diagrammatic method is that it allows rather detailed information about the solutions, hence it could be more convenient than other techniques to study problems where the underlying structure is not known or too poor to exploit general abstract arguments. Another advantage is the following. If one is interested not only in proving the existence of the solutions, but also in explicitly constructing them with any prefixed precision, this requires performing analytical or numerical computations with arbitrarily high accuracy. Then high perturbation orders have to be reached, and the easiest and most direct way to proceed is just through perturbation theory: so the approach illustrated here allows a unified treatment for both theoretical investigations and computational ones. The resummation technique described in Sect. “Resummation” can also be used for computational purposes. With respect to the naive power series expansion it can reduce the computation time required to approximate the solution within a prefixed precision. It can also provide accurate information on the analyticity properties of the solution. For instance, for the Kepler equation, Levi-Civita at the beginning of the last century described a resummation rule (see [49]), which gives immediately the radius of convergence of the perturbation series. Of course, in the case of small divisor problems, everything becomes much more complicated.
Diagrammatic Methods in Classical Perturbation Theory
Bibliography Primary Literature 1. Arnold VI (1963) Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian (Russian). Uspehi Mat Nauk 18(5):13–40 2. Arnold VI, Kozlov VV, Ne˘ıshtadt AI (1988) Dynamical systems III, Encyclopaedia of Mathematical Sciences, vol 3. Springer, Berlin 3. Bartuccelli MV, Gentile G (2002) Lindstedt series for perturbations of isochronous systems: a review of the general theory. Rev Math Phys 14(2):121–171 4. Berretti A, Gentile G (1999) Scaling properties for the radius of convergence of a Lindstedt series: the standard map. J Math Pures Appl 78(2):159–176 5. Berretti A, Gentile G (2000) Scaling properties for the radius of convergence of a Lindstedt series: generalized standard maps. J Math Pures Appl 79(7):691–713 6. Berretti A, Gentile G (2001) Bryuno function and the standard map. Comm Math Phys 220(3):623–656 7. Bonetto F, Gallavotti G, Gentile G, Mastropietro V (1998) Quasi linear flows on tori: regularity of their linearization. Comm Math Phys 192(3):707–736 8. Bonetto F, Gallavotti G, Gentile G, Mastropietro V (1998) Lindstedt series, ultraviolet divergences and Moser’s theorem. Ann Scuola Norm Sup Pisa Cl Sci 26(3):545–593 9. Bourgain J (1998) Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann of Math 148(2):363–439 10. Bourgain J (2005) Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies 158. Princeton University Press, Princeton 11. Bricmont J, Gawedzki ˛ K, Kupiainen A (1999) KAM theorem and quantum field theory. Comm Math Phys 201(3):699–727 12. Chierchia L, Falcolini C (1994) A direct proof of a theorem by Kolmogorov in Hamiltonian systems. Ann Scuola Norm Sup Pisa Cl Sci 21(4):541–593 13. Costin O, Gallavotti G, Gentile G, Giuliani A (2007) Borel summability and Lindstedt series. Comm Math Phys 269(1):175–193 14. Craig W, Wayne CE (1993) Newton’s method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math 46(11):1409–1498 15. Davie AM (1994) The critical function for the semistandard map. Nonlinearity 7(1):219–229 16. Eliasson LH (1996) Absolutely convergent series expansions for quasi periodic motions. Math Phys Electron J 2(4):1–33 17. Gallavotti G (1983) The elements of mechanics. Texts and Monographs in Physics. Springer, New York 18. Gallavotti G (1994) Twistless KAM tori. Comm Math Phys 164(1):145–156 19. Gallavotti G (1994) Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review. Rev Math Phys 6(3):343–411 20. Gallavotti G (2001) Renormalization group in statistical mechanics and mechanics: gauge symmetries and vanishing beta functions, Renormalization group theory in the new millennium III. Phys Rep 352(4–6):251–272
21. Gallavotti G (2006) Classical mechanics. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 1. Elsevier, Oxford 22. Gallavotti G, Gentile G (2002) Hyperbolic low-dimensional invariant tori and summations of divergent series. Comm Math Phys 227(3):421–460 23. Gallavotti G, Bonetto F, Gentile G (2004) Aspects of ergodic, qualitative and statistical theory of motion. Texts and Monographs in Physics. Springer, Berlin 24. Gallavotti G, Gentile G, Giuliani A (2006) Fractional Lindstedt series. J Math Phys 47(1):1–33 25. Gallavotti G, Gentile G, Mastropietro V (1999) A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems. J Math Phys 40(12):6430–6472 26. Gentile G (1995) Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dyn Stab Syst 10(3):269–308 27. Gentile G (2003) Quasi-periodic solutions for two-level systems. Comm Math Phys 242(1-2):221–250 28. Gentile G (2006) Brjuno numbers and dynamical systems. Frontiers in number theory, physics, and geometry. Springer, Berlin 29. Gentile G (2006) Diagrammatic techniques in perturbation theory. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 2. Elsevier, Oxford 30. Gentile G (2006) Resummation of perturbation series and reducibility for Bryuno skew-product flows. J Stat Phys 125(2): 321–361 31. Gentile G (2007) Degenerate lower-dimensional tori under the Bryuno condition. Ergod Theory Dyn Syst 27(2):427–457 32. Gentile G, Bartuccelli MV, Deane JHB (2005) Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms. J Math Phys 46(6):1–21 33. Gentile G, Bartuccelli MV, Deane JHB (2006) Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J Math Phys 47(7):1–10 34. Gentile G, Bartuccelli MV, Deane JHB (2007) Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies. Rev Math Phys 19(3):307–348 35. Gentile G, Cortez DA, Barata JCA (2005) Stability for quasiperiodically perturbed Hill’s equations. Comm Math Phys 260(2):403–443 36. Gentile G, Gallavotti G (2005) Degenerate elliptic tori. Comm Math Phys 257(2):319–362 37. Gentile G, Mastropietro V (1996) Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev Math Phys 8(3):393–444 38. Gentile G, Mastropietro V (2001) Renormalization group for one-dimensional fermions. A review on mathematical results. Renormalization group theory in the new millennium III. Phys Rep 352(4–6):273–437 39. Gentile G, Mastropietro V (2004) Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method. J Math Pures Appl 83(8):1019–1065 40. Gentile G, Mastropietro V, Procesi M (2005) Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Comm Math Phys 256(2):437– 490 41. Gentile G, Procesi M (2006) Conservation of resonant peri-
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Discrete Control Systems
Discrete Control Systems TAEYOUNG LEE1 , MELVIN LEOK2 , HARRIS MCCLAMROCH1 1 Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA 2 Department of Mathematics, Purdue University, West Lafayette, USA Article Outline Glossary Definition of the Subject Introduction Discrete Lagrangian and Hamiltonian Mechanics Optimal Control of Discrete Lagrangian and Hamiltonian Systems Controlled Lagrangian Method for Discrete Lagrangian Systems Future Directions Acknowledgments Bibliography Glossary Discrete variational mechanics A formulation of mechanics in discrete-time that is based on a discrete analogue of Hamilton’s principle, which states that the system takes a trajectory for which the action integral is stationary. Geometric integrator A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such as symplecticity, momentum preservation, and the structure of the configuration space. Lie group A differentiable manifold with a group structure where the composition is differentiable. The corresponding Lie algebra is the tangent space to the Lie group based at the identity element. Symplectic A map is said to be symplectic if given any initial volume in phase space, the sum of the signed projected volumes onto each position-momentum subspace is invariant under the map. One consequence of symplecticity is that the map is volume-preserving as well. Definition of the Subject Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader
field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control systems aims to address this deficiency by restricting the approximation to the choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure. Introduction The dynamics of Lagrangian and Hamiltonian systems have unique geometric properties; the Hamiltonian flow is symplectic, the total energy is conserved in the absence of non-conservative forces, and the momentum maps associated with symmetries of the system are preserved. Many interesting dynamics evolve on a non-Euclidean space. For example, the configuration space of a spherical pendulum is the two-sphere, and the configuration space of rigid body attitude dynamics has a Lie group structure, namely the special orthogonal group. These geometric features determine the qualitative behavior of the system, and serve as a basis for theoretical study. Geometric numerical integrators are numerical integration algorithms that preserve structures of the continuous dynamics such as invariants, symplecticity, and the configuration manifold (see [14]). The exact geometric properties of the discrete flow not only generate improved qualitative behavior, but also provide accurate and efficient numerical techniques. In this article, we view a geometric integrator as an intrinsically discrete dynamical system, instead of concentrating on the numerical approximation of a continuous trajectory.
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Numerical integration methods that preserve the simplecticity of a Hamiltonian system have been studied (see [28,36]). Coefficients of a Runge–Kutta method are carefully chosen to satisfy a simplecticity criterion and order conditions to obtain a symplectic Runge–Kutta method. However, it can be difficult to construct such integrators, and it is not guaranteed that other invariants of the system, such as a momentum map, are preserved. Alternatively, variational integrators are constructed by discretizing Hamilton’s principle, rather than discretizing the continuous Euler–Lagrange equation (see [31,34]). The resulting integrators have the desirable property that they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long times (see [2]). Lie group methods are numerical integrators that preserve the Lie group structure of the configuration space (see [18]). Recently, these two approaches have been unified to obtain Lie group variational integrators that preserve the geometric properties of the dynamics as well as the Lie group structure of the configuration space without the use of local charts, reprojection, or constraints (see [26,29,32]). Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. Discrete optimal control problems involve finding a control input for a discrete dynamic system such that an optimality objective is achieved with prescribed constraints. Optimality conditions are derived from the discrete equations of motion, described by a set of discrete equations. This approach is in contrast to traditional techniques where a discretization appears at the last stage to solve the optimality condition numerically. Discrete mechanics and optimal control approaches determine optimal control inputs and trajectories more accurately with less computational load (see [19]). Combined with an indirect optimization technique, they are substantially more efficient (see [17,23,25]). The geometric approach to mechanics can provide a theoretical basis for innovative control methodologies in geometric control theory. For example, these techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion. While the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. By applying geometric control algorithms to discrete mechanics that preserve geometric properties, we
obtain an exact numerical implementation of the geometric control theory. In particular, the method of controlled Lagrangian systems is based on the idea of adopting a feedback control to realize a modification of either the potential energy or the kinetic energy, referred to as potential shaping or kinetic shaping, respectively. These ideas are applied to construct a real-time digital feedback controller that stabilizes the inverted equilibrium of the cart-pendulum (see [10,11]). In this article, we will survey discrete Lagrangian and Hamiltonian mechanics, and their applications to optimal control and feedback control theory. Discrete Lagrangian and Hamiltonian Mechanics Mechanics studies the dynamics of physical bodies acting under forces and potential fields. In Lagrangian mechanics, the trajectory of the object is derived by finding the path that extremizes the integral of a Lagrangian over time, called the action integral. In many classical problems, the Lagrangian is chosen as the difference between kinetic energy and potential energy. The Legendre transformation provides an alternative description of mechanical systems, referred to as Hamiltonian mechanics. Discrete Lagrangian and Hamiltonian mechanics has been developed by reformulating the theorems and the procedures of Lagrangian and Hamiltonian mechanics in a discrete time setting (see, for example, [31]). Therefore, discrete mechanics has a parallel structure with the mechanics described in continuous time, as summarized in Fig. 1 for Lagrangian mechanics. In this section, we describe discrete Lagrangian mechanics in more detail, and we derive discrete Euler–Lagrange equations for several mechanical systems. Consider a mechanical system on a configuration space Q, which is the space of possible positions. The Lagrangian depends on the position and velocity, which are elements of the tangent bundle to Q, denoted by TQ. Let L : TQ ! R be the Lagrangian of the system. The discrete Lagrangian, Ld : Q Q ! R is an approximation to the exact discrete Lagrangian, Z h (q ; q ) D L(q01 (t); q˙01 (t)) dt ; (1) Lexact 0 1 d 0
where q01 (0) D q0 , q01 (h) D q1 ; and q01 (t) satisfies the Euler–Lagrange equation in the time interval (0; h). A discrete action sum Gd : Q NC1 ! R, analogous to the action integral, is given by Gd (q0 ; q1 ; : : : ; q N ) D
N1 X kD0
Ld (q k ; q kC1 ) :
(2)
Discrete Control Systems
Discrete Control Systems, Figure 1 Procedures to derive continuous and discrete equations of motion
The discrete Hamilton’s principle states that ıGd D 0 for any ıq k , which yields the discrete Euler–Lagrange (DEL) equation, D2 Ld (q k1 ; q k ) C D1 Ld (q k ; q kC1 ) D 0 :
(3)
This yields a discrete Lagrangian flow map (q k1 ; q k ) 7! (q k ; q kC1 ). The discrete Legendre transformation, which from a pair of positions (q0 ; q1 ) gives a position-momentum pair (q0 ; p0 ) D (q0 ; D1 Ld (q0 ; q1 )) provides a discrete Hamiltonian flow map in terms of momenta. The discrete equations of motion, referred to as variational integrators, inherit the geometric properties of the continuous system. Many interesting Lagrangian and Hamiltonian systems, such as rigid bodies evolve on a Lie group. Lie group variational integrators preserve the nonlinear structure of the Lie group configurations as well as geometric properties of the continuous dynamics (see [29,32]). The basic idea for all Lie group methods is to express the update map for the group elements in terms of the group operation, g1 D g0 f0 ;
(4)
where g0 ; g1 2 G are configuration variables in a Lie group G, and f0 2 G is the discrete update represented
by a right group operation on g 0 . Since the group element is updated by a group operation, the group structure is preserved automatically without need of parametrizations, constraints, or re-projection. In the Lie group variational integrator, the expression for the flow map is obtained from the discrete variational principle on a Lie group, the same procedure presented in Fig. 1. But, the infinitesimal variation of a Lie group element must be carefully expressed to respect the structure of the Lie group. For example, it can be expressed in terms of the exponential map as ˇ d ˇˇ ıg D g exp D g ; d ˇ D0 for a Lie algebra element 2 g. This approach has been applied to the rotation group SO(3) and to the special Euclidean group SE(3) for dynamics of rigid bodies (see [24,26,27]). Generalizations to arbitrary Lie groups gives the generalized discrete Euler–Poincaré (DEP) equation, Te L f0 D2 Ld (g0 ; f0 ) Adf0 Te L f1 D2 Ld (g1 ; f1 ) (5) CTe L g 1 D1 Ld (g1 ; f1 ) D 0 ; for a discrete Lagrangian on a Lie group, Ld : G G ! R. Here L f : G ! G denotes the left translation map given by L f g D f g for f ; g 2 G, Tg L f : Tg G ! T f g G is the
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tangential map for the left translation, and Ad g : g ! g is the adjoint map. A dual map is denoted by a superscript (see [30] for detailed definitions). We illustrate the properties of discrete mechanics using several mechanical systems, namely a mass-spring system, a planar pendulum, a spherical pendulum, and a rigid body. Example 1 (Mass-spring System) Consider a mass-spring system, defined by a rigid body that moves along a straight frictionless slot, and is attached to a linear spring. Continuous equation of motion: The configuration space is Q D R, and the Lagrangian L : R R ! R is given by ˙ D L(q; q)
1 2 1 2 m q˙ q ; 2 2
(6)
where q 2 R is the displacement of the body measured from the point where the spring exerts no force. The mass of the body and the spring constant are denoted by m; 2 R, respectively. The Euler–Lagrange equation yields the continuous equation of motion. m q¨ C q D 0 :
(7)
Discrete equation of motion: Let h > 0 be a discrete time step, and a subscript k denotes the kth discrete variable at t D kh. The discrete Lagrangian Ld : RR ! R is an approximation of the integral of the continuous Lagrangian (6) along the solution of (7) over a time step. Here, we choose the following discrete Lagrangian.
q k C q kC1 q kC1 q k Ld (q k ; q kC1 ) D hL ; 2 h 1 h D m(q kC1 q k )2 (q k C q kC1 )2 : 2h 8 (8) Direct application of the discrete Euler–Lagrange equation to this discrete Lagrangian yields the discrete equations of motion. We develop the discrete equation of motion using the discrete Hamilton’s principle to illustrate the principles more explicitly. Let Gd : R NC1 ! R be the discrete P N1 action sum defined as Gd D kD0 Ld (q k ; q kC1 ), which approximates the action integral. The infinitesimal variation of the action sum can be written as ıGd D
m h (q kC1 q k ) (q k C q kC1 ) h 4 kD0
m h C ıq k (q kC1 q k ) (q k C q kC1 ) : h 4 N1 X
ıq kC1
Since ıq0 D ıq N D 0, the summation index can be rewritten as ıGd D
N1 X kD1
n m ıq k (q kC1 2q k C q k1 ) h
o h (q kC1 C 2q k C q k1 ) : 4
From discrete Hamilton’s principle, ıGd D 0 for any ıq k . Thus, the discrete equation of motion is given by m (q kC1 2q k C q k1 ) h h (q kC1 C 2q k C q k1 ) D 0 : C 4
(9)
For a given (q k1 ; q k ), we solve the above equation to obtain q kC1 . This yields a discrete flow map (q k1 ; q k ) 7! (q k ; q kC1 ), and this process is repeated. The discrete Legendre transformation provides the discrete equation of motion in terms of the velocity as h2 h2 ˙ q kC1 D h q k C 1 qk ; 1C 4m 4m h h q˙ kC1 D q˙ k qk q kC1 : 2m 2m
(10) (11)
For a given (q k ; q˙ k ), we compute q kC1 and q˙ kC1 by (10) and (11), respectively. This yields a discrete flow map (q k ; q˙k ) 7! (q kC1 ; q˙kC1 ). It can be shown that this variational integrator has second-order accuracy, which follows from the fact that the discrete action sum is a second-order approximation of the action integral. Numerical example: We compare computational properties of the discrete equations of motion given by (10) and (11) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg, D 1 kg/s2 so that the natural p frequency is 1 rad/s. The initial conditions are q0 D 2 m, q˙0 D 0, and the total energy is E D 1 Nm. The simulation time is 200 sec, and the stepsize h D 0:035 of the discrete equations of motion is chosen such that the CPU times are the same for both methods. Figure 2a shows the computed total energy. The variational integrator preserves the total energy well. The mean variation is 2:73271013 Nm. But, there is a notable dissipation of the computed total energy for the Runge–Kutta method Example 2 (Planar Pendulum) A planar pendulum is a mass particle connected to a frictionless, one degree-offreedom pivot by a rigid massless link under a uniform gravitational potential. The configuration space is the one-
Discrete Control Systems
Discrete Control Systems, Figure 2 Computed total energy (RK45: blue, dotted, VI: red, solid)
sphere S1 D fq 2 R2 j kqk D 1g. While it is common to parametrize the one-sphere by an angle, we develop parameter-free equations of motion in the special orthogonal group SO(2), which is a group of 22 orthogonal matrices with determinant 1, i. e. SO(2) D fR 2 R22 j RT R D I22 ; det[R] D 1g. SO(2) is diffeomorphic to the one-sphere. It is also possible to develop global equations of motion on the one-sphere directly, as shown in the next example, but here we focus on the special orthogonal group in order to illustrate the key steps to develop a Lie group variational integrator. We first exploit the basic structures of the Lie group SO(2). Define a hat map ˆ, which maps a scalar ˝ to a 22 skew-symmetric matrix ˝ˆ as ˝ˆ D
0 ˝
˝ : 0
The set of 22 skew-symmetric matrices forms the Lie algebra so(2). Using the hat map, we identify so(2) with R. An inner product on so(2) can be induced from the inner product on R as h˝ˆ 1 ; ˝ˆ 2 i D 1/2tr[˝ˆ 1T ˝ˆ 2 ] D ˝1 ˝2 for any ˝1 ; ˝2 2 R. The matrix exponential is a local diffeomorphism from so(2) to SO(2) given by
cos ˝ exp ˝ˆ D sin ˝
sin ˝ cos ˝
ˆ D 1 ml 2 ˝ 2 C mg l e T Re2 L(R; ˝) 2 2 (13) 1 2 ˆ ˆ D ml h˝; ˝i C mg l e2T Re2 ; 2 where the constant g 2 R is the gravitational acceleration. The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The second expression is used to define a discrete Lagrangian later. We choose the bases of the inertial frame and the body-fixed frame such that the unit vector along the gravity direction in the inertial frame, and the unit vector along the pendulum axis in the body-fixed frame are represented by the same vector e2 D [0; 1] 2 R2 . Thus, for example, the hanging attitude is represented by R D I22 . Here, the rotation matrix R 2 SO(2) represents the linear transformation from a representation of a vector in the body-fixed frame to the inertial frame. Since the special orthogonal group is not a linear vector space, the expression for the variation should be carefully chosen. The infinitesimal variation of a rotation matrix R 2 SO(2) can be written in terms of its Lie algebra element as ˇ ˇ ˇ d ˇˇ R exp ˆ D Rˆ exp ˆ ˇˇ D Rˆ ; (14) ıR D d ˇ D0
:
The kinematics equation for R 2 SO(2) can be written in terms of a Lie algebra element as R˙ D R˝ˆ :
as
(12)
Continuous equations of motion: The Lagrangian for a planar pendulum L : SO(2)so(2) ! R can be written
D0
where 2 R so that ˆ 2 so(2). The infinitesimal variation of the angular velocity is induced from this expression and (12) as ı ˝ˆ D ıRT R˙ C RT ı R˙ ˆ˙ ˆ T R˙ C RT (R˙ ˆ C R) D (R)
(15)
D ˆ ˝ˆ C ˝ˆ ˆ C ˆ˙ D ˆ˙ ; where we used the equality of mixed partials to compute ı R˙ as d/dt(ıR).
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RT ˆ Define the action integral to be G D 0 L(R; ˝)dt. The infinitesimal variation of the action integral is obtained by using (14) and (15). Hamilton’s principle yields the following continuous equations of motion. g ˝˙ C e2T Re1 D 0 ; (16) l R˙ D R˝ˆ :
(17)
where we use the fact that F ˆ F T D ˆ for any F 2 SO(2) and ˆ 2 so(2). Define an action sum Gd : SO(2) NC1 ! R as Gd D P N1 kD0 Ld (R k ; F k ). Using (21) and (22), the variation of the action sum is written as ıGd D
N1 X kD1
If we parametrize the rotation matrix as
cos sin RD ; sin cos
1 T ml 2 F k1 F k1 2h
these equations are equivalent to g (18) ¨ C sin D 0 : l Discrete equations of motion: We develop a Lie group variational integrator on SO(2). Similar to (4), define F k 2 SO(2) such that R kC1 D R k F k :
(19)
Thus, F k D RTk R kC1 represents the relative update between two integration steps. If we find F k 2 SO(2), the orthogonal structure is preserved through (19) since multiplication of orthogonal matrices is also orthogonal. This is a key idea of Lie group variational integrators. Define the discrete Lagrangian Ld : SO(2)SO(2) ! R to be 1 ml 2 hF k I22 ; F k I22 i Ld (R k ; F k ) D 2h h h C mg l e2T R k e2 C mg l e2T R kC1 e2 2 2 (20) 1 h 2 D ml tr[I22 F k ] C mg l e2T R k e2 2h 2 h C mg l e2T R kC1 e2 ; 2 which is obtained by an approximation h˝ˆ k ' RT (R kC1
2
1 F k F kT hmg l e2T R k e1 ; ˆ k : 2h
From the discrete Hamilton’s principle, ıGd D 0 for any ˆ k . Thus, we obtain the Lie group variational integrator on SO(2) as
3
2h2 g T T F k F kT F kC1 F kC1 e R kC1 e1 D 0; (23) l 2 R kC1 D R k F k :
(24)
For a given (R k ; F k ) and R kC1 D R k F k , (23) is solved to find F kC1 . This yields a discrete map (R k ; F k ) 7! (R kC1 ; F kC1 ). If we parametrize the rotation matrices R and F with and and if we assume that 1, these equations are equivalent to 1 hg ( kC1 2 k C k ) C sin k D 0 : h l The discrete version of the Legendre transformation provides the discrete Hamiltonian map as follows. F k F kT D 2h˝ˆ
2
h2 g T e R k e1 ; l 2
R kC1 D R k F k ; ˝ kC1 D ˝ k
(25) (26)
hg T hg T e2 R k e1 e R kC1 e1 : 2l 2l 2
(27)
k
R k ) D F k I22 , applied to the continuous Lagrangian given by (13). As for the continuous time case, expressions for the infinitesimal variations should be carefully chosen. The infinitesimal variation of a rotation matrix is the same as (14), namely ıR k D R k ˆ k ;
(21)
for k 2 R, and the constrained variation of F k is obtained from (19) as ıF k D ıRTk R kC1 C RTk ıR kC1 D ˆ k F k C F k ˆ kC1 D F k (ˆ kC1 F kT ˆ k F k ) (22) kC1 ˆ k ) ; D F k (ˆ
For a given (R k ; ˝ k ), we solve (25) to obtain F k . Using this, (R kC1 ; ˝ kC1 ) is obtained from (26) and (27). This yields a discrete map (R k ; ˝ k ) 7! (R kC1 ; ˝ kC1 ). Numerical example: We compare the computational properties of the discrete equations of motion given by (25)–(27) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg, l D 9:81 m. The initial conditions are 0 D /2 rad, ˝ D 0, and the total energy is E D 0 Nm. The simulation time is 1000 sec, and the step-size h D 0:03 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 2b shows the computed total energy for both methods. The variational integrator preserves the total energy well. There is no drift in the computed total energy, and the
Discrete Control Systems
mean variation is 1:0835102 Nm. But, there is a notable dissipation of the computed total energy for the Runge– Kutta method. Note that the computed total energy would further decrease as the simulation time increases.
Since q˙ D ! q for some angular velocity ! 2 R3 satisfying ! q D 0, this can also be equivalently written as
Example 3 (Spherical Pendulum) A spherical pendulum is a mass particle connected to a frictionless, two degree-offreedom pivot by a rigid massless link. The mass particle acts under a uniform gravitational potential. The configuration space is the two-sphere S2 D fq 2 R3 j kqk D 1g. It is common to parametrize the two-sphere by two angles, but this description of the spherical pendulum has a singularity. Any trajectory near the singularity encounters numerical ill-conditioning. Furthermore, this leads to complicated expressions involving trigonometric functions. Here we develop equations of motion for a spherical pendulum using the global structure of the two-sphere without parametrization. In the previous example, we develop equations of motion for a planar pendulum using the fact that the one-sphere S1 is diffeomorphic to the special orthogonal group SO(2). But, the two-sphere is not diffeomorphic to a Lie group. Instead, there exists a natural Lie group action on the two-sphere. That is the 3-dimensional special orthogonal group SO(3), a group of 33 orthogonal matrices with determinant 1, i. e. SO(3) D fR 2 R33 j RT R D I33 ; det[R] D 1g. The special orthogonal group SO(3) acts on the two-sphere in a transitive way; for any q1 ; q2 2 S2 , there exists a R 2 SO(3) such that q2 D Rq1 . Thus, the discrete update for the two-sphere can be expressed in terms of a rotation matrix as (19). This is a key idea to develop a discrete equations of motion for a spherical pendulum. Continuous equations of motion: Let q 2 S2 be a unit vector from the pivot point to the point mass. The Lagrangian for a spherical pendulum can be written as
(32)
1 ˙ D ml 2 q˙ q˙ C mg l e3 q ; L(q; q) 2
(28)
where the gravity direction is assumed to be e3 D [0; 0; 1] 2 R3 . The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The infinitesimal variation of the unit vector q can be written in terms of the vector cross product as ıq D q ;
(29)
where 2 R3 is constrained to be orthogonal to the unit vector, i. e. q D 0. Using this expression for the infinitesimal variation, Hamilton’s principle yields the following continuous equations of motion. g ˙ C (q (q e3 )) D 0 : q¨ C (q˙ q)q l
(30)
g q e3 D 0 ; l q˙ D ! q :
!˙
(31)
These are global equations of motion for a spherical pendulum; these are much simpler than the equations expressed in term of angles, and they have no singularity. Discrete equations of motion: We develop a variational integrator for the spherical pendulum defined on S2 . Since the special orthogonal group acts on the two-sphere transitively, we can define the discrete update map for the unit vector as q kC1 D F k q k
(33)
for F k 2 SO(3). The rotation matrix F k is not uniquely defined by this condition, since exp(qˆ k ), which corresponds to a rotation about the qk direction fixes the vector qk . Consequently, if F k satisfies (33), then F k exp(qˆ k ) does as well. We avoid this ambiguity by requiring that F k does not rotate about qk , which can be achieved by letting F k D exp( fˆk ), where f k q k D 0. Define a discrete Lagrangian Ld : S2 S2 ! R to be Ld (q k ; q kC1 ) D
1 ml 2 (q kC1 q k ) (q kC1 q k ) 2h h h C mg l e3 q k C mg l e3 q kC1 : 2 2
The variation of qk is the same as (29), namely ıq k D k q k
(34)
for k 2 R3 with a constraint k q k D 0. Using this discrete Lagrangian and the expression for the variation, discrete Hamilton’s principle yields the following discrete equations of motion for a spherical pendulum. h2 g q k e3 q k q kC1 D h! k C 2l 2 !1/2 h2 g C 1 h! k C qk ; q k e3 2l hg hg ! kC1 D ! k C q k e3 C q kC1 e3 : 2l 2l
(35) (36)
Since an explicit solution for F k 2 SO(3) can be obtained in this case, the rotation matrix F k does not appear in the equations of motion. This variational integrator on S2 exactly preserves the unit length of qk , the constraint
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Discrete Control Systems, Figure 3 Numerical simulation of a spherical pendulum (RK45: blue, dotted, VI: red, solid)
q k ! k D 0, and the third component of the angular velocity ! k e3 which is conserved since gravity exerts no moment along the gravity direction e3 . Numerical example: We compare the computational properties of the discrete equations of motion given by (35) and (36) with a 4(5)th order variable step size Runge–Kutta method for (31) and (32). We choose m D 1p kg, l D 9:81 m. The p initial conditions are q0 D [ 3/2; 0; 1/2], !0 D 0:1[ 3; 0; 3] rad/sec, and the total energy is E D 0:44 Nm. The simulation time is 200 sec, and the step-size h D 0:05 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 3 shows the computed total energy and the unit length errors. The variational integrator preserves the total energy and the structure of the two-sphere well. The mean total energy deviation is 1:5460104 Nm, and the mean unit length error is 3:2476 1015 . But, there is a notable dissipation of the computed total energy for the Runge–Kutta method. The Runge–Kutta method also fails to preserve the structure of the two-sphere. The mean unit length error is 1:0164102 . Example 4 (Rigid Body in a Potential Field) Consider a rigid body under a potential field that is dependent on the position and the attitude of the body. The configuration space is the special Euclidean group, which is a semidirect product of the special orthogonal group and Eus R3 . clidean space, i. e. SE(3) D SO(3) Continuous equations of motion: The equations of motion for a rigid body can be developed either from Hamilton’s principle (see [26]) in a similar way as Example 2, or directly from the generalized discrete Euler–Poincaré equation given at (5). Here, we summarize the results. Let m 2 R and J 2 R33 be the mass and the moment of inertia matrix of a rigid body. For (R; x) 2 SE(3), the linear
transformation from the body-fixed frame to the inertial frame is denoted by the rotation matrix R 2 SO(3), and the position of the mass center in the inertial frame is denoted by a vector x 2 R3 . The vectors ˝; v 2 R3 are the angular velocity in the body-fixed frame, and the translational velocity in the inertial frame, respectively. Suppose that the rigid body acts under a configuration-dependent potential U : SE(3) ! R. The continuous equations of motion for the rigid body can be written as R˙ D R˝ˆ ;
(37)
x˙ D v ;
(38)
J ˝˙ C ˝ J˝ D M ;
(39)
@U ; @x
(40)
m˙v D
where the hat map ˆ is an isomorphism from R3 to 3 3 skew-symmetric matrices so(3), defined such that xˆ y D x y for any x; y 2 R3 . The moment due to the potential M 2 R3 is obtained by the following relationship: T ˆ D @U R RT @U : M @R @R
(41)
The matrix @U/@R 2 R33 is defined such that [@U/@R] i j D@U/@[R] i j for i; j 2 f1; 2; 3g, where the i, jth element of a matrix is denoted by [] i j . Discrete equations of motion: The corresponding discrete equations of motion are given by
b
h J ˝k C
h2 ˆ M k D F k Jd Jd F kT ; 2
R kC1 D R k F k ;
(42) (43)
Discrete Control Systems
Discrete Control Systems, Figure 4 Numerical simulation of a dumbbell rigid body (LGVI: red, solid, RK45 with rotation matrices: blue, dash-dotted, RK45 with quaternions: black, dotted)
h2 @U k ; 2m @x k h h D F kT J˝ k C F kT M k C M kC1 ; 2 2 h Uk h @U kC1 D mv k ; 2 @x k 2 @x kC1
x kC1 D x k C hv k
(44)
J˝ kC1
(45)
mv kC1
(46)
where Jd 2 R33 is a non-standard moment of inertia matrix defined as Jd D1/2tr[J]I33 J. For a given (R k ; x k ; ˝ k ; v k ), we solve the implicit Eq. (42) to find F k 2 SO(3). Then, the configuration at the next step R kC1 ;x kC1 is obtained by (43) and (44), and the moment and force M kC1;(@U kC1 )/(@x kC1 ) can be computed. Velocities ˝ kC1 ;v kC1 are obtained from (45) and (46). This defines a discrete flow map, (R k ; x k ; ˝ k ; v k ) 7! (R kC1 ; x kC1 ; ˝ kC1 ; v kC1 ), and this process can be repeated. This Lie group variational integrator on SE(3) can be generalized to multiple rigid bodies acting under their mutual gravitational potential (see [26]). Numerical example: We compare the computational properties of the discrete equations of motion given by (42)–(46) with a 4(5)th order variable step size Runge– Kutta method for (37)–(40). In addition, we compute the attitude dynamics using quaternions on the unit threesphere S3 . The attitude kinematics Eq. (37) is rewritten in terms of quaternions, and the corresponding equations are integrated by the same Runge–Kutta method. We choose a dumbbell spacecraft, that is two spheres connected by a rigid massless rod, acting under a central gravitational potential. The resulting system is a restricted full two body problem. The dumbbell spacecraft model has an analytic expression for the gravitational potential, resulting in a nontrivial coupling between the attitude dynamics and the orbital dynamics. As shown in Fig. 4a, the initial conditions are chosen such that the resulting motion is a near-circular orbit
combined with a rotational motion. Figure 4b and c show the computed total energy and the orthogonality errors of the rotation matrix. The Lie group variational integrator preserves the total energy and the Lie group structure of SO(3). The mean total energy deviation is 2:5983104 , and the mean orthogonality error is 1:8553 1013 . But, there is a notable dissipation of the computed total energy and the orthogonality error for the Runge–Kutta method. The mean orthogonality errors for the Runge– Kutta method are 0.0287 and 0.0753, respectively, using kinematics equation with rotation matrices, and using the kinematics equation with quaternions. Thus, the attitude of the rigid body is not accurately computed for Runge– Kutta methods. It is interesting to see that the Runge– Kutta method with quaternions, which is generally assumed to have better computational properties than the kinematics equation with rotation matrices, has larger total energy error and orthogonality error. Since the unit length of the quaternion vector is not preserved in the numerical computations, orthogonality errors arise when converted to a rotation matrix. This suggests that it is critical to preserve the structure of SO(3) in order to study the global characteristics of the rigid body dynamics. The importance of simultaneously preserving the symplectic structure and the Lie group structure of the configuration space in rigid body dynamics can be observed numerically. Lie group variational integrators, which preserve both of these properties, are compared to methods that only preserve one, or neither, of these properties (see [27]). It is shown that the Lie group variational integrator exhibits greater numerical accuracy and efficiency. Due to these computational advantages, the Lie group variational integrator has been used to study the dynamics of the binary near-Earth asteroid 66391 (1999 KW 4 ) in joint work between the University of Michigan and the Jet Propulsion Laboratory, NASA (see [37]).
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Optimal Control of Discrete Lagrangian and Hamiltonian Systems Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. The equations of motion for a system are constrained by Lagrange multipliers, and necessary conditions for optimality is obtained by the calculus of variations. The solution for the corresponding two point boundary value problem provides the optimal control input. Alternatively, a sub-optimal control law is obtained by approximating the control input history with finite data points. Discrete optimal control problems involve finding a control input for a given system described by discrete Lagrangian and Hamiltonian mechanics. The control inputs are parametrized by their values at each discrete time step, and the discrete equations of motion are derived from the discrete Lagrange–d’Alembert principle [21], ı
N1 X kD0
Ld (q k ; q kC1 ) D
N1 X
[Fd (q k ; q kC1 ) ıq k
kD0
C FdC (q k ; q kC1 ) ıq kC1 ] ; which modifies the discrete Hamilton’s principle by taking into account the virtual work of the external forces. Discrete optimal control is in contrast to traditional techniques such as collocation, wherein the continuous equations of motion are imposed as constraints at a set of collocation points, since this approach induces constraints on the configuration at each discrete timestep. Any optimal control algorithm can be applied to the discrete Lagrangian or Hamiltonian system. For an indirect method, our approach to a discrete optimal control problem can be considered as a multiple stage variational problem. The discrete equations of motion are derived by the discrete variational principle. The corresponding variational integrators are imposed as dynamic constraints to be satisfied by using Lagrange multipliers, and necessary conditions for optimality, expressed as discrete equations on multipliers, are obtained from a variational principle. For a direct method, control inputs can be optimized by using parameter optimization tools such as a sequential quadratic programming. The discrete optimal control can be characterized by discretizing the optimal control problem from the problem formulation stage. This method has substantial computational advantages when used to find an optimal control law. As discussed in the previous section, the discrete dynamics are
more faithful to the continuous equations of motion, and consequently more accurate solutions to the optimal control problem are obtained. The external control inputs break the Lagrangian and Hamiltonian system structure. For example, the total energy is not conserved for a controlled mechanical system. But, the computational superiority of the discrete mechanics still holds for controlled systems. It has been shown that the discrete dynamics is more reliable even for controlled system as it computes the energy dissipation rate of controlled systems more accurately (see [31]). For example, this feature is extremely important in computing accurate optimal trajectories for long term spacecraft attitude maneuvers using low energy control inputs. The discrete dynamics does not only provide an accurate optimal control input, but also enables us to find it efficiently. For the indirect optimal control approach, optimal solutions are usually sensitive to a small variation of multipliers. This causes difficulties, such as numerical ill-conditioning, when solving the necessary conditions for optimality expressed as a two point boundary value problem. Sensitivity derivatives along the discrete necessary conditions do not have numerical dissipation introduced by conventional numerical integration schemes. Thus, they are numerically more robust, and the necessary conditions can be solved computationally efficiently. For the direct optimal control approach, optimal control inputs can be obtained by using a larger discrete step size, which requires less computational load. We illustrate the basic properties of the discrete optimal control using optimal control problems for the spherical pendulum and the rigid body model presented in the previous section. Example 5 (Optimal Control of a Spherical Pendulum) We study an optimal control problem for the spherical pendulum described in Example 3. We assume that an external control moment u 2 R3 acts on the pendulum. Control inputs are parametrized by their values at each time step, and the discrete equations of motion are modified to include the effects of the external control inputs by using the discrete Lagrange–d’Alembert principle. Since the component of the control moment that is parallel to the direction along the pendulum has no effect, we parametrize the control input as u k D q k w k for w k 2 R3 . The objective is to transfer the pendulum from a given initial configuration (q0 ; !0 ) to a desired configuration (qd ; ! d ) during a fixed maneuver time N, while minimizing the square of the weighted l2 norm of the control moments. N N X X h T h min J D uk uk D (q k w k )T (q k w k ) : wk 2 2 kD0
kD0
Discrete Control Systems
Discrete Control Systems, Figure 5 Optimal control of a spherical pendulum
We solve this optimal control problem by using a direct numerical optimization method. The terminal boundary condition is imposed as an equality constraint, and the 3(N C 1) control input parameters fw k g N kD0 are numerically optimized using sequential quadratic programming. This method is referred to as a DMOC (Discrete Mechanics and Optimal Control) approach (see [19]). Figure 5 shows a optimal solution transferring the spherical pendulum from a hanging configuration given by (q0 ; !0 ) D (e3 ; 031 ) to an inverted configuration (qd ; ! d ) D (e3 ; 031 ) during 1 second. The time step size is h D 0:033. Experiment have shown that the DMOC approach can compute optimal solutions using larger step sizes than typical Runge–Kutta methods, and consequently, it requires less computational load. In this case, using a second-order accurate Runge–Kutta method, the same optimization code fails while giving error messages of inaccurate and singular gradient computations. It is presumed that the unit length errors of the Runge– Kutta method, shown in Example 3, cause numerical instabilities for the finite-difference gradient computations required for the sequential quadratic programming algorithm. Example 6 (Optimal Control of a Rigid Body in a Potential Field) We study an optimal control problem of a rigid body using a dumbbell spacecraft model described in Example 4 (see [25] for detail). We assume that external control forces u f 2 R3 , and control moment u m 2 R3 act on the dumbbell spacecraft. Control inputs are parametrized by their values at each time step, and the Lie group variational integrators are modified to include the effects of the external control inputs by using discrete Lagrange– d’Alembert principle. The objective is to transfer the dumbbell from a given initial configuration (R0 ; x0 ; ˝0 ; v0 ) to a desired configuration (R d ; x d ; ˝ d ; v d ) during a fixed maneuver
time N, while minimizing the square of the l2 norm of the control inputs. N1 X
min J D
u kC1
kD0
T h f T h f Wm u m u kC1 Wf u kC1C u m kC1 kC1 ; 2 2
where Wf ; Wm 2 R33 are symmetric positive definite matrices. Here we use a modified version of the discrete equations of motion with first order accuracy, as it yields a compact form for the necessary conditions. Necessary conditions for optimality: We solve this optimal control problem by using an indirect optimization method, where necessary conditions for optimality are derived using variational arguments, and a solution of the corresponding two-point boundary value problem provides the optimal control. This approach is common in the optimal control literature; here the optimal control problem is discretized at the problem formulation level using the Lie group variational integrator presented in Sect. “Discrete Lagrangian and Hamiltonian Mechanics”. Ja D
N1 X kD0
h f T f f h m T m m u W u kC1 C W u kC1 u 2 kC1 2 kC1
C 1;T k fx kC1 C x k C hv k g
@U kC1 f 2;T mv kC1 C mv k h C hu kC1 C k @x kC1 _ logm(F k RTk R kC1 ) C 3;T k ˚ C 4;T J˝ kC1 C F kT J˝ k C h M kC1 C u m kC1 ; k where 1k ; 2k ; 3k ; 4k 2 R3 are Lagrange multipliers. The matrix logarithm is denoted by logm : SO(3) ! so(3) and the vee map _ : so(3) ! R3 is the inverse of the hat map introduced in Example 4. The logarithm form of (43) is used, and the constraint (42) is considered implicitly using constrained variations. Using similar expressions for the
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Discrete Control Systems, Figure 6 Optimal control of a rigid body
variation of the rotation matrix and the angular velocity given in (14) and (15), the infinitesimal variation can be written as N1 n o X f ;T f hıu k W f u k C 2k1 ıJa D kD1
˚ 4 C hıu m;T Wm u m k C k1 k ˚ C zTk k1 C ATk k ; where k D [1k ; 2k ; 3k ; 4k ] 2 R12 , and z k 2 R12 represents the infinitesimal variation of (R k ; x k ; ˝ k ; v k ), given by z k D [logm(RTk ıR k )_ ; ıx k ; ı˝ k ; ıv k ]. The matrix A k 2 R1212 is defined in terms of (R k ; x k ; ˝ k ; v k ). Thus, necessary conditions for optimality are given by f
u kC1 D Wf1 2k ;
(47)
1 4 um kC1 D Wm k ;
(48)
k D ATkC1 kC1
(49)
together with the discrete equations of motion and the boundary conditions. Computational approach: Necessary conditions for optimality are expressed in terms of a two point boundary problem. The problem is to find the optimal discrete flow, multipliers, and control inputs to satisfy the equations of motion, optimality conditions, multiplier equations, and boundary conditions simultaneously. We use a neighboring extremal method (see [12]). A nominal solution satisfying all of the necessary conditions except the boundary conditions is chosen. The unspecified initial multiplier is updated by successive linearization so as to satisfy the specified terminal boundary conditions in the limit. This is also referred to as the shooting method. The main advantage of the neighboring extremal method is that the number of iteration variables is small.
The difficulty is that the extremal solutions are sensitive to small changes in the unspecified initial multiplier values. The nonlinearities also make it hard to construct an accurate estimate of sensitivity, thereby resulting in numerical ill-conditioning. Therefore, it is important to compute the sensitivities accurately to apply the neighboring extremal method. Here the optimality conditions (47) and (48) are substituted into the equations of motion and the multiplier equations, which are linearized to obtain sensitivity derivatives of an optimal solution with respect to boundary conditions. Using this sensitivity, an initial guess of the unspecified initial conditions is iterated to satisfy the specified terminal conditions in the limit. Any type of Newton iteration can be applied. We use a line search with backtracking algorithm, referred to as Newton–Armijo iteration (see [22]). Figure 6 shows optimized maneuvers, where a dumbbell spacecraft on a reference circular orbit is transferred to another circular orbit with a different orbital radius and inclination angle. Figure 6a shows the violation of the terminal boundary condition according to the number of iterations on a logarithmic scale. Red circles denote outer iterations in the Newton–Armijo iteration to compute the sensitivity derivatives. The error in satisfaction of the terminal boundary condition converges quickly to machine precision after the solution is close to the local minimum at around the 20th iteration. These convergence results are consistent with the quadratic convergence rates expected of Newton methods with accurately computed gradients. The neighboring extremal method, also referred to as the shooting method, is numerically efficient in the sense that the number of optimization parameters is minimized. But, in general, this approach may be prone to numerical ill-conditioning (see [3]). A small change in the initial multiplier can cause highly nonlinear behavior of the terminal attitude and angular momentum. It is difficult
Discrete Control Systems
to compute the gradient for Newton iterations accurately, and the numerical error may not converge. However, the numerical examples presented in this article show excellent numerical convergence properties. The dynamics of a rigid body arises from Hamiltonian mechanics, which have neutral stability, and its adjoint system is also neutrally stable. The proposed Lie group variational integrator and the discrete multiplier equations, obtained from variations expressed in the Lie algebra, preserve the neutral stability property numerically. Therefore the sensitivity derivatives are computed accurately. Controlled Lagrangian Method for Discrete Lagrangian Systems The method of controlled Lagrangians is a procedure for constructing feedback controllers for the stabilization of relative equilibria. It relies on choosing a parametrized family of controlled Lagrangians whose corresponding Euler–Lagrange flows are equivalent to the closed loop behavior of a Lagrangian system with external control forces. The condition that these two systems are equivalent results in matching conditions. Since the controlled system can now be viewed as a Lagrangian system with a modified Lagrangian, the global stability of the controlled system can be determined directly using Lyapunov stability analysis. This approach originated in Bloch et al. [5] and was then developed in Auckly et al. [1]; Bloch et al. [6,7,8,9]; Hamberg [15,16]. A similar approach for Hamiltonian controlled systems was introduced and further studied in the work of Blankenstein, Ortega, van der Schaft, Maschke, Spong, and their collaborators (see, for example, [33,35] and related references). The two methods were shown to be equivalent in Chang et al. [13] and a nonholonomic version was developed in Zenkov et al. [40,41], and Bloch [4]. Since the method of controlled Lagrangians relies on relating the closed-loop dynamics of a controlled system with the Euler–Lagrange flow associated with a modified Lagrangian, it is natural to discretize this approach through the use of variational integrators. In Bloch et al. [10,11], a discrete theory of controlled Lagrangians was developed for variational integrators, and applied to the feedback stabilization of the unstable inverted equilibrium of the pendulum on a cart. The pendulum on a cart is an example of an underactuated control problem, which has two degrees of freedom, given by the pendulum angle and the cart position. Only the cart position has control forces acting on it, and the stabilization of the pendulum has to be achieved indirectly through the coupling between the pendulum and the cart. The controlled Lagrangian is obtained by modifying the
kinetic energy term, a process referred to as kinetic shaping. Similarly, it is possible to modify the potential energy term using potential shaping. Since the pendulum on a cart model involves both a planar pendulum, and a cart that translates in one-dimension, the configuration space is a cylinder, S1 R. Continuous kinetic shaping: The Lagrangian has the form kinetic minus potential energy ˙ D L(q; q)
1 ˙2 ˛ C 2ˇ( ) ˙ s˙ C s˙2 V(q); 2
(50)
and the corresponding controlled Euler–Lagrange dynamics is d @L @L D0; dt @ ˙ @ d @L Du; dt @˙s
(51) (52)
where u is the control input. Since the potential energy is translation invariant, i. e., V(q) D V( ), the relative equilibria D e , s˙ D const are unstable and given by non-degenerate critical points of V( ). To stabilize the relative equilibria D e , s˙ D const with respect to , kinetic shaping is used. The controlled Lagrangian in this case is defined by 1 ˙ D L( ; ˙ ; s˙ C ( ) ˙ ) C (( ) ˙ )2 ; (53) L; (q; q) 2 where ( ) D ˇ( ). This velocity shift corresponds to a new choice of the horizontal space (see [8] for details). The dynamics is just the Euler–Lagrange dynamics for controlled Lagrangian (53), d @L; @L; D0; dt @ ˙ @ d @L; D0: dt @˙s
(54) (55)
The Lagrangian (53) satisfies the simplified matching conditions of Bloch et al. [9] when the kinetic energy metric coefficient in (50) is constant. Setting u D d ( ) ˙ /dt defines the control input, makes Eqs. (52) and (55) identical, and results in controlled momentum conservation by dynamics (51) and (52). Setting D 1/ makes Eqs. (51) and (54) identical when restricted to a level set of the controlled momentum. Discrete kinetic shaping: Here, we adopt the following notation: q kC1/2 D
q k C q kC1 ; 2 q k D q kC1 q k ;
q k D ( k ; s k ) :
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Then, a second-order accurate discrete Lagrangian is given by, Ld (q k ; q kC1 ) D hL(q kC1/2 ; q k /h) : The discrete dynamics is governed by the equations @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D0; @ k @ k @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D u k ; @s k @s k
(56) (57)
where u k is the control input. Similarly, the discrete controlled Lagrangian is, ; (q kC1/2 ; q k /h) ; L; d (q k ; q kC1 ) D hL
with discrete dynamics given by, @L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0; @ k @ k
@L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0: @s k @s k
Numerical example: Simulating the behavior of the discrete controlled Lagrangian system involves viewing Eqs. (58)–(59) as an implict update map (q k2 ; q k1 ) 7! (q k1 ; q k ). This presupposes that the initial conditions are given in the form (q0 ; q1 ); however it is generally preferable to specify the initial conditions as (q0 ; q˙0 ). This is achieved by solving the boundary condition,
(58) (59)
@L (q0 ; q˙0 ) C D1 Ld (q0 ; q1 ) C Fd (q0 ; q1 ) D 0 ; @q˙ for q1 . Once the initial conditions are expressed in the form (q0 ; q1 ), the discrete evolution can be obtained using the implicit update map. We first consider the case of kinetic shaping on a level surface, when is twice the critical value, and without dissipation. Here, h D 0:05 sec, m D 0:14 kg, M D 0:44 kg, and l D 0:215 m. As shown in Fig. 7, the dynamics is stabilized, but since there is no dissipation, the oscillations are sustained. The s dynamics exhibits both a drift and oscillations, as potential shaping is necessary to stabilize the translational dynamics.
Equation (59) is equivalent to the discrete controlled momentum conservation: Future Directions pk D ; where @L; d (q k ; q kC1 ) @s k (1 C )ˇ( kC1/2 ) k C s k D : h
pk D
Setting uk D
k ( kC1/2 ) k1 ( k1/2 ) h
makes Eqs. (57) and (59) identical and allows one to represent the discrete momentum equation (57) as the discrete momentum conservation law p k D p. The condition that (56)–(57) are equivalent to (58)– (59) yield the discrete matching conditions. The dynamics determined by Eqs. (56)–(57) restricted to the momentum level p k D p is equivalent to the dynamics of Eqs. (58)– (59) restricted to the momentum level p k D if and only if the matching conditions D hold.
1 ;
D
p ; 1 C
Discrete Receding Horizon Optimal Control: The existing work on discrete optimal control has been primarily focused on constructing the optimal trajectory in an open loop sense. In practice, model uncertainty and actuation errors necessitate the use of feedback control, and it would be interesting to extend the existing work on optimal control of discrete systems to the feedback setting by adopting a receding horizon approach. Discrete State Estimation: In feedback control, one typically assumes complete knowledge regarding the state of the system, an assumption that is often unrealistic in practice. The general problem of state estimation in the context of discrete mechanics would rely on good numerical methods for quantifying the propagation of uncertainty by solving the Liouville equation, which describes the evolution of a phase space distribution function advected by a prescribed vector field. In the setting of Hamiltonian systems, the solution of the Liouville equation can be solved by the method of characteristics (Scheeres et al. [38]). This implies that a collocational approach (Xiu [39]) combined with Lie group variational integrators, and interpolation based on noncommutative harmonic analysis on Lie groups could yield an efficient means of propagating uncertainty, and serve as the basis of a discrete state estimation algorithm.
Discrete Control Systems
Discrete Control Systems, Figure 7 Discrete controlled dynamics with kinetic shaping and without dissipation. The discrete controlled system stabilizes the motion about the equilibrium, but the s dynamics is not stabilized; since there is no dissipation, the oscillations are sustained
Forced Symplectic–Energy-Momentum Variational Integrators: One of the motivations for studying the control of Lagrangian systems using the method of controlled Lagrangians is that the method provides a natural candidate Lyapunov function to study the global stability properties of the controlled system. In the discrete theory, this approach is complicated by the fact that the energy of a discrete Lagrangian system is not exactly conserved, but rather oscillates in a bounded fashion. This can be addressed by considering the symplectic-energy-momentum [20] analogue to the discrete Lagrange-d’Alembert principle, ı
N1 X kD0
Ld (q k ; q kC1 ; h k ) D
N1 X
[Fd (q k ; q kC1 ; h k ) ıq k
kD0
C FdC (q k ; q kC1 ; h k ) ıq kC1 ] ; where the timestep hk is allowed to vary, and is chosen to satisfy the variational principle. The variations in hk yield an Euler–Lagrange equation that reduces to the conservation of discrete energy in the absence of external forces. By developing a theory of controlled Lagrangians around a geometric integrator based on the symplecticenergy-momentum version of the Lagrange–d’Alembert principle, one would potentially be able to use Lyapunov techniques to study the global stability of the resulting numerical control algorithms.
Acknowledgments TL and ML have been supported in part by NSF Grant DMS-0504747 and DMS-0726263. TL and NHM have been supported in part by NSF Grant ECS-0244977 and CMS-0555797. Bibliography Primary Literature 1. Auckly D, Kapitanski L, White W (2000) Control of nonlinear underactuated systems. Commun Pure Appl Math 53:354–369 2. Benettin G, Giorgilli A (1994) On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J Stat Phys 74:1117–1143 3. Betts JT (2001) Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia, PA 4. Bloch AM (2003) Nonholonomic Mechanics and Control. In: Interdisciplinary Applied Mathematics, vol 24. Springer, New York 5. Bloch AM, Leonard N, Marsden JE (1997) Matching and stabilization using controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Hyatt Regency San Diego, San Diego, CA, 10–12 December 1997, pp 2356–2361 6. Bloch AM, Leonard N, Marsden JE (1998) Matching and stabilization by the method of controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Hyatt Regency Westshore, Tampa, FL, 16–18 December 1998, pp 1446– 1451 7. Bloch AM, Leonard N, Marsden JE (1999) Potential shaping and the method of controlled Lagrangians. In: Proceedings of the
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IEEE Conference on Decision and Control. Crowne Plaza Hotel and Resort, Phoenix, AZ, 7–10 December 1999, pp 1652–1657 Bloch AM, Leonard NE, Marsden JE (2000) Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Trans Syst Control 45:2253–2270 Bloch AM, Chang DE, Leonard NE, Marsden JE (2001) Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Trans Autom Contr 46:1556– 1571 Bloch AM, Leok M, Marsden JE, Zenkov DV (2005) Controlled Lagrangians and stabilization of the discrete cart-pendulum system. In: Proceedings of the IEEE Conference on Decision and Control. Melia Seville, Seville, Spain, 12–15 December 2005, pp 6579–6584 Bloch AM, Leok M, Marsden JE, Zenkov DV (2006) Controlled Lagrangians and potential shaping for stabilization of discrete mechanical systems. In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 3333–3338 Bryson AE, Ho Y (1975) Applied Optimal Control. Hemisphere, Washington, D.C. Chang D-E, Bloch AM, Leonard NE, Marsden JE, Woolsey C (2002) The equivalence of controlled Lagrangian and controlled Hamiltonian systems. Control Calc Var (special issue dedicated to Lions JL) 8:393–422 Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin Hamberg J (1999) General matching conditions in the theory of controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Crowne Plaza Hotel and Resort, Phoenix, AZ, 7–10 December 1999, pp 2519–2523 Hamberg J (2000) Controlled Lagrangians, symmetries and conditions for strong matching. In: Lagrangian and Hamiltonian Methods for Nonlinear Control. Elsevier, Oxford Hussein II, Leok M, Sanyal AK, Bloch AM (2006) A discrete variational integrator for optimal control problems in SO(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 6636–6641 Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365 Junge D, Marsden JE, Ober-Blöbaum S (2005) Discrete mechanics and optimal control. In: IFAC Congress, Praha, Prague, 3–8 July 2005 Kane C, Marsden JE, Ortiz M (1999) Symplectic-energymomentum preserving variational integrators. J Math Phys 40(7):3353–3371 Kane C, Marsden JE, Ortiz M, West M (2000) Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int J Numer Meth Eng 49(10):1295– 1325 Kelley CT (1995) Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA Lee T, Leok M, McClamroch NH (2005) Attitude maneuvers of a rigid spacecraft in a circular orbit. In: Proceedings of the American Control Conference. Portland Hilton, Portland, OR, 8–10 June 2005, pp 1742–1747 Lee T, Leok M, McClamroch NH (2005) A Lie group variational integrator for the attitude dynamics of a rigid body with appli-
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cations to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications. Toronto, Canada, 28–31 August 2005, pp 962–967 Lee T, Leok M, McClamroch NH (2006) Optimal control of a rigid body using geometrically exact computations on SE(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 2710–2715 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem. Comput Method Appl Mech Eng 196:2907–2924 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem in orbital mechanics. Celest Mech Dyn Astron 98(2):121–144 Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Leok M (2004) Foundations of Computational Geometric Mechanics. Ph D thesis, California Instittute of Technology Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer, New York Marsden JE, West M (2001) Discrete mechanics and variational integrators. In: Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Marsden JE, Pekarsky S, Shkoller S (1999) Discrete Euler–Poincarée and Lie–Poisson equations. Nonlinearity 12(6):1647–1662 Maschke B, Ortega R, van der Schaft A (2001) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Autom Contr 45:1498–1502 Moser J, Veselov AP (1991) Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun Math Phys 139:217–243 Ortega R, Spong MW, Gómez-Estern F, Blankenstein G (2002) Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans Autom Contr 47:1218–1233 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286 Scheeres DJ, Fahnestock EG, Ostro SJ, Margot JL, Benner LAM, Broschart SB, Bellerose J, Giorgini JD, Nolan MC, Magri C, Pravec P, Scheirich P, Rose R, Jurgens RF, De Jong EM, Suzuki S (2006) Dynamical configuration of binary near-Earth asteroid (66391) 1999 KW4. Science 314:1280–1283 Scheeres DJ, Hsiao F-Y, Park RS, Villac BF, Maruskin JM (2006) Fundamental limits on spacecraft orbit uncertainty and distribution propagation. J Astronaut Sci 54:505–523 Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Comm Comput Phys 2:293–309 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2000) Matching and stabilization of low-dimensional nonholonomic systems. In: Proceedings of the IEEE Conference on Decision and Control. Sydney Convention and Exhibition Centre, Sydney, NSW Australia; 12–15 December 2000, pp 1289–1295 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2002) Flat nonholonomic matching. In: Proceedings of the American Control Conference. Hilton Anchorage, Anchorage, AK, 8–10 May 2002, pp 2812–2817
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Books and Reviews Bloch AM (2003) Nonholonomic Mechanics and Control. Interdisciplinary Appl Math, vol 24. Springer Bullo F, Lewis AD (2005) Geometric control of mechanical systems. Texts in Applied Mathematics, vol 49. Springer, New York Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365
Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer Marsden JE, West M (2001) Discrete mechanics and variational integrators. In Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286
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Dispersion Phenomena in Partial Differential Equations
Dispersion Phenomena in Partial Differential Equations PIERO D’ANCONA Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Roma, Italy
kukH s D (1 )s/2 uL q : Recall that this definition does not reduce to the preceding one when q D 1. We shall also use the homogeneous space H˙ qs , with norm kukH˙ s D ()s/2 u L q :
Article Outline Glossary Definition of the Subject Introduction The Mechanism of Dispersion Strichartz Estimates The Nonlinear Wave Equation The Nonlinear Schrödinger Equation Future Directions Bibliography Glossary Notations Partial derivatives are written as ut or @ t u, @˛ D @˛x 11 @˛x nn , the Fourier transform of a function is defined as Z F f () D b f () D ei x f (x)dx and we frequently use the mute constant notation A . B to mean A CB for some constant C (but only when the precise dependence of C from the other quantities involved is clear from the context). Evolution equations Partial differential equations describing physical systems which evolve in time. Thus, the variable representing time is distinguished from the others and is usually denoted by t. Cauchy problem A system of evolution equations, combined with a set of initial conditions at an initial time t D t0 . The problem is well posed if a solution exists, is unique and depends continuously on the data in suitable norms adapted to the problem. Blow up In general, the solutions to a nonlinear evolution equation are not defined for all times but they break down after some time has elapsed; usually the L1 norm of the solution or some of its derivatives becomes unbounded. This phenomenon is called blow up of the solution. Sobolev spaces we shall use two instances of Sobolev space: the space W k;1 with norm kukW k;1 D
and the space H qs with norm
X j˛jk
k@˛ ukL 1
Dispersive estimate a pointwise decay estimate of the form ju(t; x)j Ct ˛ , usually for the solution of a partial differential equation. Definition of the Subject In a very broad sense, dispersion can be defined as the spreading of a fixed amount of matter, or energy, over a volume which increases with time. This intuitive picture suggests immediately the most prominent feature of dispersive phenomena: as matter spreads, its size, defined in a suitable sense, decreases at a certain rate. This effect should be contrasted with dissipation, which might be defined as an actual loss of energy, transferred to an external system (heat dissipation being the typical example). This rough idea has been made very precise during the last 30 years for most evolution equations of mathematical physics. For the classical, linear, constant coefficient equations like the wave, Schrödinger, Klein–Gordon and Dirac equations, the decay of solutions can be measured in the Lp norms, and sharp estimates are available. In addition, detailed information on the profile of the solutions can be obtained, producing an accurate description of the evolution. For nonlinear equations, the theory now is able to explain and quantify several complex phenomena such as the splitting of solutions into the sum of a train of solitons, plus a radiation part which disperses and decays as time increases. Already in the 1960s it was realized that precise information on the decay of free waves could be a powerful tool to investigate the effects of linear perturbations (describing interactions with electromagnetic fields) and even nonlinear perturbations of the equations. One of the first important applications was the discovery that a nonlinear PDE may admit global small solutions, provided the rate of decay is sufficiently strong compared with the concentration effect produced by the nonlinear terms. This situation is very different from the ODE setting, where dispersion is absent. Today, dispersive and Strichartz estimates represent the backbone of the theory of nonlinear evolution equations. Their applications include local and global existence
Dispersion Phenomena in Partial Differential Equations
results for nonlinear equations, existence of low regularity solutions, scattering, qualitative study of evolution equations on manifolds, and many others. Introduction To a large extent, mathematical physics is the study of qualitative and quantitative properties of the solutions to systems of differential equations. Phenomena which evolve in time are usually described by nonlinear evolution equations, and their natural setting is the Cauchy problem: the state of the system is assigned at an initial time, and the following evolution is, or should be, completely determined by the equations. The most basic question concerns the local existence and uniqueness of the solution; we should require that, on one hand, the set of initial conditions is not too restrictive, so that a solution exists, and on the other hand it is not too lax, so that a unique solution is specified. For physically meaningful equations it is natural to require in addition that small errors in the initial state propagate slowly in time, so that two solutions corresponding to slightly different initial data remain close, at least for some time (continuous dependence on the data). This set of requirements is called local well posedness of the system. Global well posedness implies the possibility of extending these properties to arbitrary large times. However, local well posedness is just the first step in understanding a nonlinear equation; basically, it amounts to a check that the equation and its Cauchy problem are meaningful from the mathematical and physical point of view. But the truly interesting questions concern global existence, asymptotic behavior and more generally the structure and shape of solutions. Indeed, if an equation describes a microscopic system, all quantities measured in an experiment correspond to asymptotic features of the solution, and local properties are basically useless in this context. The classical tool to prove local well posedness is the energy method; it can be applied to many equations and leads to very efficient proofs. But the energy method is intrinsically unable to give answers concerning the global existence and behavior of solutions. Let us illustrate this point more precisely. Most nonlinear evolution equations are of the form Lu D N(u) with a linear part L (e. g., the d’Alembert or Schrödinger operator) and a nonlinear term N(u). The two terms are in competition: the linear flow is well behaved, usually with
several conserved quantities bounded by norms of the initial data, while the nonlinear term tends to concentrate the peaks of u and make them higher, increasing the singularity of the solution. At the same time, a small function u is made even smaller by a power-like nonlinearity N(u). The energy method tries to use only the conserved quantities, which remain constant during the evolution of the linear flow. Essentially, this means to regard the equation as an ODE of the form y 0 (t) D N(y(t)). This approach is very useful if the full (nonlinear) equation has a positive conserved quantity. But in general the best one can prove using the energy method is local well posedness, while a blow up in finite time can not be excluded, even if the initial data are assumed to be small, as the trivial example y 0 D y 2 clearly shows. To improve on this situation, the strength of the linear flow must be used to a full extent. This is where dispersive phenomena enter the picture. A finer study of the operator L shows that there are quantities, different from the standard L2 type norms used in the energy estimates, which actually decay during the evolution. Hence the term L has an additional advantage in the competition with N which can lead to global existence of small solutions. See Sect. “The Nonlinear Wave Equation” for more details. This circle of ideas was initiated at the beginning of the 1970s and rapidly developed during the 1980s in several papers, (see [26,27,31,32,33,34,38,40,41,42,50] to mention just a few of the fundamental contributions on the subject). Global existence of small solutions was proved for many nonlinear evolution equations, including the nonlinear wave, Klein–Gordon and Schrödinger equations. The proof of the stability of vacuum for the Einstein equations [15,35] can be regarded as an extreme development of this direction of research. Another early indication of the importance of Lp estimates came from the separate direction of harmonic analysis. In 1970, at a time when general agreement in the field was that the L2 framework was the only correct one for the study of the wave equation, R. Strichartz ([44,45], see also [10,11,46]) proved the estimate u t t u D F ;
u(0; x) D u t (0; x) D 0
H)
kukL p (RnC1 ) . kFkL p0 (RnC1 )
for p D 2(n C 1)/(n 1). This was a seminal result, both for the emphasis on the Lp approach, and for the idea of regarding all variables including time on the same level, an idea which would play a central role in the later developments. At the same time, it hinted at a deep connection
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between dispersive properties and the techniques of harmonic analysis. The new ideas were rapidly incorporated in the mainstream theory of partial differential equations and became an important tool ([22,23,52]); the investigation of Strichartz estimates is still in progress. The freedom to work in an Lp setting was revealed as useful in a wide variety of contexts, including linear equations with variable coefficients, scattering, existence of low regularity solutions, and several others. We might say that this point of view created a new important class of dispersive equations, complementing and overlapping with the standard classification into elliptic, hyperbolic, parabolic equations. Actually this class is very wide and contains most evolution equations of mathematical physics: the wave, Schrödinger and Klein–Gordon equations, Dirac and Maxwell systems, their nonlinear counterparts including wave maps, Yang–Mills and Einstein equations, Korteweg de Vries, Benjamin–Ono, and many others. In addition, many basic physical problems are described by systems obtained by coupling these equations; among the most important ones we mention the Dirac–Klein–Gordon, Maxwell–Klein–Gordon, Maxwell– Dirac, Maxwell–Schrödinger and Zakharov systems. Finally, a recent line of research pursues the extension of the dispersive techniques to equations with variable coefficients and equations on manifolds. The main goal of this article is to give a quick overview of the basic dispersive techniques, and to show some important examples of their applications. It should be kept in mind however that the field is evolving at an impressive speed, mainly due to the contribution of ideas from Fourier and harmonic analysis, and it would be difficult to give a self-contained account of the recent developments, for which we point at the relevant bibliography. In Sect. “The Mechanism of Dispersion” we analyze the basic mechanism of dispersion, with special attention given to two basic examples (Schrödinger and wave equation). Section “Strichartz Estimates” is a coincise exposition of Strichartz estimates, while Sects. “The Nonlinear Wave Equation” and “The Nonlinear Schrödinger Equation” illustrate some typical applications of dispersive techniques to problems of global existence for nonlinear equations. In the last Sect. “Future Directions” we review some of the more promising lines of research in this field.
The same mechanism lies behind dispersive phenomena, and indeed two dual explanations are possible: a) in physical space, dispersion is characterized by a finite speed of propagation of the signals; data concentrated on some region tend to spread over a larger volume following the evolution flow. Correspondingly, the L1 and other norms of the solution tend to decrease; b) in Fourier space, dispersion is characterized by oscillations in the Fourier variable , which increase for large jj and large t. This produces cancelation and decay in suitable norms. We shall explore these dual points of view in two fundamental cases: the Schrödinger and the wave equation. These are the most important and hence most studied dispersive equations, and a fairly complete theory is available for both. The Schrödinger Equation in Physical Space The solution u(t; x) of the Schrödinger equation iu t C u D 0 ;
u(0; x) D f (x)
admits several representations which are not completely equivalent. From spectral theory we have an abstract representation as a group of L2 isometries u(t; x) D e i t f ;
(1)
then we have an explicit representation using the fundamental solution 2 1 i jxj 4t f e (4 it)n/2 Z jx yj2 1 i 4t e f (y)dy ; D (4 it)n/2 Rn
u(t; x) D
finally, we can represent the solution via Fourier transform as Z 2 n f ()d : (3) e i(tjj Cx)b u(t; x) D (2) Notice for instance that the spectral and Fourier representations are well defined for f 2 L2 , while it is not immediately apparent how to give a meaning to (2) if f 62 L1 . From (1) we deduce immediately the dispersive estimate for the Schrödinger equation, which has the form 1
The Mechanism of Dispersion
je i t f j (4)n/2
The Fourier transform puts in duality oscillations and translations according to the elementary rule
Since the L2 norm of u is constant in time
F ( f (x C h)) D ei xh F f :
(2)
t n/2
ke i t f kL 2 D k f kL 2 ;
k f kL 1 :
(4)
(5)
Dispersion Phenomena in Partial Differential Equations
this might suggest that the constant L2 mass spreads on a region of volume t n i. e. of diameter t. Apparently, this is in contrast with the notion that the Schrödinger equation has an infinite speed of propagation. Indeed, this is a correct but incomplete statement. To explain this point with an explicit example, consider the evolution of a wave packet 1
from zero, and this already suggests a decay of the order t n/2 . This argument can be made precise and gives an alternative proof of (4). We also notice that the Fourier transform in both space and time of u(t; x) D e i t f is a temperate distribution in S0 (RnC1 ) which can be written, apart from a constant, ˜ ) D ı( jj2) u(;
2
x 0 ;0 D e 2 jxx 0 j e i(xx 0 )0 : A wave packet is a mathematical representation of a particle localized near position x0 , with frequency localized near 0 ; notice indeed that its Fourier transform is 1
2
b x 0 ;0 D ce 2 j0 j e ix 0 for some constant c. Exact localization both in space and frequency is of course impossible by Heisenberg’s principle; however the approximate localization of the wave packet is a very good substitute, since the gaussian tails decay to 0 extremely fast. The evolution of a wave packet is easy to compute explicitly: 1 (x x0 2t0 )2 i i t n2 e x 0 ;0 D (1C2it) exp e 2 (1 C 4t 2 )
and is supported on the paraboloid D jj2. This representation of the solution gives additional insight to its behavior in certain frequency regimes, and has proved to be of fundamental importance when studying the existence of low regularity solutions to the nonlinear Schrödinger equation [6,9]. The Wave Equation in Physical Space For the wave equation, the situation is reversed: the most efficient proof of the dispersive estimate is obtained via the Fourier representation of the solution. This is mainly due to the complexity of the fundamental solution; we recall the relevant formulas briefly. The solution to the homogeneous wave equation
where the real valued function is given by D
t(x x0 )2 C (x x0 t0 ) 0 : (1 C 4t 2 )
We notice that the particle “moves” with velocity 20 and the height of the gaussian spreads precisely at a rate t n/2 . The mass is essentially concentrated in a ball of radius t, as suspected. This simple remark is at the heart of one of the most powerful techniques to have appeared in harmonic analysis in recent years, the wave packet decomposition, which has led to breakthroughs in several classical problems (see e. g. [7,8,48,51]).
w t t w D 0 ;
2
(t; x; ) D tjj C x
H)
r D 2t C x
so that for each (t; x) we have a unique stationary point D x/2t; the n curvatures at that point are different
w t (0; x) D g(x)
can be written as u(t; x) D cos(tjDj) f C
sin(tjDj) g jDj
where we used the symbolic notations cos(tjDj) f D F 1 cos(tjj)F f ; sin(tjDj) sin(tjj) Fg g D F 1 jDj jj (recall that F denotes the Fourier transform in space variables). Notice that
The Schrödinger Equation in Fourier Space In view of the ease of proof of (4), it may seem unnecessary to examine the solution in Fourier space, however a quick look at this other representation is instructive and prepares the discussion for the wave equation. If we examine (3), we see that the integral is a typical oscillatory integral which can be estimated using the stationary phase method. The phase here is
w(0; x) D f (x) ;
cos(tjDj) f D
@ sin(tjDj) f; @t jDj
hence the properties of the first operator can be deduced from the second. Moreover, by Duhamel’s formula, we can express the solution to the nonhomogeneous wave equation w t t w D F(t; x) ;
w(0; x) D 0 ;
w t (0; x) D 0
as Z
t
u(t; x) D 0
sin((t s)jDj) F(u(s; ))ds : jDj
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Dispersion Phenomena in Partial Differential Equations
of the solution remains constant, in view of the energy estimate
In other words, a study of the operator S(t)g D
sin(tjDj) g jDj
ku t (t; )k2L 2 Ckru(t; )k2L 2 D ku t (0; )k2L 2 Ckru(0; )k2L 2 ;
is sufficient to deduce the behavior of the complete solution to the nonhomogeneous wave equation with general data. The operator S(t) can be expressed explicitly using the fundamental solution of the wave equation. We skip the one-dimensional case since then from the expression of the general solution u(t; x) D (t C x) C
(t x)
it is immediately apparent that no dispersion can occur. In odd space dimension n 3 we have ! n3 Z 2 1 sin(tjDj) 1 g(x y)d y ; g D cn @t jDj t t jyjDt
on the other hand, the velocity of propagation for the wave equation is exactly 1, so that the constant L2 energy is spread on a volume which increases at a rate t n ; as a consequence, we would expect the same rate of decay t n/2 as for the Schrödinger equation. However, a more detailed study of the shape of the solution reveals that it tends to concentrate in a shell of constant thickness around the light cone. For instance, in odd dimension larger than 3 initial data with support in a ball B(0; R) produce a solution which at time t is supported in the shell B(0; t C R) n B(0; t R). Also in even dimension the solution tends to concentrate along the light cone jtj D jxj, with a faster decreasing tail far from the cone. Thus, the energy spreads over a volume of size t n1 , in accordance with the rate of decay in the dispersive estimate.
and in even space dimension n 2 sin(tjDj) g D cn jDj
1 @t t
n2 2
1 t
Z jyj jtj/2 we obtain je i tjDj f j C( f )jtj
n1 2
:
It is also possible to make precise the form of the constant C( f ) appearing in the estimate. This can be done in several ways; one of the most efficient exploits the scaling properties of the wave equation (see [10,23]). In order to state the optimal estimate we briefly recall the definition of some function spaces. The basic ingredient is the homogeneous Paley–Littlewood partition of unity which has the form X j () D 1 ; j () D 0 (2 j ) ; j2Z
sup j D f : 2 j1 jj 2 jC1 g for a fixed 0 2 C01 (Rn ) (the following definitions are independent of the precise choice of 0 ). To each symbol j () we associate by the standard calculus the operator j (D) D F 1 j ()F . Now the homogeneous Besov norm s can be defined as follows: B˙ 1;1 X s D 2 js k j (D) f kL 1 (8) k f kB˙ 1;1 j2Z
and the corresponding Besov space can be obtained by completing C01 (see [4] for a thorough description of these definitions). Then the optimal dispersive estimate for the wave equation is je i tjDj f j Ck f k
n1
B˙ 1;12
jtj
n1 2
:
(9)
We might add that in odd space dimension a slightly better estimate can be proved involving only the standard ˙ n1 2 instead of the Besov norm. Sobolev norm W The singularity in t at the right hand side of (9) can be annoying in applications, thus it would be natural to try to replace jtj by (1 C jtj) by some different proof. However, this is clearly impossible since both sides have the same scaling with respect to the transformation u(t; x) ! u(t; x) which takes a solution of the homogeneous wave equation into another solution. Moreover, if such an estimate were true, taking the limit as t ! 0 we would obtain a false Sobolev embedding. Thus, the only way to get a nonsingular estimate is to replace the Besov norm of the data by some stronger norm. Indeed, for jtj < 1 we can use the L2 conservation which implies, for any > 0, je i tjDj f j ke i tjDj f kH n/2C D k f kH n/2C : Combining the two estimates, we arrive at n1 je i tjDj f j . k f k n1 C k f k (1Cjtj) 2 : (10) n/2C H 2 B˙ 1;1
Other Dispersive Equations Dispersive estimates can be proved for many other equations; see for instance [3] for general results concerning the equations of the form iu t D P(D)u ;
P polynomial
which correspond to integrals of the form Z e i(xx iCP())b f ()d : Here we would like to add some more details on two equations of particular importance for physics. The Klein– Gordon equation u t t u C u D 0 ;
u(0; x) D f (x) ;
u t (0; x) D g(x)
shows the same decay rate as the Schrödinger equation, although for most other properties it is very close to the wave
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Dispersion Phenomena in Partial Differential Equations
equation. This phenomenon can be easily explained. In physical space, we notice that the solution has finite speed of propagation less or equal to 1, so that for compactly supported initial data the solution spreads uniformly on a volume of order tn , and there is no concentration along the light cone as for the wave equation. On the other hand, the Fourier representation of the solution is sin(thDi) u(t; x) D cos(thDi) f C g hDi
cos(thDi) f D F 1 cos(tjhij)F f ;
with hi D (1 C i. e. hDi D (1 Thus, we see that, similarly to the wave equation, the study of the solutions can be reduced to the basic operator Z f ()d : e i thDi f D F 1 e i thi F f D (2)n e i(thiCx)b It is easy to check that the phase has an isolated stationary point if and only if jtj < jxj, and no stationary point if jtj jj: Cx; hi
in accord with the claimed dispersion rate of order t n/2 . However, the proof is not completely elementary (for a very detailed study, see Section 7.2 of [25]). The optimal estimate is now je
i thDi
f j Ck f k
n
2 B 1;1
jtj
n2
sup j D f : 2
and the 4 4 Dirac matrices can be written I 0 k 0 ˛k D ; ˇD 2 ; k 0 0 I2 in terms of the Pauli matrices 1 0 0 I2 D ; 1 D 0 1 1 0 i 1 2 D ; 3 D i 0 0
j 0;
jj 2 jC1 g ;
sup 1 D fjxj 1g for some 1 ; 0 2 C01 (Rn ). A second important example is the Dirac system. From a mathematical point of view (and setting the values of the
1 0
k D 1; 2; 3
;
0 : 1
Then the solution u(t; x) D e i t D f of the massless Dirac system with initial value u(0; x) D f (x) satisfies the dispersive estimate ju(t; x)j
C k f kB˙ 2 1;1 t
while in the massive case we have ju(t; x)j
j2N
j1
3 1X ˛k @k i
(11)
(see the Appendix to [18]). Here we used the nonhomoges which is defined exactly as the honeous Besov norm B1;1 mogeneous one (8) by using the inhomogeneous version of the Palye–Littlewood partition of unity: X 1 () C j () D 1 ; j () D 0 (2 j ) for
iu t C Du C ˇu D 0
kD1
)1/2 .
r D t
in the massless case, and
DD
sin(thDi) sin(tjhij) Fg g D F 1 hDi jhij
(t; x; ) D thi C x ;
iu t C Du D 0
in the massive case. Here u : R t R3x ! C 4 , the operator D is defined as
where we used the symbolic notations
jj2)1/2
speed of light and Planck’s constant equal to 1 for simplicity of notation) this is a 4 4 constant coefficient system of the form
C 3
t2
kfk
5
2 B 1;1
(see Sect. 7 of [16]). Strichartz Estimates Since the Schrödinger flow e i t f is an L2 isometry, it may be tempting to regard it as some sort of “rotation” of the Hilbert space L2 . This picture is far from true: there exist small subspaces of L2 which contain the flow for almost all times. This surprising phenomenon, which holds in greater generality for any unbounded selfadjoint operator on an abstract Hilbert space, is known as Kato smoothing and was discovered by T. Kato in 1966 [29], see also [39]. This fact is just a corollary of a quantitative estimate, the Kato smoothing estimate, which in the case of the Laplace
Dispersion Phenomena in Partial Differential Equations
operator takes the following form: there exist closed unbounded operators A on L2 such that kAe
i t
f kL 2 L 2 . k f kL 2 :
As a consequence, the flow belongs for a. e. t to the domain of the operator A, which can be a very small subspace of L2 . Two simple examples of smoothing estimates are the following: V(x) 2 L2 \ L n \ L nC H)
kV(x)e i t f kL 2 L 2 . k f kL 2
(12)
and kjxj1 e i t f kL 2 L 2 . k f kL 2 :
(13)
On the other hand, Strichartz proved in 1977 [46] that pD
2(n C 2) n H)
2n
L 2 L n2
ke
i t
f kL p (RnC1 ) . k f kL 2
(14)
. k f kL 2
(15)
while (13) would follow from the slightly stronger ke i t f k
2n
L 2 L n2 ;2
The homogeneous Strichartz estimates have the general form ke i t f kL p L q . k f kL 2 for suitable values of the couple (p; q). Notice that we can restrict the L p L q norm at the right hand side to an arbitrary time interval I (i. e. to L p (I; L q (Rn ))). Notice also that by the invariance of the flow under the scaling u(t:x) 7! u(2 t; x) the possible couples (p; q) are bound to satisfy n n 2 C D : p q 2
. k f kL 2
(16)
where L p;2 denotes a Lorentz space. Both (15) and (16) are true, although their proof is far from elementary. Such estimates are now generally called Strichartz estimates. They can be deduced from the dispersive estimates, hence they are in a sense weaker and contain less information. However, for this same reason they can be obtained for a large number of equations, even with variable coefficients, and their flexibility and wide range of applications makes them extremely useful. The full set of Strichartz estimates, with the exception of the endpoint (see below) was proved in [22], where it was also applied to the problem of global well posedness for the nonlinear Schrödinger equation. The corresponding estimates for the wave equation were obtained in [23], however they turned out to be less useful because of the loss of derivatives. A fundamental progress was made in [30], where not only the most difficult endpoint case was solved, but it also outlined a general procedure to deduce the Strichartz estimates from the dispersive estimates, with applications to any evolution equation. In the following we shall focus on the two main examples, the Schrödinger and the wave equation, and we refer to the original papers for more details and the proofs.
(17)
The range of possible values of q is expressed by the inequalities 2q
thus showing that the smoothing effect could be measured also in the Lp norms. If we try to reformulate (12) in terms of an Lp norm, we see that it would follow from ke i t f k
The Schrödinger Equation
2n n2
2 q 1 if
if
n 2;
n D1;
q ¤1;
(18) (19)
the index p varies accordingly between 1 and 2 (between 4 and 2 in one space dimension). Such couples (p; q) are called (Schrödinger) admissible. It is easy to visualize the admissible couples by plotting the usual diagram in the (1/p; 1/q) space (see Fig 1–3). Note that the choice (p; q) D (1; 2), the point E in the diagrams, corresponds to the conservation of the L2 norm, while at the other extreme we have for n 2 point P corresponding to the couple 2n (p; q) D 2; : (20) n2 This is usually called the endpoint and is excluded in dimension 2, allowed in dimension 3. The original Strichartz estimate corresponds to the case p D q and is denoted by S. After the proof in [22] of the general Strichartz estimates, it was not clear if the endpoint case P could be reached or not; the final answer was given in [30] where the endpoint estimates were proved to be true for all dimension larger than three. For the two-dimensional case and additional estimates, see [47]. Notice that all admissible couples can be obtained by interpolation between the endpoint and the L2 conservation, thus in a sense the endpoint contains the maximum of information concerning the Lp smoothing properties of the flow. There exist an inhomogeneous form of the estimates, which is actually equivalent to the homogeneous one, and can be stated as follows: Z t i(ts) e F(s)ds . kFkL p˜ 0 L q˜0 (21) 0
L p Lq
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168
Dispersion Phenomena in Partial Differential Equations
Dispersion Phenomena in Partial Differential Equations, Figure 3 Schrödinger admissible couples, n 3 Dispersion Phenomena in Partial Differential Equations, Figure 1 Schrödinger admissible couples, n D 1
neous global estimates are optimal since other combination of indices are excluded by scale invariance; however, the local Strichartz estimates can be slightly improved allowing for indices outside the admissible region (see [21]). We also mention that a much larger set of estimates can be deduced from the above ones by combining them with Sobolev embeddings; in the above diagrams, all couples in the region bounded by the line of admissibility, the axes and the line 1/p D 2 can be reached, while the external region is excluded by suitable counterexamples (see [30]). Strichartz Estimates for the Wave Equation
Dispersion Phenomena in Partial Differential Equations, Figure 2 Schrödinger admissible couples, n D 2
Most of the preceding section has a precise analogue for the wave equation. Since the decay rate for the dispersive estimates is t (n1)/2 for the wave equation instead of t n/2 as for the Schrödinger equation, this causes a shift n ! n 1 in the numerology of indices. A more serious difference is the loss of derivatives: this was already apparent in the original estimate by Strichartz [44], which can be written ke i tjDj f k
˜ which can be for all admissible couples (p; q) and ( p˜ ; q), chosen independently. Here p0 denotes the conjugate exponent to p. These estimates continue to be true in their local version, with the norms at both sides restricted to a fixed time interval I R. Notice that the inhomoge-
L
2(nC1) n1
. k f kH˙ 1/2 ;
n 2:
The general homogeneous Strichartz estimates for the wave equation take the form ke i tjDj f kL p L q . k f k
1 nC1 n1
˙p H
;
Dispersion Phenomena in Partial Differential Equations
now the (wave) admissible couples of indices must satisfy the scaling condition 2 n1 n1 C D p q 2
(22)
and the constraints 2 p 1, 2q
2(n 1) n3
2q1
if
n 3;
if
q¤1
(23)
n D2;
(24)
The Nonlinear Wave Equation In this section we shall illustrate, using the semilinear wave equation as a basic example, what kind of improvements can be obtained over the classical energy method via the additional information given by dispersive techniques. From the linear theory it is well known that the correct setting of the Cauchy problem for the wave equation u t t u D F(t; x) ;
requires two initial data at t D 0: u(0; x) D u0 (x) ;
while p varies accordingly between 2 and 1 (between 4 and 1 in dimension 2). Recall that in dimension 1 the solutions of the wave equation do not decay since they can be decomposed into traveling waves. We omit the graphic representation of these sets of conditions since it is completely analogous to the Schrödinger case (apart from the shift n ! n 1). The inhomogeneous estimates may be written Z t i(ts) e F(s)ds 0
nC1 1p n1
˙q LpH
. kFk
nC1 1 n1
0 ˙ p˜ L p˜ H q˜0
(25)
u(t; x) : R Rn ! C
u t (0; x) D u1 (x) :
For this equation the energy method is particularly simple. If we multiply the equation by u t we can recast it in the form @ t ju t j2 C jruj2 D 2
n 2
H)
ku1 u2 k X T
. C(kv1 k X T C kv2 k X T ) kv1 v2 k X T T : At this point the philosophy of the method should be clear: if we agree to differentiate n/2 times the equation, we can prove a nonlinear estimate, with a coefficient which can be made small for T small. Thus, for T 1 the mapping v 7! u is a contraction, and we obtain a local (in time) solution of the nonlinear equation. The energy method is quite elementary, yet it has the remarkable property that it can be adapted to a huge number of equations: it can be applied to equations with variable coefficients, and even to fully nonlinear equations. An outstanding example of its flexibility is the proof of the local solvability of Einstein’s equations ([14]). The main drawbacks of the energy method are: 1) A high regularity of the data and of the solution is required; 2) The nonlinear nature of the estimate leads to a local existence result, leaving completely open the question of global existence.
k f kW N;1 ;
N>n
will be sufficient. This suggests modification of the energy norm by adding a term that incorporates the decay information. To this end, we introduce the time-dependent quantity M(t) D sup
n
0t
kD
hni 2
ku(; )kH kC1 C (1 C )
n1 2
o ku(; )kL 1 ;
:
Consider now the nonlinear Eq. (28) with initial data u0 ; u1 . We already know that a local solution exists, provided the initial data are smooth enough; we shall now try to extend this solution to a global one. Using Duhamel’s principle we rewrite the equation as Z u(t; x) D w(t; x) C 0
t
sin((t s)jDj) F(u(s; ))ds jDj
where we are using the symbolic notations of Sect. “Other Dispersive Equations”. F being the space Fourier transform. Note that the above dispersive estimate can be applied in particular to the last term; we obtain ˇ ˇ ˇ sin((t s)jDj) ˇ ˇ ˇ C(1 C t s) n1 2 kFk N;1 : F(s) W ˇ ˇ jDj
Dispersion Phenomena in Partial Differential Equations
By this inequality and the energy estimate already proved, in a few steps we arrive at the estimate n1 2
M(T) . C0 C kFkL 1 H k C (1 C T) T Z T n1 (1 C T s) 2 kF(u(s))kW N;1 ds 0
where C0 D ku0 kW N;1 C ku1 kW N;1 and the mute constant is independent of T. By suitable nonlinear estimates of Moser type, we can bound the derivatives of the nonlinear term using only the L1 and the H k norm of u, i. e., the quantity M(t) itself. If we assume that F(u) juj for small u, we obtain
Consider the Cauchy problem iu t u D juj ;
u(0; x) D f (x)
(31)
for some 1. We want to study the local and global solvability in L2 of this Cauchy problem. Recall that this is a difficult question, still not completely solved; however, at least in the subcritical range, a combination of dispersive techniques and the contraction mapping principle gives the desired result easily. Instead of Eq. (31), we are actually going to solve the corresponding integral equation Z t e i(ts) ju(s)j ds : u(t; x) D e i t f C i 0
n1
M(T) . C0 C M(T) (1 C T) 2 Z T n1 n1 (1 C T s) 2 (1 C s) 2 ( 2) ds : 0
Our standard approach is to look for a fixed point of the mapping u D ˚(v) defined by Z t ˚(v) D e i t f C i e i(ts) jv(s)j ds ; 0
Now we have Z
t
(1 C t )
n1 2
(1 C )
n1 2 ( 2)
d
0
C(1 C t)
n1 2
k˚(v)kL p L q . k f kL 2 C kj˚(v)j kL p˜ 0 L q˜0
provided
> 2C
it is easy to check, using Strichartz estimates, that the mapping ˚ is well defined, provided the function v is in the suitable L p L q . Indeed we can write, for all admissible cou˜ ples (p; q) and ( p˜; q)
2 n1
so that, for such values of , we obtain M(T) . C0 CM(T) : For small initial data, which means small values of the constant C0 , (30) implies that M(T) is uniformly bounded as long as the smooth solution exists. By a simple continuation argument, this implies that the solution is in fact global, provided the initial data are small enough.
and by Hölder’s inequality this implies 0 0 L p˜ L q˜
k˚(v)kL p L q C1 k f kL 2 C C1 kvk
:
(32)
Thus, we see that the mapping ˚ operates on the space L p L q , provided we can satisfy the following set of conditions: (30) p˜0 D p ; q˜0 D q; 2 n n n n 2˜ C D ; C D ; ˜ p q˜ p q 2 2
2n p; p˜ 2 [2; 1] ; q˜ 2 2; : n2 Notice that from the first two identities we have
The Nonlinear Schrödinger Equation Strichartz estimates for the Schrödinger equation are expressed entirely in terms of Lp norms, without loss of derivatives, in contrast with the case of the wave equations. For this reason they represent a perfect tool to handle semilinear perturbations. Using Strichartz estimates, several important results can be proved with minimal effort. As an example, we chose to illustrate in this Section the critical and subcritical well posedness in L2 in the case of power nonlinearities (see [13] for a systematic treatment).
n 2
n
2 C D˛ Cn p˜ q˜ p q and by the admissibility conditions this implies n n
D2Cn ; 2 2 which forces the value of to be
D1C
4 : n
This value of is called L2 critical.
(33)
171
172
Dispersion Phenomena in Partial Differential Equations
Let us assume for the moment that is exactly the critical exponent (33). Then it is easy to check that there exist ˜ which satisfy all the above conchoices of (p; q) and ( p˜ ; q) straints. To prove that ˚ is a contraction on the space
p˜0 < p ;
we take any v1 ; v2 2 X and compute k˚(v1 ) ˚(v2 )k X . jv1 j jv2 j L p˜ 0 L q˜0 . jv1 v2 j(jv1 j 1 C jv2 j 1 )
0
L p˜ L q˜
1
k˚ (v1 ) ˚(v2 )k X C2 kv1 v2 k X kjv1 j C jv2 jk X
0
;
(34)
for some constant C2 . Now assume that
C2 T kv1 v2 k X T kjv1 j C jv2 jkX1 : T
provided "; ı are so small that
k f kL 2
0. Analogously, estimate (34) will be replaced by
using (32) we can write
C2 (2") 1
B c :D B0;1 the unforced rod is stable to self-weight buckling (a result known to Greenhill [26]). Moreover, it can be shown [14,15] that each eigenmode n (s) retains a qualitatively similar mode shape for B > B0;n with the nth mode having n D 1 internal zeros, and that the corresponding loci n (B) are approximately straight lines as shown in the upper part of Fig. 11. The lower part of Fig. 11 shows the results of a parametric analysis of the Eq. (27) for D 0. Basically, each time one of the eigenvalue loci passes through times the square of a half-integer ( j/2)2 we reach a B-value for which there is the root point of an instability tongue. Essentially we have a Mathieu-equation-like Ince–Strutt diagram in the (B; ") plane for each mode n. These diagrams are overlaid on top of each other to produce the overall stability
plot. However, note from Fig. 11 that the slope of each eigenvalue locus varies with the frequency ratio . Hence as increases, the values of all the B j/2;n , j D 0; : : : ; 1 for a given n increase with the same fixed speed. But this speed varies with the mode number n. Hence the instability tongues slide over each other as the frequency varies. Note the schematic diagram in the lower panel of Fig. 11 indicates a region of the stability for B < B0;1 for small positive ". Such a region would indicate the existence of rod of (marginally) lower bending stiffness than that required for static stability, that can be stabilized by the introduction of the parametric excitation. Essentially, the critical bending stiffness for the buckling instability is reduced (or, equivalently since B / `3 , the critical length for buckling is increased). This effect has been dubbed the ‘Indian rod trick’ because the need for bending stiffness in the wire means that mathematically this is a rod, not a rope (or string); see, e. g. the account in [4]. To determine in theory the precise parameter values at which the rod trick works turns out to be an involved process. That the instability boundary emerging from B0;1 in the lower panel of Fig. 11 should usually bend back to the left can be argued geometrically [7]. However, for -values at which B0;1 B1;n for some n, it has been shown that there is actually a singularity in the coefficient of the quadratic part of this buckling curve causing the instability boundary to bend to the right [20]. A complete un-
Dynamics of Parametric Excitation
Dynamics of Parametric Excitation, Figure 11 Summarizing schematically the results of the parametric resonance analysis of the parametric excited vertical column in the absence of damping. The upper part shows eigenvalue loci n (B) and the definition of the points Bj/2;n , j D 0; 1; 2; 3; : : : In the lower part, instability tongues in the (B; ")-plane are shown to occur with root points Bj/2;n and to have width "j . The shaded regions correspond to the where the vertical solution is stable. Solid lines represent neutral stability curves with Floquet multiplier + 1 (where ˛ is an integer) and dashed lines to multipliers 1 (where ˛ is half an odd integer). Reproduced from [14] with permission from the Royal Society
folding of this so-called resonance tongue interaction was performed in [15], also allowing for the presence of material damping. The resulting asymptotic analysis needs to be carried out in the presence of four small parameters; ",
, B B0;1 and c where c is the frequency-ratio value at which the resonance-tongue interaction occurs. The results produce the shaded region shown in Fig. 10b, which can be seen to agree well with the experimental results. Faraday Waves and Pattern Formation When a vessel containing a liquid is vibrated vertically, it is well known that certain standing wave patterns form on the free surface. It was Faraday in 1831 [19] who first noted that the frequency of the liquid motion is half that of the vessel. These results were confirmed by Rayleigh [50] who suggested that what we now know as parametric resonance was the cause. This explanation was shown to be
correct by Benjamin and Ursell [6] who derived a system of uncoupled Mathieu equations from the Navier–Stokes equations describing the motion of an ideal (non-viscous) fluid. There is one Mathieu equation (4), for each wave number i that fits into the domain, each with its own parameters ˛ i and ˇ i . They argue that viscous effects would effectively introduce a linear damping term, as in (8) and they found good agreement between the broad principal (subharmonic) instability tongue and the results of a simple experiment. Later, Kumer and Tuckerman [36] produced a more accurate Ince–Strutt diagram (also known as a dispersion relation in this context) for the instabilities of free surfaces between two viscous fluids. Through numerical solution of appropriate coupled Mathieu equations that correctly take into account the boundary conditions, they showed that viscosity acts in a more complicated way than the simple velocity proportional damping term in (8). They produced the diagram shown in Fig. 12b, which showed a good match with the experimental results. Note that, compared with the idealized theory, Fig. 12a, the correct treatment of viscosity causes a small shift in the critical wavelength of each instability, a broadening of the higher-harmonic resonance tongues and a lowering of the amplitude required for the onset of the principal subharmonic resonance if damping is introduced to the idealized theory (see the inset to Fig. 12b). More interest comes when Faraday waves are excited in a vessel where two different wave modes have their subharmonic resonance at nearly the same applied frequency, or when the excitation contains two (or more) frequencies each being close to twice the natural frequency of a standing wave. See for example the book by Hoyle [30] and references therein, for a glimpse at the amazing complexity of the patterns of vibration that can result and for their explanation in terms of dynamical systems with symmetry. Recently a new phenomenon has also been established, namely that of localized spots of pattern that can form under parametric resonance, so-called oscillons see e. g. [61] for a review and [39] for the first steps at an explanation. Fluid-Structure Interaction Faraday waves are parametric effects induced by the gross mechanical excitation of a fluid. In Subsect. “Autoparametric Resonance” we briefly mentioned that fluid-structure interaction can go the other way, namely that external fluid flow effects such as vortex shedding can excite (auto)parametric resonances in simple mechanical systems. Such flow-induced parametric resonance can also occur for continuous structures, in which waves can be excited in the structure due to external parametric loading of
199
200
Dynamics of Parametric Excitation
Dynamics of Parametric Excitation, Figure 12 Stability boundaries for Faraday waves from: a Benjamin and Ursell’s theory for ideal fluids; b the results of full hydrodynamic stability analysis by Kumar and Tuckerman. The horizontal axis is the frequency and the vertical scale the amplitude of the excitation. The inset to panel b shows the difference between the true instability boundary for the principal subharmonic resonance (lower curve) and the idealized boundary in the presence of simple proportional damping. After [36], to which the reader is referred for the relevant parameter values; reproduced with permission from Cambridge University Press
the fluid; see for example the extensive review by Païdousis and Li [47]. They also treat cases where the flow is internal to the structure: we are all familiar with the noisy ducts and pipes, for example in central heating or rumbling digestive systems, and it could well be that many of these effects are a result of parametric resonance. Semler and Païdousis [52] consider in detail parametric resonance effects in the simple paradigm of a cantilevered (free at one end, clamped at the other) flexible pipe conveying fluid. Think of this as a hosepipe resting on a garden lawn. Many authors have considered such an apparatus as a paradigm for internal flow-induced oscillation
problems (see the extensive references in [47,52]), starting from Benjamin’s [5] pioneering derivation of equations of motion from the limit of a segmented rigid pipe. The equations of motion for such a situation are similar to those of the rod (27), but with significant nonlinear terms arising from inertial and fluid-structure interaction effects. In the absence of periodic fluctuations in the flow, then it is well known that there is a critical flow rate beyond which the pipe will become oscillatory, through a Hopf bifurcation (flutter instability). This effect can be seen in the way that hosepipes writhe around on a lawn, and indeed forms the basis for a popular garden toy. Another form of instability can occur when the mean flow rate is beneath the critical value for flutter, but the flow rate is subject to a periodic variation, see [52] and references therein. There, a parametric instability can occur when the pulse rate is twice the natural frequency of small amplitude vibrations of the pipe. In [52] this effect was observed in theory, in numerical simulation and in a careful experimental study, with good agreement between the three. Parametric resonance could also be found for super-critical flow rates. In this case, the effect was that the periodic oscillation born at the Hopf bifurcation was excited into a quasi-periodic response. Another form of fluid structure interaction can occur when the properties of an elastic body immersed in a fluid are subject to a temporal periodic variation. With a view to potential biological applications such as active hearing processes and heart muscles immersed in blood, Cortez et al. [16] consider a two-dimensional patch of viscous fluid containing a ring filament that is excited via periodic variation of its elastic modulus. Using a mixture of immersed boundary numerical methods and analytic reductions, they are able to find an effective Ince–Strutt diagram for this system and demonstrate the existence of parametric instability in which the excitation of the structure causes oscillations in the fluid. Taking a fixed wave number p of the ring in space, they find a sequence of harmonic and subharmonic instabilities as a parameter effectively like ˛ in the Mathieu equation is increased. Interestingly though, even for low viscosity, the principal subharmonic resonance tongue (corresponding to ˛ D 1/4) does not occur for low amplitude forcing, and the most easily excited instability is actually the fundamental, harmonic resonance (corresponding to ˛ D 1). Dispersion Managed Optical Solitons The stabilizing effect of high-frequency parametric excitation has in recent years seen a completely different application, in an area of high technological importance, namely
Dynamics of Parametric Excitation
optical communications. Pulses of light sent along optical fibers are a crucial part of global communication systems. A pulse represents a packet of linear waves, with a continuum of different frequencies. However, optical fibers are fundamentally dispersive, which means that waves with different wavelengths travel at different group velocities. In addition, nonlinearity, such as the Kerr effect of intensity-dependent refractive index change, and linear losses mean that there is a limit to how far a single optical signal can be sent without the need for some kind of amplifier. Depending on the construction of the fiber, dispersion can be either normal which means that lower frequencies travel faster, or anomalous in which case higher frequencies travel faster. One idea to overcome dispersive effects, to enable pulses to be travel over longer distances without breaking up, is to periodically vary the medium with which the fiber is constructed so that the dispersion is alternately normally and anomalously dispersive [38]. Another idea, first proposed by Hasegawa and Tappert [28], is to balance dispersion with nonlinearity. This works because the complex wave envelope A of the electric field of the pulse can be shown to leading order to satisfy the nonlinear Schrödinger (NLS) equation i
dA 1 d2 A C jAj2 A D 0 ; C dz 2 dt 2
(30)
in which time and space have been rescaled so that the anomalous dispersion coefficient (multiplying the secondderivative term) and the coefficient of the Kerr nonlinearity are both unity. Note the peculiarity in optics compared to other wave-bearing system in that time t plays the role of the spatial coordinate, and the propagation distance z is the evolution variable. Thus we think of pulses as being time-traces that propagate along the fiber. Now, the NLS equation is completely integrable and has the well-known soliton solution 2
A(z; t) D sech(t cz)e i z ;
(31)
which represents a smooth pulse with amplitude , which propagates at ‘speed’ c. Moreover, Eq. (30) is completely integrable [69] and so arbitrary initial conditions break up into solitons of different speed and amplitude. Optical solitons were first realized experimentally by Mollenauer et al. [43]. However, true optical fibers still suffer linear losses, and the necessary periodic amplification of the optical solitons to enable them to survive over long distances can cause them to jitter [25] and to interact with each other as they move about in time. The idea of combining nonlinearity with dispersion management in order to stabilize soliton propagation was
first introduced by Knox, Forysiak and Doran [35], was demonstrated experimentally by Suzuki et al. [54] and has shown surprisingly advantageous performance that is now being exploited technologically (see [60] for a review). Essentially, dispersion management introduces a periodic variation of the dispersion coefficient d(z), that can be used to compensate the effects of periodic amplification, which with the advent of modern erbium-doped optical amplifiers can be modeled as a periodic variation of the nonlinearity. As an envelope equation we arrive at the modified form of (30), i
dA d(z) d2 A C (z)jAj2 A D 0 ; C dz 2 dt 2
(32)
where the period L of the dispersion map d(z) and the amplifier spacing (the period of (z)) Z may not necessarily be the same. Two particular regimes are of interest. One, of application to long-haul, sub-sea transmission systems where the cable is already laid with fixed amplifiers, is to have L Z. Here, one modifies an existing cable by applying bits of extra fiber with large dispersion of the opposite sign only occasionally. In this case, one can effectively look only at the long scale and average out the effect of the variation of . Then the parametric term only multiplies the second derivative. Another interesting regime, of relevance perhaps when designing new terrestrial communication lines, is to have L D Z so that one adds loops of extra opposite-dispersion fiber at each amplification site. In both of these limits we have an evolution equation whose coefficients are periodic functions of the evolution variable z with period L. In either regime, the periodic (in z) parameter variation has a profound stabilizing influence, such that the effects of linear losses and jitter are substantially reduced. A particular observation is that soliton solutions appear to (32) breathe during each period; that is, their amplitude and phase undergo significant periodic modulation. Also, with increase in the amplitude of the oscillatory part of d, so the average shape of the breathing soliton becomes less like a sech-pulse and more like a Gaussian. An advantage is that the Gaussian has much greater energy the regular NLS soliton (31) for the same average values of d and , hence there is less of a requirement for amplification. Surprisingly, it has been found that the dispersion managed solitons can propagate stably even if the average of d is slightly positive, i. e. in the normal dispersion region where there is no corresponding NLS soliton. This has a hugely advantageous consequence. The fact that the local dispersion can be chosen to be high, but the average dispersion can be chosen to be close to zero, is the key to enabling the jitter introduced by the amplifiers to be greatly suppressed
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Dynamics of Parametric Excitation
compared with the regular solitons. Finally the large local modulation of phase that is a consequence of the breathing, means that neighboring solitons are less likely to interact with each other. A mathematical analysis of why these surprising effects arise is beyond the scope of this article, see e. g. [32,60] for reviews, but broadly-speaking the stable propagation of dispersion-managed solitons is due to the stabilizing effects of high frequency parametric excitation. Future Directions This article has represented a whistle-stop tour of a wide areas of science. One of the main simplifications we have taken is that we have assumed the applied excitation to the system to be periodic; in fact, in most examples it has been pure sinusoidal. There is a growing literature on quasiperiodic forcing, see e. g. [51], and it would be interesting to look at those effects that are unique to parametrically excited systems that contain two or more frequencies. The inclusion of noise in the excitation also needs to be investigated. What effect does small additive or multiplicative noise have on the shape of the resonance tongues? And what is the connection with the phenomenon of stochastic resonance [66]? There is a mature theory of nonlinear resonance in systems with two degrees of freedom which, when in an integrable limit, undergoes quasiperiodic motion with two independent frequencies. Delicate results, such as the KAM theorem, see e. g. [55], determine precisely which quasiperiodic motions (invariant tori) survive as a function of two parameters, the size of the perturbation from integrability and the frequency detuning between the two uncoupled periodic motions. For the tori that break up (inside so-called Arnol’d tongues) we get heteroclinic tangles and bands of chaotic motion. Parametric resonance tends to be slightly different though, since in the uncoupled limit, one of the two modes typically has zero amplitude (what we referred to earlier as a semi-trivial solution). It would seem that there is scope for a deeper connection to be established between the Arnol’d tongues of KAM theory and the resonance tongues that occur under parametric excitation. In terms of applications, a large area that remains to be explored further is the potential exploitation of parametric effects in fluid-structure interaction problems. As I write, the heating pipes in my office are buzzing at an extremely annoying frequency that seems to arise on warm days when many radiators have been switched off. Is this a parametric effect? It would also seem likely that nature uses parametric resonance in motion. Muscles contract
longitudinally, but motion can be in a transverse direction. Could phenomena such as the swishing of bacterial flagella, fish swimming mechanisms and the pumping of fluids through vessels, be understood in terms of exploitation of natural parametric resonances? How about motion of vocal chords, the Basilar membrane in the inner ear, or the global scale pumping of the heart? Does nature naturally try to tune tissue to find the primary subharmonic resonance tongue? Perhaps other effects of parametric excitation that we have uncovered here, such as the quenching of resonant vibrations, structural stiffening, and frequency tuned motion, are already being usefully exploited in nature. If so, there could be interesting consequences for nature inspired design in the engineered world. Finally, let us return to our opening paradigm of how children effectively excite a parametric instability in order to induce motion in a playground swing. In fact, it was carefully shown by Case and Swanson [11] that the mechanism the child uses when in the sitting position is far more accurately described as an example of direct forcing rather than parametric excitation. By pushing on the ropes of the swing during the backwards flight, and pulling on them during the forward flight, the child is predominately shifting his center of gravity to and fro, rather than up and down; effectively as in Fig. 1c rather than Fig. 1b. Later, Case [10] argued that, even in the standing position, the energy gained by direct forcing from leaning backwards and forwards is likely to greatly outweigh any gained from parametric up-and-down motion of the child. The popular myth that children swinging is a suitable playground science demonstrator of parametric resonance was finally put to bed by Post et al. [49]. They carried out experiments on human patients and analyzed the mechanisms they use to move, reaching the conclusion that even in the standing position, typically 95% of the energy comes from direct forcing, although parametric effects do make a slightly more significant contribution for larger amplitudes of swing.
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Dynamics on Fractals
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Dynamics on Fractals RAYMOND L. ORBACH Department of Physics, University of California, Riverside, USA Article Outline Glossary Definition of the Subject Introduction Fractal and Spectral Dimensions Nature of Dynamics on Fractals – Localization Mapping of Physical Systems onto Fractal Structures Relaxation Dynamics on Fractal Structures Transport on Fractal Structures Future Directions Bibliography Glossary Fractal Fractal structures are of two types: deterministic fractals and random fractals. The former involves repeated applications of replacing a given structural element by the structure itself. The process proceeds indefinitely, leading to dilation symmetry: if we magnify part of the structure, the enlarged portion looks the same as the original. Examples are the Mandelbrot– Given fractal and the Sierpinski gasket. A random fractal obeys the same properties (e. g. dilation symmetry), but only in terms of an ensemble average. The stereotypical example is the percolating network. Fractals can be constructed in any dimension, d, with for example, d D 6 being the mean-field dimension for percolating networks. Fractal dimension The fractal dimension represents the “mass” dependence upon length scale (measuring length). It is symbolized by Df , with the number of sites on a fractal as a function of the measurement length L being proportional to L D f , in analogy to a homogeneous structure embedded in a dimension d having mass proportional to the volume spanned by a length L proportional to Ld . Spectral (or fracton) dimension The spectral (or fracton) dimension refers to the dynamical properties of fractal networks. It is symbolized by d˜s and can be most easily thought of in terms of the density of states of a dynamical fractal structure (e. g. vibrations of a fractal network). Thus, if the excitation spectrum is measured as a function of energy !, the density of states for excitations of a fractal network would be propor-
tional to ! (d s 1) , in analogy to a homogeneous structure embedded in a dimension d having a density of states proportional to ! (d1) . Localization exponent Excitations on a fractal network are in general strongly localized in the sense that wave functions fall off more rapidly than a simple exponential, (r) exp[f[r/(!)]d g] where (!) is an energy dependent localization length, and the exponent d is in general greater than unity. Definition of the Subject The dynamical properties of fractal networks are very different from homogeneous structures, dependent upon strongly localized excitations. The thermal properties of fractals depend upon a “spectral dimension” d˜s less than the “Euclidean” or embedding dimension d. Fractal randomness introduces statistical properties into relaxation dynamics. Transport takes place via hopping processes, reminiscent of Mott’s variable range-rate hopping transport for electronic impurity states in semi-conductors. Fractal dynamics serve as a guide for the behavior of random structures where the short length scale excitations are localized. Introduction The study of the dynamics on and of fractal networks [29, 30,31] is not an arcane investigation, with little application to real physical systems. Rather, there are many examples of real materials that exhibit fractal dynamics. Examples are the vibrational properties of silica aerogels [21,45], and magnetic excitations in diluted antiferromagnets [18,19,43]. Beyond these explicit physical realizations, one can learn from the very nature of the excitations on fractal structures how localized excitations in homogeneous materials behave. That is, the dynamics of fractal networks serve as a calculable model for transport of localized excitations in random structures that are certainly not mass fractals. Examples are thermal transport in glasses above the so-called “plateau” temperature [2,20,26], and the lifetime of high-energy lattice vibrations in a-Si [32,33,36,37]. As we shall show in subsequent sections, the former is an example of vibrational hopping-type transport (similar in nature to Mott’s variable-range hopping [22,23] for localized electronic states), while the latter exhibits counter-intuitive energy dependences for vibrational lifetimes. The following Sect. “Fractal and Spectral Dimensions” describes in brief terms the concept of the fractal (or mass) dimension, Df [7,8,9,10,15,40,41,46], and the spectral (or fracton) dimension d˜s [5,14,17,34,35]. The latter is re-
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lated to the anomalous diffusion characteristics of fractals (the so-called “ant in a labyrinth” introduced by de Gennes [8]), and is at the heart of the thermal properties of fractal structures. A conjecture introduced by Alexander and Orbach [5] suggested that the mean-field value (exact at d D 6) for d˜s for percolating networks, d˜s D 4/3, might be universal, independent of dimension. It is now generally regarded that this conjecture is only approximate, though a very good approximation for percolating networks in 2 D d < 6. Were it to be exact, it would mean that the dynamical properties of percolating networks could be expressed in terms of their geometrical properties. Section “Nature of Dynamics on Fractals–Localization” introduces the nature of the wave function for an excitation embedded in a fractal structure. In general, random self-similar structures generate “super-localized” wave functions, falling off faster than exponential. We will characterize the envelope of the wave function by the functional form [1] (r) exp[f[r/(!)]d g] ; where (!) is an energy-dependent localization length, and the exponent d is in general greater than unity. For excitations on fractal structures, the localization length (!) can be calculated analytically, allowing for explicit computation of physical phenomena. This allows analytic expressions for scattering lifetimes and excitation transport. Mapping of the diffusion problem onto the secular equation for scalar lattice vibrations allows the extraction of the density of vibrational states [5]. It is given ˜ by the simple relationship D(!) ! (d s 1) displaying the utility of the spectral dimension d˜s . A similar though slightly more complicated mapping can be applied to diluted Heisenberg antiferromagnets. The silica aerogels [13,16,42,44,45], and Mnx Zn1x F2 [43] and RbMnx Mg1x F3 [18,19] are realizations of such systems, respectively, and have been extensively investigated experimentally. The random nature of fractal structures introduces statistical complexity into relaxation processes. The direct (single vibration) [38] and Raman (scattering of two vibrations) [24,39] spin-lattice relaxation processes are discussed for random fractal networks in Sect. “Relaxation Dynamics on Fractal Structures”. The theoretical results are complex, related to the “Devil’s staircase” for Raman relaxation processes. Experiments probing these rather bizarre behaviors would be welcome. Given the “super-localization” of excitations on fractal networks [1,42], transport properties need to be ex-
pressed according to the same concepts as introduced by Mott [22,23] for the insulating phase of doped semiconductors—the so-called “variable range-rate hopping”. This is developed in Sect. “Transport on Fractal Structures”, and applied to the thermal transport of fractal structures [2,20,26]. The “lessons learned” from these applications suggest an interpretation of the thermal transport of glassy materials above the so-called “plateau” temperature range. This instance of taking the insights from fractal structures, and applying it to materials that are certainly not fractal yet possess localized vibrational states, is an example of the utility of the analysis of fractal dynamics. Finally, some suggestions, hinted at above, for future experimental investigations are briefly discussed in Sect. “Future Directions”. Fractal and Spectral Dimensions Fractals can be defined by the nature of the symmetry they exhibit: self-similar geometry or equivalently dilation symmetry [7,8,9,10,15,40,41,46]. This symmetry is most assuredly not translational, leading to a very different set of behaviors for physical realizations that shall be explored in subsequent Sections. Stated most simply, self-similar structures “look the same” on any length scale. Conversely, one cannot extract a length scale from observation of a self-similar structure. There are clearly limits to self-similarity for physical realizations. For example, at the atomic level, dilation symmetry must come to an end. And for finite structures, there will be a largest length scale. There are other length scales that bracket the self-similar regime. Thus, for percolating networks, an example of a random fractal [38,39], the percolation correlation length, (p), where p is the concentration of occupied sites or bonds, defines an upper limit to the regime of fractal geometry. In general, for percolating networks, the range of length scales ` over which the structure is fractal is sandwiched between the atomic length a, and the percolation correlation length (p) : a < ` < (p). Fractal structures can be deterministic or random. The former is a result of applying a structural rule indefinitely, replacing an element of a design by the design itself. Examples are the Mandelbrot–Given fractal and the Sierpinski gasket. The latter arises from a set of probabilistic rules. A simple example is the percolating network where a site or bond (e. g. in d D 2, a square grid) is occupied randomly with probability p. The resulting clusters contain a random number of sites or bonds, where the distribution function for the number of finite clusters of size s is denoted by ns (p). There exists a critical value p D pc where a connected cluster extends over all space (the “in-
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finite” cluster). For p < pc only finite clusters exist, with the largest cluster spanned by the length scale (p). (p) diverges at p D pc as jp pc j . For p > pc finite clusters and the infinite cluster coexist, the number of finite clusters vanishing at p approaches unity. The probability that a site belongs to the infinite cluster varies with occupation probability p as (p pc )ˇ . The characteristic length (p) for p > pc is a measure of the largest finite cluster size. Hence, for length scales ` > (p) the systems looks homogeneous. Thus, the fractal regime is sandwiched between the lattice constant a and the percolation correlation length (p). The remarkable utility of the percolating network becomes clear from these properties: there is a crossover length scale between fractal and homogeneous structural behavior that can be chosen at will by choosing a value for the site or bond occupation probability p > pc . Excitations of structures that map onto percolating networks can change their nature from fractal to continuous as a function of their length scale `, the transition occurring when `c (p). This property makes the percolating network the “fruit fly” of random structures. The flexibility of adjusting the crossover length `c enables a fit between those properties calculated for a percolating network with the properties of those physical systems that map onto percolating networks. The fractal dimension, Df , is defined from the dependence of the “mass” on length scale. For fractals, the number of points or bonds on the infinite cluster within a length scale ` (hence, the “mass”), for ` (p), depends upon ` as M(`) ` D f . From above it is straight forward to show Df D d (ˇ/). For a percolation network [29] in d D 2; Df D 91/48. For d D 3; Df D 2:48 ˙ 0:09. In mean field, d D 6; Df D 4. The spectral dimension, d˜s , follows from the analysis of diffusion on a fractal network, as first postulated by de Gennes [14]: “An ant parachutes down onto an occupied site of the infinite cluster of a percolating network. At every time unit, the ant makes one attempt to jump to one of its adjacent sites. If that site is occupied, it moves there. If it is empty, the ant stays at its original site. What is the ensemble-averaged square distance that the ant travels in time t?” Gefen et al. [17] found that hr2 (t)i t 2C where depends upon the scaling exponent (conductivity exponent), the probability that a site belongs to the infinite cluster scaling exponent ˇ, and the scaling exponent for (p) : D ( ˇ)/. Alexander and Orbach [5] defined the spectral (or fracton) dimension as d˜s D 2Df /(2 C ). The importance of d˜s can be seen from the calculation of the probability of finding a diffusing particle at the starting point at time t, P0 (t), which for compact diffusion is the inverse of the number of visited sites in
˜
time t, V(t) t (d s /2) . For scalar elasticity, the vibrational density of states ˜
D(!) D (2!/)Im P˜0 (! 2 C i0C ) ! (d s 1) ; where P˜0 (!) is the Laplace transform of P0 (t). Noting that for homogeneous systems, D(!) ! (d1) , the name spectral dimension for d˜s becomes evident. Alexander and Orbach [5] named the vibrational excitations of fractal structures, fractons, in analogy with the vibrational excitations of homogeneous structures, phonons. Hence, they termed d˜s the fracton dimension. Alexander and Orbach noted that for percolation structures in the mean-field limit, d D 6, d˜s D 4/3 precisely. At that time (1982), values for d˜s for d < 6 appeared close to that value, and they conjectured that d˜s D 4/3 for 2 d 6. We now know that this is only approximate (but remarkably close [24]): d˜s D 1:325 ˙ 0:002 in d D 2, and 1:317 ˙ 0:003 in d D 3. Were the conjecture exact, a numerical relationship between structure and transport would be exact [ D (3Df /2) 2], dictated solely by fractal geometry. For percolating networks, the structure appears fractal for length scales ` (p) and homogeneous for ` (p). The vibrational excitations in the fractal regime are termed fractons, and in the homogeneous regime phonons. We define the crossover frequency !c as that for which the excitation length scale equals (p). Then !c (p ˜ pc )( D f /d s ) . Continuity leads to a phonon velocity v(p) (p pc ) [(Df /d˜s ) 1]. This then leads to the dispersion relations: [` (p); ! !c ; phonon regime]! v(p)k ; and ˜
[` (p); ! !c ; fracton regime]! k (D f /d s ) : In the latter case, as we shall show in the next Section, vibrational excitations in the fractal regime are localized, so that k should not be thought of as a wave vector but rather the inverse of the localization length (!). Nature of Dynamics on Fractals – Localization The conductance of a d dimensional percolating network of size L; G(L), can be shown to be proportional to Lˇ where the exponent ˇ D (Df /d˜s )(d˜s 2). In the Anderson sense [6], localization occurs when ˇ 0 (marginal at ˇ D 0). The Alexander–Orbach conjecture [5], d˜s 4/3, leads to ˇ well less than zero for all embedding dimensions d. One can think of this form of localization as geometrical, as opposed to the scattering localization that traditionally is associated with Anderson localization. The lo-
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calization is strong, the wave function falling off faster than simple exponential. The wave function has the form [1]: h (r)i exp[ f[r/(!)]d g]. The exponent d is a geometrical exponent describing the fact that an exponential decay along the fractal will be distorted when viewed in real space. In general, 1 d dmin , where dmin is defined by ` R d min , with ` being the shortest path along the network between two points separated by a Pythagorean distance R. Bunde and Roman [11] have found that for percolating networks d D 1. The decay length (!) is the localization length ˜ from Sect. “Introduction”, (!) ! (d s /D f ) . For a random system such as a percolating network, a given realization of (r) is badly behaved, depending upon the particular choice of origin taken for the position coordinate r. The ensemble average takes the average over all realizations of (r), and is denoted by h (r)i. This has its dangers, as the calculation of physical properties should be performed using a specific realization for (r), and then ensemble averaged. Calculation of physical properties utilizing an ensemble averaged wave function h (r)i will not yield the same result, and could be very misleading. Specific realizations for (r) for two-dimensional bond percolation vibrational networks have been calculated by Nakayama and Yakubo [25,28,47]. The fracton “core” (or largest amplitude) possesses very clear boundaries for the edges of the excitation, with an almost steplike character and a long tail. The tail extends over large distances and the amplitude oscillates in sign. This is required for orthogonality to the uniform translational mode with ! D 0. Clearly these modes do not look like simple exponentials, hence the warning about the use of h (r)i as compared to (r) for the calculation of matrix elements. Mapping of Physical Systems onto Fractal Structures The foregoing description of dynamics of fractal networks would be interesting in and of itself, but its applicability to physical systems makes it of practical importance. Further, the nature of localization, and computation of “hopping-type” transport on fractal structures, has application to materials that are not fractal. Thus, it is worthwhile to consider what systems are fractal, the nature of dynamics on such systems, and the lessons we can learn from them for structures that possess similar characteristics but are not fractal. Specifically, Mott’s [22,23] variable range-rate hopping for localized electronic impurity states has direct application to transport of vibrational energy via fracton hopping on fractal networks [2], and we shall argue to
thermal transport in glassy materials above the “plateau” temperature [20,26]. The fundamental equations for fractal dynamics spring from the equation for diffusion of a random walker P on a network: dPi /dt D j¤I w i j (P j Pi ) where Pi is the probability that the ith site is occupied, wij is the probability per unit time for hopping from site i to site j, and w i j D w ji . Introducing new quantities Wi j D w i j for P i ¤ j, and Wi i D j¤I w i j , one can rewrite the difP fusion equation as dPi /dt D j Wi j P j with the defined P relation j Wi j D 0. For scalar elasticity, the equation of motion for atomic P vibrations is d2 u i /dt 2 D j K i j u j , with ui the atomic displacement at position i, and K i j (i ¤ j) is given by K i j D k i j /m i with kij the force constant connecting two sites i and j, and mi the mass of the ith site. The diagonal element W ii is obtained from the balancing of forces at P the ith site: j K i j D 0. The only difference between these two equations is the order of the time derivatives, leading to the equivalence of the eigenvalue problems. We have used this equivalence to extract the density of vibrational states for scalar elasticity, and the definition of the spectral (or fracton) dimension in Sect. “Introduction”. The most direct physical example of a vibrating fractal network is the silica aerogel [13,16,42,44]. Different preparation conditions can change the density of these materials to very small values, indicative of the open porous structure of these materials. Two length scales are present: the fundamental building block of the structure (the silica), and the correlation length of the gel. In between, the clusters possess a fractal structure, while at length scales beyond the correlation length the gel behaves as a homogeneous porous glass. This crossover from fractal to homogeneous is the physical example of our previous discussion for percolating networks when the length scale of excitations passes through the percolation correlation length (p). The thermal, transport, and scattering properties of aerogels have been well studied, and are the classic example of fractal behavior. There are other examples of structures that map onto the eigenvalue spectrum of the diffusion equation. Ferromagnetic and antiferromagnetic materials can be diluted by non-magnetic impurities. This leads to random site dilution, mapping directly onto the site-diluted percolating network for magnetic interactions. This can be seen from the Heisenberg Hamiltonian for spin systems: P H D 1/2 i; j J i j S i S j , where S i is the vector spin at the ith site, and J ij the isotropic exchange interaction between sites i and j. The equation of motion for the spin operx x ator S C i D S i C iS i becomes for diluted ferromagnets
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P z C z C i„@S C j¤i J i j (S i S j S j S i ). For low levels of i /@t D spin wave excitations, S zi S zj S, so that one obP tains the linear equation of motion i„@S C j¤I J i j i /@t D C C (S j S i ). This is the same form as that for diffusion of a random walker on a network, so that, in analogy with the scalar vibrational network, the eigenvalue spectrum is the same. This mapping allows the spin wave spectrum to exhibit magnons and fracton waves obeying the same properties that phonons and fractons exhibit for scalar vibrations. For antiferromagnets, the change in sign of S zi for the two sublattices complicates the linearized equations of P C C motion. They become, i„@S C j¤I J i j (S j CS i ), i /@t D i where i D 1 for down spins and i D 1 for up spins. This alters the dynamical equations for diluted antiferromagnets, and requires a separate calculation of the eigenvalue spectrum. Neutron diffraction experiments exhibit both conventional magnon and fracton excitations. In the case of the site-diluted antiferromagnet Mnx Zn1x F2 both excitations are observed simultaneously for length scales in the vicinity of (p) [43]. Relaxation Dynamics on Fractal Structures Electron spin resonance and non-radiative decay experiments can be performed in fractal materials. It is of interest to compare these dynamical properties with those found in translationally ordered structures. At first sight, for vibrational relaxation, it might be thought that the only difference would originate with the differences in the densities of vibrational states D(!). However, the localized nature of the vibrational states (fractons) and their random positions in the fractal network, introduces properties that are specific to fractal lattices. There are two cases of interest: one-fracton relaxation, and two-fracton inelastic scattering. The former is referred to as the “direct” relaxation process; the latter the “Raman” relaxation process. In both cases, the main effect of vibrational localization is on the relaxation time profile. Different spatial sites can have very different relaxation rates because of the randomness of their distance from the appropriate fracton sites. The result is a strongly non-exponential time decay. For one-fracton relaxation [1], a straightforward generalization of the direct relaxation process rate, W(!0 ; L), for relaxation energy ! 0 arising from interaction with a fracton of the same energy centered a distance L away, is proportional to 2q1
W(!0 ; L) !0
f[(!0 )](D f ) g
[coth(ˇ!0 /2)](1/ıL ) expf[L/(!0 )]d g :
Here, ˇ D 1/kB T and the factor ı L represents the energy width of the fracton state. In the homogeneous limit, where phonons represent the extended vibrational states, an energy conserving delta function would replace ı L . There are two limits for ı L , case (a) when the fracton relaxation rate ı caused by anharmonicity dominates; and case (b) when the electronic relaxation rate caused by the electron-fracton interaction is greater than ı. In the former case, ıL D ı; while in the latter, ıL D W(!0 ; L) itself, leading to a self-consistent determination of the relaxation rate. The latter case requires explicit consideration of the L dependence of ı L , further complicating the calculation of W(!0 ; L). The calculation of the time profile of the electronic state population and the average relaxation rate are complex, and the reader is referred to [32] for details. For case (a), the population of the initial electronic state is found to vary as h i [(D f /d )1] g : (ln t)[(D f d )/2d ] t fc 1 (ln t) Here, c1 is a constant independent of time, but dependent upon ! 0 and ı. The population of the initial electronic state decays is thus found to be faster than a power law, but slower than exponential or stretched exponential. For case (b), the population of the initial electronic state is found to vary as n o (1/t) (ln t)[(D f /d )1] : The decay is slower than in case (a), and is closer to a power law. The average relaxation rates for cases (a) and (b) have the same functional form: hWi D(!0 )[(!0 )(2q1) ] coth(ˇ!0 /2) where q D d˜s (d /Df ). The temperature dependence is the usual one found for the direct process, but the energy dependence differs, depending on the values of the dimensions and parameters for fractal networks. Two fracton relaxation is considerably more complex [3,4]. In summary, the time profile of the initial electronic state in the long time regime begins as a stretched exponential, then crosses over into a form similar to case (a) for the one fracton decay process. In the presence of rapid electronic cross relaxation, the time profile is exponential, with a low-temperature relaxation time h1/T1 i ˜ proportional to T f2d s [1C2(d /D f )]1g for Kramers transitions (between half-integer time-reversed spin states), and ˜ T f2d s [1C2(d /D f )]3g for non-Kramers transitions (between integer time-reversed spin states). The complexity of random systems adds an interesting consideration: the nature of the relaxation rate at a specific site as compared to the average relaxation rate calculated above. The decay profile of single-site spin-lattice
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relaxation is always exponential. However, the temperature dependence does not simply follow that of the average relaxation rate. Instead, it exhibits irregular statistical fluctuations reflecting the environment of the chosen site. As the temperature is raised, new relaxation channels, involving higher-frequency localized vibrations, will become activated, adding to the relaxation processes responsible for the relaxation at low temperatures. The temperature dependence of the relaxation rate W(T) at different sites will fluctuate with strong correlations in the temperature dependence between them, underlying large accumulative fluctuations. Using a step function for the vibrational Bose functions, the relaxation rate has a T 2 temperature dependence between randomly spaced steps that occur when new relaxation channels are opened: a “devil’s staircase” structure. The full Bose function smoothes out these steps, but the qualitative features remain. All of these features point to the complexity and richness of electronic relaxation associated with localized vibrational spectra. Their remarkable complexity is worthy of investigation. Transport on Fractal Structures The localization of excitations discussed in Sect. “Fractal and Spectral Dimensions” has profound influence on the transport properties of and on fractal structures. For example, in the case of the silica aerogels, the “conventional” form for glasses or amorphous materials for the thermal conductivity (T) is found [12]: a rapid rise at low temperatures leveling off to a “plateau” as a function of temperature T, and then a rise in (T) roughly as T 2 . At first sight, this is to be “expected” as the aerogels are random, and one could make an analogy with glassy systems. However, there is a fundamental inconsistency in such an analysis. The fracton excitations are localized, and therefore cannot contribute to transport. Thus, one would expect instead a rapid rise in (T) at low temperatures where the long length scale excitations are phonon-like (and hence delocalized), flattening off to a constant value when all the extended state density of states have reached their duLong– Petit value. The specific heat for these states is then a constant, and (T) would be a constant independent of temperature T. So far, this behavior is consistent with experiment. But what happens for temperatures T greater than the plateau temperature region? How can (T) increase when the only excitations are fractons and therefore localized? The solution to this conundrum lies in the anharmonic forces connecting the silica clusters in the aerogel network. The
presence of the anharmonicity allows for a process of fracton hopping [2,20,26], in precisely the same fashion as in Mott’s [22,23] variable range-rate hopping for localized electronic impurity states in semiconductors. The fracton can absorb/emit a lower frequency extended state phonon, and hop to another fracton location, thereby transmitting excitation energy spatially. This possibility is amusing, for it means that for fractal networks the anharmonicity contributes to thermal transport, whereas in translationally invariant systems the anharmonicity inhibits thermal transport. In fact, were there no anharmonicity in a fractal network, the thermal conductivity (T) would stay at the plateau value for all T greater than the temperature T p , the onset of the plateau, for all T > Tp . In this sense, anharmonicity stands on its head. It is essential for transport in the fracton regime, defined by T Tp „!c /kB . The localized vibrational excitation hopping contribution to the thermal conductivity is hop (T) D (kB /2V)˙0 R2 (!0 )/sl (!0 ; T). Here sl means strongly localized modes, R(!0 ) is the hopping distance associated with the sl mode, 0 ; sl (!0 ; T) the hopping lifetime of the sl mode 0 caused by anharmonic interactions at temperature T, and V the volume of the system. We have already derived the ensemble averaged wave function for fractons, h (r)i, in Sect. “Fractal and Spectral Dimensions”. We adopt the sl notation [26] because the hopping transport arguments apply to strongly localized states, independent of whether or not they are fractons. The dynamical process for hot (T) is an sl mode at one site coupling with a low energy extended state to hop to an sl mode at another site, thereby transferring excitation energy spatially. The evaluation of sl (!0 ; T) will depend upon the hopping distance for transport of the sl modes, R(!0 ). The argument of Mott [22,23], derived by him for localized electronic states, can be taken over for sl vibrational modes [2]. The volume that contains at least one sl mode is given by (4/3)D(!0 )!sl [R(!0 )]3 D 1, where D(!0 ) is the density of sl states per unit energy. This condition assures that, for an sl mode at the origin, a second sl mode can be found within a hopping distance R(!0 ). We find R(!0 ) (!0 /!sl )(!0 ). Thus, the most probable hopping distance is at least the sl localization length. The hopping rate 1/sl (!0 ; T) caused by anharmonicity, with anharmonic coupling constant Ceff can be found from the Golden Rule, leading to the hopping contribution to the thermal conductivity, with ( M )3 the volume for finding a single sl mode, hop (T) D [43 3 (Ceff )2 (kB )2 T]/[ 3 (vs )5 3 ( M )3 h(! )i2 ]. Here h(! )i is the average sl length scale, vs the velocity of sound, and the density. There are very few undetermined parameters in this expression for hop (T). The anharmonic coupling
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constant can be found from the shift of the “boson peak” in the Raman spectra with pressure [48,49], typically a factor of 25 times larger than the long length scale third-order elastic coupling constant for extended modes. Making use of experiments on amorphous a-GeS2 one finds [26,27] hop (T) 0:0065T (W/mK) using the appropriate values for and vs , and M h(! )i D 15 Å. This contribution is comparable to that observed for other networkforming glasses above the plateau value. There is a quantum phenomenon associated with sl mode hopping. The hopping vertex is associated with a low energy extended mode coupling with an sl mode to hop to another (spatially separated) sl mode. As the temperature increases, the lifetime of the sl mode will become shorter than 1/! of the low energy extended mode. This quantum effect leads to a breakdown of the Golden Rule expression, causing a leveling off of the linear-in-T thermal conductivity above the plateau value. The sl mode hopping contribution to (T) above the plateau temperature for glasses and amorphous materials appears to provide a quantitative explanation of the observed phenomena. It is a specific example of how dynamics on a fractal network can be used to determine the properties of materials that, while certainly not fractal, have properties that map onto the behavior of fractal structures. Future Directions As stated in the Introduction, studying the dynamics on fractal networks is a convenient structure for analyzing the properties of localized states, with all of the frequency and temperature-dependent physical properties determined without arbitrary parameters. For example, the frequency dependence of the localization length scale, (!), is known precisely on fractal networks, and is used for precise calculation of the hopping transport in Sect. “Relaxation Dynamics on Fractal Structures”, without adjustable parameters. If one extrapolates the properties of fractal networks to those random systems that are certainly not fractal, but which exhibit localization, the transport and relaxation properties can be understood. Examples are the thermal transport of glasses and amorphous materials above the plateau temperature, and the lifetime of high energy vibrational states in a-Si. Measurements of these properties may offer opportunities for practical devices in frequency regimes beyond extended state frequencies. For example, the lifetime of high energy vibrational states in a-Si has been shown to increase with increasing vibrational energies [36,37]. Such behavior is opposite to that expected from extended vibrational states interacting anharmoni-
cally. But it is entirely consistent if these states are localized and behave as shown for fractal networks [32,33]. The thesis presented here points to a much broader application of the consequences of fractal geometry than simply of those physical systems which exhibit such structures. The claim is made that using what has been learned from fractal dynamics can point to explication of physical phenomena in disordered systems that are certainly not fractal, but display properties analogous to fractal structures. In that sense, the future lies in application of the ideas contained in the study of fractal dynamics to that of random systems in general. This very broad class of materials may well have properties of great practical importance, predicted from the dynamics on fractals.
Bibliography Primary Literature 1. Alexander S, Entin-Wohlman O, Orbach R (1985) Relaxation and nonradiative decay in disordered systems, I. One-fracton emission. Phys Rev B 32:6447–6455 2. Alexander S, Entin-Wohlman O, Orbach R (1986) Phonon-Fracton anharmonic interaction: the thermal conductivity of amorphous materials. Phys Rev B 34:2726–2734 3. Alexander S, Entin-Wohlman O, Orbach R (1986) Relaxation and non-radiative decay in disordered systems, II. Two-fracton inelastic scattering. Phys Rev B 33:3935–3946 4. Alexander S, Entin-Wohlman O, Orbach R (1987) Relaxation and non-radiative decay in disordered systems, III. Statistical character of Raman (two-quanta) spin-lattice relaxation. Phys Rev B 35:1166–1173 5. Alexander S, Orbach R (1982) Density of states on fractals, “fractons”. J Phys Lett (Paris) 43:L625–L631 6. Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109:1492–1505 7. Avnir D (ed) (1989) The Fractal Approach to Heterogeneous Chemistry. Wiley, Chichester 8. Barabaási A-L, Stanley HE (1995) Fractal Concepts in Crystal Growth. Cambridge University Press, Cambridge 9. Bunde A, Havlin S (eds) (1991) Fractals and Disordered Systems. Springer, Berlin 10. Bunde A, Havlin S (eds) (1994) Fractals in Science. Springer-Verlag, Berlin 11. Bunde A, Roman HE (1992) Vibrations and random walks on random fractals: anomalous behavior and multifractality. Philos Magazine B 65:191–211 12. Calemczuk R, de Goer AM, Salce B, Maynard R, Zarembowitch A (1987) Low-temperature properties of silica aerogels. Europhys Lett 3:1205–1211 13. Courtens E, Pelous J, Phalippou J, Vacher R, Woignier T (1987) Brillouin-scattering measurements of phonon-fracton crossover in silica aerogels. Phys Rev Lett 58:128–131 14. de Gennes PG (1976) La percolation: un concept unificateur. Recherche 7:919–927 15. Feder J (1988) Fractals. Plenum Press, New York
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16. Freltoft T, Kjems J, Richter D (1987) Density of states in fractal silica smoke-particle aggregates. Phys Rev Lett 59:1212–1215 17. Gefen Y, Aharony A, Alexander S (1983) Anomalous diffusion on percolating clusters. Phys Rev Lett 50:77–80 18. Ikeda H, Fernandez-Baca JA, Nicklow RM, Takahashi M, Iwasa I (1994) Fracton excitations in a diluted Heisenberg antiferromagnet near the percolation threshold: RbMn0.39 Mg0.61 F3 . J Phys: Condens Matter 6:10543–10549 19. Ikeda H, Itoh S, Adams MA, Fernandez-Baca JA (1998) Crossover from Homogeneous to Fractal Excitations in the Near-Percolating Heisenberg Antiferromagnet RbMn0.39 Mg0.61 F3 . J Phys Soc Japan 67:3376–3379 20. Jagannathan A, Orbach R, Entin-Wohlman O (1989) Thermal conductivity of amorphous materials above the plateau. Phys Rev B 39:13465–13477 21. Kistler SS (1932) Coherent Expanded–Aerogels. J Phys Chem 36:52–64 22. Mott NF (1967) Electrons in disordered structures. Adv Phys 16:49–144 23. Mott NF (1969) Conduction in non-crystalline materials III. Localized states in a pseudogap and near extremities of conduction and valence bands. Philos Mag 19:835–852 24. Nakayama T (1992) Dynamics of random fractals: large-scale simulations. Physica A 191:386–393 25. Nakayama T (1995) Elastic vibrations of fractal networks. Japan J Appl Phys 34:2519–2524 26. Nakayama T, Orbach R (1999) Anharmonicity and thermal transport in network glasses. Europhys Lett 47:468–473 27. Nakayama T, Orbach RL (1999) On the increase of thermal conductivity in glasses above the plateau region. Physica B 263– 264:261–263 28. Nakayama T, Yakubo K (2001) The forced oscillator method: eigenvalue analysis and computing linear response functions. Phys Rep 349:239–299 29. Nakayama T, Yakubo K (2003) Fractal Concepts in Condensed Matter Physics. Solid-State Sciences. Springer, Berlin 30. Nakayama T, Yakubo K, Orbach R (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 31. Orbach R (1986) Dynamics of Fractal Networks. Science 231:814–819 32. Orbach R (1996) Transport and vibrational lifetimes in amorphous structures. Physica B 219–220:231–234 33. Orbach R, Jagannathan A (1994) High energy vibrational lifetimes in a-Si:H. J Phys Chem 98:7411–7413 34. Rammal R (1984) Spectrum of harmonic excitations on fractals. J Phys (Paris) 45:191–206 35. Rammal R, Toulouse G (1983) Random walks on fractal structures and percolation clusters. J Phys Lett (Paris) 44:L13–L22
36. Scholten AJ, Dijkhuis JI (1996) Decay of high-frequency phonons in amorphous silicon. Phys Rev B 53:3837–3840 37. Scholten AJ, Verleg PAWE, Dijkjuis JI, Akimov AV, Meltzer RS, Orbach R (1995) The lifetimes of high-frequency phonons in amorphous silicon: evidence for phonon localization. Solid State Phenom 44–46:289–296 38. Stanley HE (1977) Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents. J Phys A 10:L211–L220 39. Stauffer D, Aharony A (1992) Introduction to Percolation Theory, 2nd edn. Taylor and Francis, London/Philadelphia 40. Stauffer D, Stanley HE (1995) From Newton to Mandelbrot: A primer in Theoretical Physics, 2nd edn. Springer, Berlin 41. Takayasu H (1990) Fractals in the Physical Sciences. Manchester University Press, Manchester 42. Tsujimi Y, Courtens E, Pelous J, Vacher R (1988) Raman-scattering measurements of acoustic superlocalization in silica aerogels. Phys Rev Lett 60:2757–2760 43. Uemura YJ, Birgeneau RJ (1987) Magnons and fractons in the diluted antiferromagnet Mnx Zn1x F2 . Phys Rev B 36:7024– 7035 44. Vacher R, Courtens E, Coddens G, Pelous J, Woignier T (1989) Neutron-spectroscopy measurement of a fracton density of states. Phys Rev B 39:7384–7387 45. Vacher R, Woignier T, Pelous J, Courtens E (1988) Structure and self-similarity of silica aerogels. Phys Rev B 37:6500–6503 46. Vicsek T (1993) Fractal Growth Phenomena. World Scientific, Singapore 47. Yakubo K, Nakayama T (1989) Direct observation of localized fractons excited on percolating nets. J Phys Soc Japan 58:1504–1507 48. Yamaguchi M, Nakayama T, Yagi T (1999) Effects of high pressure on the Bose peak in a-GeS2 studied by light scattering. Physica B 263–264:258–260 49. Yamaguchi M, Yagi T (1999) Anharmonicity of low-frequency vibrations in a-GeS2 studied by light scattering. Europhys Lett 47:462–467
Recommended Reading While many of the original articles are somewhat difficult to access, the reader is nevertheless recommended to read them as there are subtle issues that tend to disappear as references are quoted and re-quoted over time. For a very helpful introduction to percolation theory, Ref. 39 is highly recommended. A thorough review of dynamics of/on fractal structures can be found in Ref. 30. Finally, a comprehensive treatment of localization and multifractals, and their relevance to dynamics of/on fractal networks, can be found in Ref. 29.
Entropy in Ergodic Theory
Entropy in Ergodic Theory JONATHAN L. F. KING University of Florida, Gainesville, USA Article Outline Glossary Definition of the Subject Entropy example: How many questions? Distribution Entropy A Gander at Shannon’s Noisy Channel Theorem The Information Function Entropy of a Process Entropy of a Transformation Determinism and Zero-Entropy The Pinsker–Field and K-Automorphisms Ornstein Theory Topological Entropy Three Recent Results Exodos Bibliography Glossary Some of the following definitions refer to the “Notation” paragraph immediately below. Use mpt for ‘measure-preserving transformation’. Measure space A measure space (X; X ; ) is a set X, a field (that is, a -algebra) X of subsets of X, and a countably-additive measure : X ! [0; 1]. (We often just write (X; ), with the field implicit.) For a collection C X , use Fld(C) C for the smallest field including C. The number (B) is the “-mass of B”. Measure-preserving map A measure-preserving map : (X; X ; ) ! (Y; Y; ) is a map : X ! Y such that the inverse image of each B 2 Y is in X , and ( 1 (B)) D (B). A (measure-preserving) transformation is a measure-preserving map T : (X; X ; ) ! (X; X ; ). Condense this notation to (T : X; X ; ) or (T : X; ). Probability space A probability space is a measure space (X; ) with (X) D 1; this is a probability measure. All our maps/transformations in this article are on probability spaces. Factor map A factor map : (T : X; X ; ) ! (S : Y; Y; ) is a measure-preserving map : X ! Y which intertwines the transformations, ıT D S ı . And is an
isomorphism if – after deleting a nullset (a mass-zero set) in each space – this is a bijection and 1 is also a factor map. Almost everywhere (a.e.) A measure-theoretic statement holds almost everywhere, abbreviated a.e., if it holds off of a nullset. (Eugene Gutkin once remarked to me that the problem with Measure Theory is . . . that you have to say “almost everywhere”, almost everywhere.) a:e: For example, B A means that (B X A) is zero. The a.e. will usually be implicit. Probability vector A probability vector vE D (v1 ; v2 ; : : : ) is a list of non-negative reals whose sum is 1. We generally assume that probability vectors and partitions (see below) have finitely many components. Write “countable probability vector/partition”, when finitely or denumerably many components are considered. Partition A partition P D (A1 ; A2 ; : : : ) splits X into pairwise disjoint subsets A i 2 X so that the disjoint union F # i A i is all of X. Each Ai is an atom of P. Use jPj or P for the number of atoms. When P partitions a probability space, then it yields a probability vector vE, where v j :D (A j ). Lastly, use Phxi to denote the P-atom that owns x. Fonts We use the font H ; E ; I for distribution-entropy, entropy and the information function. In contrast, the script font ABC : : : will be used for collections of sets; usually subfields of X . Use E() for the (conditional) expectation operator. Notation Z D integers, ZC D positive integers, and N D natural numbers D 0; 1; 2; : : :. (Some wellmeaning folk use N for ZC , saying ‘Nothing could be more natural than the positive integers’. And this is why 0 2 N. Use de and bc for the ceiling and floor functions; bc is also called the “greatest-integer function”. For an interval J :D [a; b) [1; C1], let [a : : : b) denote the interval of integers J \ Z (with a similar convention for closed and open intervals). E. g., (e : : : ] D (e : : : ) D f3g. For subsets A and B of the same space, ˝, use A B for inclusion and A ¦ B for proper inclusion. The difference set B X A is f! 2 B j ! … Ag. Employ Ac for the complement ˝ X A. Since we work in a probability space, if we let x :D (A), then a convenient convention is to have xc
denote 1 x ;
since then (Ac ) equals xc . Use A4B for the symmetric difference [AXB][[BXA]. For a collection C D fE j g j of sets in ˝, let the disjoint F F S union j E j or (C) represent the union j E j and also assert that the sets are pairwise disjoint.
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Use “8large n” to mean: “9n0 such that 8n > n0 ”. To refer to left hand side of an Eq. (20), use LhS(20); do analogously for RhS(20), the right hand side.
Definition of the Subject The word ‘entropy’ (originally German, Entropie) was coined by Rudolf Julius Emanuel Clausius circa 1865 [2,3], taken from the Greek o ˛, ‘a turning towards’. This article thus begins (Prolegomenon, “introduction”) and ends (Exodos1 , “the path out”) in Greek. Clausius, in his coinage, was referring to the thermodynamic notion in physics. Our focus in this article, however, will be the concept in measurable and topological dynamics. (Entropy in differentiable dynamics2 would require an article by itself.) Shannon’s 1948 paper [6] on Information Theory, then Kolmogorov’s [4] and Sinai’s [7] generalization to dynamical systems, will be our starting point. Our discussion will be of the one-dimensional case, where the acting-group is Z. “My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. John von Neumann had a better idea, he told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function goes by that name in statistical mechanics. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.’ ” (Shannon as quoted in [59]) Entropy example: How many questions? Imagine a dartboard, Fig. 1, split in five regions A; : : : ; E with known probabilities. Blindfolded, you throw a dart 1 This is
the Greek spelling. 2 For instance, see [15,18,24,25].
Entropy in Ergodic Theory, Figure 1 This dartboard is a probability space with a 5-set partition. The atoms have probabilities 12 ; 18 ; 18 ; 18 ; 18 . This probability distribution will be used later in Meshalkin’s example
at the board. What is the expected number V of Yes/No questions needed to ascertain the region in which the dart landed? Solve this by always dividing the remaining probability in half. ‘Is it A?’ – if Yes, then V D 1. Else: ‘Is it B or C?’ – if Yes, then ‘Is it B?’ – if No, then the dart landed in C, and V D 3 was the number of questions. Evidently V D 3 also for regions B; D; E. Using “log” to denote base-2 logarithm3 , the expected number of questions4 is thus 1 1 1 1 1 1C 3C 3C 3C 3 2 8 8 8 8 4 X 1 note D p j log D 2: pj
E(V ) D
(1)
jD0
Letting vE :D 12 ; 18 ; 18 ; 18 ; 18 be the probability vector, we can write this expectation as X (x) : E(V ) D x2E v
Here, : [0; 1] ! [0; 1) is the important function5 (x) :D x log(1/x) ;
so extending by continuity gives
(0) D 0 : (2) An interpretation of “(x)” is the number of questions needed to winnow down to an event of probability x. Distribution Entropy Given a probability vector vE, define its distribution entropy as X H (E v ) :D (x) : (3) x2E v
This article will use the term distropy for ‘distribution entropy’, reserving the word entropy for the corresponding 3 In this paper, unmarked logs will be to base-2. In entropy theory, it does not matter much what base is used, but base-2 is convenient for computing entropy for messages described in bits. When using the natural logarithm, some people refer to the unit of information as a nat. In this paper, I have picked bits rather than nats. 4 This is holds when each probability p is a reciprocal power of two. For general probabilities, the “expected number of questions” interpretation holds in a weaker sense: Throw N darts independently at N copies of the dartboard. Efficiently ask Yes/No questions to determine where all N darts landed. Dividing by N, then sending N ! 1, will be the p log 1p sum of Eq. (1). 5 There does not seem to be a standard name for this function. I use , since an uppercase looks like an H, which is the letter that Shannon used to denote what I am calling distribution-entropy.
Entropy in Ergodic Theory
dynamical concept, when there is a notion of time involved. Getting ahead of ourselves, the entropy of a stationary process is the asymptotic average value that its distropy decays to, as we look at larger and larger finite portions of the process. An equi-probable vector vE :D K1 ; : K: :; K1 evidently has H (E v ) D log(K). On a probability space, the “distropy of partition P”, written H (P) or H (A1 ; A2 ; : : :) shall mean the distropy of probability vector j 7! (A j ). A (finite) partition necessarily has finite distropy. A countable partition can have finite distropy, e. g. H (1/2; 1/4; 1/8; : : : ) D 2. One could also have infinite distropy: Consider a piece B X of mass 1/2 N . Splitting B into 2 k many equal-mass atoms gives an -sum of 2 k (k C N)/ (2 k 2 N ). Setting k D k N :D 2 N N makes this -sum F1 equal 1; so splitting the pieces of X D ND1 B N , with 1 (B N ) D 2N , yields an 1-distropy partition.
Entropy in Ergodic Theory, Figure 2 Using natural log, here are the graphs of: (x) in solid red, H (x; x c ) in dashed green, 1 x in dotted blue. Both (x) and H (x; x c ) are strictly convex-down. The 1 x line is tangent to (x) at x D 1
Function The (x) D x log(1/x) function6 has vertical tangent at x D 0, maximum at 1/e and, when graphed in nats slope 1 at x D 1. Consider partitions P and Q on the same space (X; ). Their join, written P _ Q, has atoms A \ B, for each pair A 2 P and B 2 Q. They are independent, written P?Q if (A\ B) D (A)(B) for each A; B pair. We write P < Q, and say that “P refines Q”, if each P-atom is a subset of some Q-atom. Consequently, each Q-atom is a union of Patoms. Recall, for ı a real number, our convention that ı c means 1 ı, in analogy with (B c ) equaling 1 (B) on a probability space. Distropy Fact
Entropy in Ergodic Theory, Figure 3 Using natural log: The graph of H (x1 ; x2 ; x3 ) in barycentric coordinates; a slice has been removed, between z D 0:745 and z D 0:821. The three arches are copies of the distropy curve from Fig. 2.
For partitions P; Q; R on probability space (X; ): (a) H (P) log(# P), with equality IFF P is an equi-mass partition. (b) H (Q _ R) H (Q) C H (R), with equality IFF Q?R. (c) For ı 2 0; 12 , the function ı 7! H (ı; ı c ) is strictly increasing. a:e: (d) R 4 P implies H (R) H (P), with equality IFF R D P. 6 Curiosity:
Just in this paragraph we compute distropy in nats, that is, using natural logarithm. Given a small probability p 2 [0; 1] and setting x :D 1/p, note that (p) D log(x)/x 1/(x), where (x) denotes the number of prime numbers less-equal x. (This approximation is a weak form of the Prime Number Theorem.) Is there any actual connection between the ‘approximate distropy’ function P H (E p) :D p2Ep 1/(1/p) and Number Theory, other than a coincidence of growth rate?
Proof Use the strict concavity of (), together with Jensen’s inequality. Remark 1 Although we will not discuss it in this paper, most distropy statements remain true with ‘partition’ replaced by ‘countable partition of finite distropy’. Binomial Coefficients The dartboard gave an example where distropy arises in a natural way. Here is a second example. For a small ı > 0, one might guess that the binomial coefficient ınn grows asymptotically (as n ! 1) like 2"n , for some small ". But what is the correct relation between " and ı?
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Well, Stirling’s formula n! [n/e]n gives nn
n! c [ın]![ı c n]! [ın][ı n] [ı c n][ı n] 1 (recall ı c D 1 ı) : D ın c ıc n [ı [ı ] ] Thus n1 log ınn H (ı; ı c ). But by means of the above distropy inequalities, we get an inequality true for all n, not just asymptotically. Lemma 2 (Binomial Lemma) Fix a ı 2 0; 12 and let H :D H (ı; ı c ). Then for each n 2 ZC : ! X n 2Hn : (4) j j2[0:::ı n]
Proof Let X f0; 1gn be the set of xE with #
Noisy Channel Shannon’s theorem says that a noisy channel has a channel capacity. Transmitting above this speed, there is a minimum error-rate (depending how much “above”) that no error-correcting code can fix. Conversely, one can transmit below – but arbitrarily close to – the channel capacity, and encode the data so as to make the error-rate less than any given ". We use Corollary 3 to show the existence of such codes, in the simplest case where the noise7 is a binary independent-process (a “Bernoulli” process, in the language later in this article). We have a channel which can pass one bit per second. Alas, there is a fixed noise-probability 2 [0; 12 ) so that a bit in the channel is perturbed into the other value. Each perturbation is independent of all others. Let H :D H (; c ). The value [1 H] bits-per-second is the channel capacity of this noise-afflicted channel.
fi 2 [1 : : : n] j x i D 1g ın :
On X, let P1 ; P2 ; : : : be the coordinate partitions; e. g. P7 D (A7 ; Ac7 ), where A7 :D fE x j x7 D 1g. Weighting each point 1 , the uniform distribution on X, gives that (A7 ) by jXj ı. So H (P7 ) H, by (c) in Sect. “Distropy Fact”. Finally, the join P1 _ _ Pn separates the points of X. So log(# X) D H (P1 _ _ Pn ) H (P1 ) C : : : C H (P n ) Hn ; making use of (a),(b) in “Distropy Fact”. And # X equals LhS (4). A Gander at Shannon’s Noisy Channel Theorem We can restate the Binomial lemma using the Hamming metric on f0; 1gn , Dist(E x ; yE) :D # fi 2 [1 : : : n] j x i ¤ y i g :
Encoding/Decoding Encode using an “k; n–blockcode”; an injective map F : f0; 1g k ! f0; 1gn . The source text is split into consecutive k-bit blocks. A block xE 2 f0; 1g k is encoded to F(E x ) 2 f0; 1gn and then sent through the channel, where it comes out perturbed to ˛E 2 f0; 1gn . The transmission rate is thus k/n bits per second. For this example, we fix a radius r > 0 to determine the decoding map, Dr : f0; 1gn ! fOopsg t f0; 1g k : We set Dr (E ˛) to zE if there is a unique zE with F(Ez) 2 Bal(E ˛; r); else, set Dr (E ˛) :D Oops. One can think of the noise as a f0; 1g-independentprocess, with Prob(1) D , which is added mod-2 to the signal-process. Suppose we can arrange that the set fF(E x )) j xE 2 f0; 1g k g of codewords, is a strongly r-separated-set. Then
Use Bal(E x ; r) for the open radius-r ball centered at xE, and Bal(E x ; r) :D f yE j Dist (E x ; yE) rg for the closed ball. The above lemma can be interpreted as saying that x ; ın)j 2H (ı;ı jBal(E
c )n
for each xE 2 f0; 1gn : (5) Corollary 3 Fix n 2 ZC and ı 2 0; 12 , and let ;
H :D H (ı; ı c ) : Then there is a set C f0; 1gn , with # C 2[1H]n , that is strongly ın-separated. I. e., Dist (E x ; yE) > ın for each distinct pair xE; yE 2 C.
The probability that a block is mis-decoded is the probability, flipping a -coin n times that we get more than r many Heads:
(6) 1 Theorem 4 (Shannon) Fix a noise-probability 2 0; 2 and let H :D H (; c ). Consider a rate R < [1 H] and an " > 0. Then 8large n there exists a k and a code F : f0; 1g k ! f0; 1gn so that: The F-code transmits bits at faster than R bits-per-second, and with error-rate < ". 7 The noise-process is assumed to be independent of the signalprocess. In contrast, when the perturbation is highly dependent on the signal, then it is sometimes called distortion.
Entropy in Ergodic Theory
Proof Let H0 :D H (ı; ı c ), where ı > was chosen so close to that ı
R :
(7)
The information function has been defined so that its expectation is the distropy of P, E(IP ) D
where
k :D b[1 H0 ]nc :
(8)
By Corollary 3, there is a strongly ın-separated-set C
0 f0; 1gn with # C 2[1H ]n . So C is big enough to permit an injection F : f0; 1g k ! C. Courtesy Eq. (6), the probability of a decoding error is that of getting more than ın many Heads in flipping a -coin n times. Since ı > , the Weak Law of Large Numbers guarantees – once n is large enough – that this probability is less than the given ". The Information Function Agree to use P D (A1 ; : : :), Q D (B1 ; : : :), R D (C1 ; : : :) for partitions, and F ; G for fields. With C a (finite or infinite) family of subfields of X , W their join G2C G is the smallest field F such that G F , for each G 2 C. A partition Q can be interpreted also as a field; namely, the field of unions of its atoms. A join of denumerably many partitions will be interpreted as a field, but a join of finitely many, P1 _ _ PN , will be viewed as a partition or as a field, depending on context. Conditioning a partition P on a positive-mass set B, let PjB be the probability vector A 7! (A\B) (B) . Its distropy is X 1 (A \ B) H (PjB) D log : (A \ B)/(B) (B) A2P So conditioning P on a partition Q gives conditional distropy X H (PjQ) D H (PjB)(B) B2Q
D
X A2P;B2Q
log
1 (A\B) (B)
Conditioning on a Field For a subfield F X , recall that each function g 2 L1 () has a conditional expectation E(gjF ) 2 L1 (). It is the unique F -measurable function with Z
8B 2 F :
E(gjF )d D B
Z gd : B
Returning to distropy ideas, use (AjF ) for the conditional probability function; it is the conditional expectation E(1A jF ). So the conditional information function is IPjF (x) :D
X A2P
log
1 1A (x) : (AjF (x))
(11)
Its integral H (PjF ) :D
Z
IPjF d ;
is the conditional distropy of P on F . When F is the field of unions of atoms from some partition Q, then the number H (PjF ) equals the H (PjQ) from Eq. (9). Write G j % F to indicate fields G1 G2 : : : S1 that are nested, and that Fld 1 G j D F , a.e. The Martingale Convergence Theorem (p. 103 in [20]) gives (c) below. Conditional-Distropy Fact Consider partitions P; Q; R and fields F and G j . Then a:e:
! (A \ B) :
(9)
A “dartboard” interpretation of H (PjQ) is The expected number of questions to ascertain the P-atom that a random dart x 2 X fell in, given that we are told its Q-atom. For a set A X, use 1A : X ! f0; 1g for its indicator function; 1A (x) D 1 IFF x 2 A. The information function of partition P, a map IP : X ! [0; 1), is X 1 IP :D log (10) 1A () : (A) A2P
IP ()d D H (P) : X
Pick a large n for which k >R; n
Z
(a) 0 H (PjF ) H (P), with equality IFF P F , respectively, P?F . (b) H (Q _ RjF ) H (QjF ) C H (RjF ). (c) Suppose G j % F . Then H (PjG j ) & H (PjF ). (d) H (Q _ R) D H (QjR) C H (R). (d’) H (Q _ R1 jR0 ) D H (QjR1 _ R0 ) C H (R1 jR0 ). Imagining our dartboard (Fig. 1) divided by superimposed partitions Q and R, equality (d) can interpreted as saying: ‘You can efficiently discover where the dart landed in both partitions, by first asking efficient questions about R, then – based on where it landed in R – asking intelligent questions about Q.’
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Entropy in Ergodic Theory
Entropy of a Process Consider a transformation (T : X; ) and partition P D (A1 ; A2 ; : : :). Each “time” n determines a partition Pn :D T n P, whose jth-atom is T n (A j ). The process T; P refers W to how T acts on the subfield 1 0 P n X . (An alternative view of a process is as a stationary sequence V0 ; V1 ; : : : of random variables Vn : X ! ZC , where Vn (x) :D j because x is in the jth-atom of Pn .) Write E (T; P) or E T (P) for the “entropy” of the T; P process”. It is the limit of the conditional-distropy-numbers c n :D H (P0 jP1 _ P2 _ _ Pn1 ) : This limit exists since H (P) D c1 c2 0. Define the average-distropy-number n1 h n , where h n :D H (P0 _ P1 _ _ Pn1 ) : Certainly h n D c n C H (P1 _ _ Pn1 ) D c n C h n1 , since T is measure preserving. Induction gives h n D Pn 1 jD1 c j . So the Cesàro averages n h n converge to the entropy. Theorem 5 The entropy of process (T; P : X; X ; ) equals
T(x) :D [n 7! xnC1 ] : The time-zero partition P separates points, under the action of the shift. This L-atom time-zero partition has Phxi D Phyi IFF x0 D y0 . So no matter what shift-invariant measure is put on X, the time-zero partition will generate under the action of T. Time Reversibility A transformation need not be isomorphic to its inverse. Nonetheless, the average-distropy-numbers show that E(T 1 ; P) D E(T; P); although this is not obvious from the conditioning-definition of entropy. Alternatively, ! ˇ_ ˇ n H P0 ˇˇ Pj 1
D H (P0 _ _ Pn ) H (P1 _ _ Pn ) D H (Pn _ _ P0 ) H (Pn _ _ P1 ) ! ˇ_ ˇ n (12) D H P0 ˇˇ P j :
1 H (P0 _ _ Pn1 ) n!1 n ! ˇ_ ˇ n ˇ D lim H P0 ˇ P j n!1 1 ˇ1 ! ˇ_ D H P0 ˇˇ P j : lim
1
Bernoulli Processes
1
Both limits are non-increasing. The entropy E T (P) 0, W T with equality IFF P 1 1 P j . And E (P) H (P), with equality IFF T; P is an independent process. Generators We henceforth only discuss invertible mpts, that is, when T 1 is itself an mpt. Viewing the atoms of P as “letters”, then, each x 2 X has a T; P-name : : : x2 x1 x0 x1 x2 x3 : : : ; PhT n (x)i,
IFF P separates points. That is, after deleting a (T-invariant) nullset, distinct points of X have distinct T; P-names. A finite set [1 : : : L] of integers, interpreted as an alphabet, yields the shift space X :D [1 : : : L]Z of doubly-infinite sequences x D (: : : x1 x0 x1 : : : ). The shift T : X ! X acts on X by
T n (x).
where xn is the P-letter owning A partition P generates (the whole field) under (T : W n 8 X; ), if 1 1 T P D X . It turns out that P generates 8 I am now at liberty to reveal that our X has always been a Lebesgue space, that is, measure-isomorphic to an interval of R together with countably many point-atoms (points with positive mass).
A probability vector vE :D (v1 ; : : : ; v L ) can be viewed as a measure on alphabet [1 : : : L]. Let vE be the resulting product measure on X :D [1 : : : L]Z , with T the shift on X and P the time-zero partition. The independent process (T; P : X; vE ) is called, by ergodic theorists, a Bernoulli process. Not necessarily consistently, we tend to refer to the underlying transformation as a Bernoulli shift. The 12 ; 12 -Bernoulli and the 13 ; 13 ; 13 -Bernoulli have different process-entropies, but perhaps their underlying transformations are isomorphic? Prior to the Kolmogorov–Sinai definition of entropy9 of a transformation, this question remained unanswered. The equivalence of generating and separating is a technical theorem, due to Rokhlin. Assuming to be Lebesgue is not much of a limitation. For instance, if is a finite measure on any Polish space, then extends to a Lebesgue measure on the -completion of the Borel sets. To not mince words: All spaces are Lebesgue spaces unless you are actively looking for trouble. 9 This is sometimes called measure(-theoretic) entropy or (perhaps unfortunately) metric entropy, to distinguish it from topological
Entropy in Ergodic Theory
Entropy of a Transformation The Kolmogorov–Sinai definition of the entropy of an mpt is E(T) :D supfE T (Q) j Q a partition on Xg :
Certainly entropy is an isomorphism invariant – but is it useful? After all, the supremum of distropies of partitions is always infinite (on non-atomic spaces) and one might fear that the same holds for entropies. The key observation (restated in Lemma 8c and proved below) was this, from [4] and [7]. Theorem 6 (Kolmogorov–Sinai Theorem) If P generates under T, then E(T) D E(T; P). Thereupon the 12 ; 12 and 13 ; 13 ; 13 Bernoulli-shifts are not isomorphic, since their respective entropies are log(2) ¤ log(3). Wolfgang Krieger later proved a converse to the Kolmogorov–Sinai theorem. Theorem 7 (Krieger Generator Theorem, 1970) Suppose T ergodic. If E(T) < 1, then T has a generating partition. Indeed, letting K be the smallest integer K > E(T), there is a K-atom generator.10 Proof See Rudolph [21], or § 5.1 in Joinings in Ergodic Theory, where Krieger’s theorem is stated in terms of joinings. Entropy Is Continuous
Then Lemma 8b says that process-entropy varies continuously with varying the partition. Lemma 8 Fix a mpt (T : X; ). For partitions P; Q; Q0 , define R :D Q4Q0 and let ı :D H (R). Then (a) jH (Q) H (Q0 )j ı. (Distropy varies continuously with the partition.) (b) jE T (Q) E T (Q0 )j ı. (Process-entropy varies continuously with the partition.) (c) For all partitions Q Fld(T; P) : E T (Q) E T (P). Proof (of (a)) Evidently Q0 _ R D Q0 _ Q D Q _ R. So H (Q0 ) H (Q _ R) H (Q) C ı. Proof (of (b)) As above, H
N _
! Q0j
N _
H
1
! Qj C H
1
N _
! :
Rj
1
Sending N ! 1 gives E T (Q0 ) E T (Q) C E T (R). Finally, E T (R) H (R) and so E T (Q0 ) E T (Q) C ı. Proof (of (c)) Let K :D jQj. Then there is a sequence of WL P` . By above, K-set partitions Q(L) ! Q with Q(L) 4 L E T (Q(L) ) ! E T (Q), so showing that E
T
L _
!
?
P` E T (P)
L
will suffice. Note that 0
(B01 ; : : :), 0
Given ordered partitions Q D (B1 ; : : :) and Q D extend the shorter by null-atoms until jQj D jQ j. Let F Fat :D j [B j \ B0j ]; this set should have mass close to 1 if Q and Q0 are almost the same partition. Define a new partition Q4Q0 :D fFatg t fB i \ B0j j with i ¤ jg : (In other words, take Q _ Q0 and coalesce, into a single atom, all the B k \ B0k sets.) Topologize the space of parti tions by saying11 that Q(L) ! Q when H Q4Q(L) ! 0. entropy. Tools known prior to entropy, such as spectral properties, did not distinguish the two Bernoulli-shifts; see Spectral Theory of Dynamical Systems for the definitions. 10 It is an easier result, undoubtedly known much earlier, that every ergodic T has a countable generating partition – possibly of 1-distropy. 11 On the set of ordered K-set partitions (with K fixed) this conver gence is the same as: Q(L) ! Q when Fat Q(L) ; Q ! 1. An alternative approach is the Rokhlin metric, Dist(P; Q) :D H (PjQ) C H (QjP), which has the advantage of working for unordered partitions.
h N :D H
N1 _
L _
Tn
1 NH
W
N1 Pj 0
0 D H@
P`
L
nD0
So N1 h N N ! 1.
!!
N1CL _
1 PjA :
jDL
C
1 N 2LH (P).
Now send
Entropy Is Not Continuous The most common topology placed on the space ˝ of mpts is the coarse topology12 that Halmos discusses in his “little red book” [14]. The Rokhlin lemma (see p. 33 in [21]) implies that the isomorphism-class of each ergodic mpt is dense in ˝, (e. g., see p. 77 in [14]) disclosing that the S 7! E(S) map is exorbitantly discontinuous. 12 i. e, S ! T IFF 8A 2 X : (S 1 (A)4T 1 (A)) ! 0; this is n n a metric-topology, since our probability space is countably generated. This can be restated in terms of the unitary operator U T on L2 (), where U T ( f ) :D f ı T. Namely, S n ! T in the coarse topology IFF U S n ! U T in the strong operator topology.
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Entropy in Ergodic Theory
Indeed, the failure happens already for process-entropy with respect to a fixed partition. A Bernoulli process T; P has positive entropy. Take mpts S n ! T, each isomorphic to an irrational rotation. Then each E(S n ; P) is zero, as shown in the later section "Determinism and ZeroEntropy". Further Results When F is a T-invariant subfield, agree to use T F for “T restricted to F ”, which is a factor (see Glossary) of T. Transformations T and S are weakly isomorphic if each is isomorphic to a factor of the other. The foregoing entropy tools make short shrift of the following. Lemma 9 (Entropy Lemma) Consider T-invariant subfields G j and F . (a) Suppose G j % F . Then E T G j % E(T F ). In particular, G F implies that E(T G ) E(T F ), so entropy is an invariant of weak-isomorphism. P (b) E(T G1 _G2 _::: ) j E(T G j ). P And E(T; Q1 _ Q2 _ : : :) j E(T; Q j ). P (c) For mpts (S j : Yj ; j ): E(S1 S2 ) D j E(S j ). (d) E(T 1 ) D E(T). More generally, E(T n ) D jnj E(T). Meshalkin’s Map In the wake of Kolmogorov’s 1958 entropy paper, for two Bernoulli-shifts to be isomorphic one now knew that they had to have equal entropies. Meshalkin provided the first non-trivial example in 1959 [45]. Let S : Y ! Y be the Bernoulli-shift over the “letter” alphabet fE; D; P; Ng, with probability distribution 14 ; 14 ; 14 ; 14 . The letters E; D; P; N stand for Even, oDd, Positive, Negative, and will be used to describe the code (isomorphism) between the processes. Use T : X ! X for the Bernoulli-shift over “digit” alphabet 1; 1 1f0;1 C1; C2; 2g, with probability 1 1 1 distribu 1 1 1 tion ; ; ; ; . Both distributions 4 ; 4 ; 4 ; 4 and 1 1 12 18 18 8 8 2 ; 8 ; 8 ; 8 ; 8 have distropy log(4). After deleting invariant nullsets from X and Y, the construction will produce a measure-preserving isomorphism : X ! Y so that T ı D ı S. The Code
The leftmost 0 is linked to the rightmost +1, as indicated by the longest-overbar. The left/right-parentheses form a 12 ; 12 -random-walk. Since this random walk is recurrent, every position in x will be linked (except for a nullset of points x). Below each 0, write “P” or “N” as the 0 is linked to a positive or negative digit. And below the other digits, write “E” or “D” as the digit is even or odd. So the upper name in X is mapped to the lower name, a point y 2 Y. map : X ! Y carries 1 This the upstairs 1 1 1 1 1 1 1 1 distribution to ; ; ; ; ; ; ; 2 8 8 8 8 4 4 4 4 , downstairs. It takes some arguing to show that independence is preserved. The inverse map 1 views D and E as right-parentheses, and P and N as left. Above D, write the odd digit C1 or 1, as this D is linked to Positive or Negative. Markov Shifts A Bernoulli process T; P has independence P(1:::0] ?P1 whereas a Markov process is a bit less aloof: The infinite Past P(1:::0] doesn’t provide any more information about Tomorrow than Today did. That is, the conditional distribution P1 jP(1:::0] equals P1 jP0 . Equivalently, note H (P1 jP0 ) D H P1 jP(1:::0] D E(T; P) : (13) The simplest non-trivial Markov process (T; P : X; ) is over a two-letter alphabet fa; bg, and has transition graph Fig. 4, for some choice of transition probabilities s and c. The graph’s Markov matrix is
s c ; M D [m i; j ] i; j D 1 0 where c D 1 s, and m i; j denotes the probability of going from state i to state j. If Today’s distribution on the two states is the probability-vector vE :D [pa pb ], then Tomorrow’s is the product vEM. So a stationary process needs vEM D vE. This equa1 c tion has the unique solution pa D 1Cc and pb D 1Cc .
In X, consider this point x:
: : : 0 0 0 1 0 0 +1 +2 1 +1 0 : : : Regard each 0 as a left-parenthesis, and each non-zero as a right-parenthesis. Link them according to the legal way of matching parentheses, as shown in the top row, below: 0 0 0 1 0 0 +1 +2 1 +1 0 P N N D P P D
E
D
D ?
Entropy in Ergodic Theory, Figure 4 Call the transition probabilities s :D Prob(a!a) for stay, and c :D Prob a!b for change. These are non-negative reals, and sCcD1
Entropy in Ergodic Theory
An example of computing the probability of a word or cylinder set; (see Sect. “The Carathéodory Construction” in Measure Preserving Systems) in the process, is
approximately 2E(T;P)n ; this, after discarding small mass from the space. But the growth of n 7! 2n is sub-exponential and so, for our rotation, E(T; P) must be zero. Theorem 10 (Shannon–McMillan–Breiman Theorem (SMB-Theorem)) Set E :D E(T; P), where the tuple (T; P : X; ) is an ergodic process. Then the average information function
s (baaaba) D pb mba maa maa mab mba c D 1ssc1: 1Cc The subscript on s indicates the dependence on the transition probabilities; let’s also mark the mpt and call it T s . Using Eq. (13), the entropy of our Markov map is
1 n!1 IP (x) ! E ; n [0:::n)
for a.e.
x2X:
(15)
The functions f n :D IP[0:::n) converge to the constant function E both in the L1 -norm and in probability. 14
E(Ts ) D pa H (s; c) C pb H (1; 0)
„ ƒ‚ … D0
1 D [s log(s) C c log(c)] : 1Cc
(14)
Determinism and Zero-Entropy Irrational rotations have zero-entropy; let’s reveal this in two different ways. Equip X :D [0; 1) with “length” (Lebesgue) measure and wrap it into a circle. With “˚” denoting addition mod-1, have T : X ! X be the rotation T(x) :D x ˚ ˛, where the rotation number ˛ is irrational. Pick distinct points y0 ; z0 2 X, and let P be the partition whose two atoms are the intervals [y0 ; z0 ) and [z0 ; y0 ), wrapping around the circle. The T-orbit of each point x is dense13 in X. In particular, y0 has dense orbit, so P separates points – hence ?
generates – under T. Our goal, thus, is E(T; P) D 0.
Proof See the texts of Karl Petersen (p. 261 in [20]), or Dan Rudolph (p. 77 in [21]). Consequences Recall that P[0:::n) means P0 _ P1 _ _ Pn1 , where P j :D T j P. As usual, P[0:::n) hxi denotes the P[0:::n) -atom owning x. Having deleted a nullset, we can restate Eq. (15) to now say that 8"; 8x; 8large n: 1/2[EC"]n P[0:::n) hxi 1/2[E"]n :
(16)
This has the following consequence. Fixing a number ı > 0, we consider any set with (B) ı and count the number of n-names of points in B. The SMB-Thm implies
8"; 8large n; 8B ı : ˇ ˇ ˇfn-names in Bgˇ 2[E"]n :
(17)
Rotations are Deterministic
Rank-1 Has Zero-Entropy
The forward T-orbit of each point is dense. This is true for y0 , and so the backward T; P-name of each x actually W n tells us which point x is. I. e., P 1 1 T P, which is our definition of “process T; P is deterministic”. Our P being finite, this determinism implies that E(T; P) is zero, by Theorem 5.
There are several equivalent definitions for “rank-1 transformation”, several of which are discussed in the introduction of [28]. (See Chap. 6 in [13] as well as [51] and [27] for examples of stacking constructions.) A rank-1 transformation (T : X; ) admits a generating partition P and a sequence of Rokhlin stacks S n X, with heights going to 1, and with (S n ) ! 1. Moreover, each of these Rokhlin stacks is P-monochromatic, that is, each level of the stack lies entirely in some atom of P. Taking a stack of some height 2n, let B D B n be the union of the bottom n levels of the stack. There are at most n many length-n names starting in Bn , by monochromaticity. Finally, (B n ) is almost 12 , so is certainly larger than ı :D 13 . Thus Eq. (17) shows that our rank-1 T has zero entropy.
Counting Names in a Rotation The P0 _ _ Pn1 partition places n translates of points y0 and of z0 , cutting the circle into at most 2n intervals. Thus H (P0 _ : : : _ Pn1 ) log(2n). And n1 log(2n) ! 0. Alternatively, the below SMB-theorem implies, for an ergodic process T; P, that the number of length-n names is an " > 0 and an N > 1/". Points x; T(x); : : : ; T N (x) have some two at distance less than N1 ; say, Dist(T i (x); T j (x)) < ", for some 0 i < j N. Since T is an isometry, " > Dist(x; T k (x)) > 0, where k :D j i. So the T k -orbit of x is "-dense. 13 Fix
14 In engineering circles, this is called the Almost-everywhere equipartition theorem.
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Entropy in Ergodic Theory
Cautions on Determinism’s Relation to Zero-Entropy
K-Processes
A finite-valued process T; P has zero-entropy iff P
W1 1 P j . Iterating gives
Kolmogorov introduced the notion of a K-process or Kolmogorov process, in which the present becomes asymptotically independent of the distant past. The asymptotic past of the T; P process is called its tail field, where
1 _ 0
Pj
1 _
Pj ;
1
i. e., the future is measurable with respect to the past. This was the case with the rotation, where a point’s past uniquely identified the point, thus telling us its future. While determinism and zero-entropy mean the same thing for finite-valued processes, this fails catastrophically for real-valued (i. e., continuum-valued) processes, as shown by an example of the author’s. A stationary realvalued process V D : : : V1 V0 V1 V2 : : : is constructed in [40] which is simultaneously strongly deterministic: The two values V0 ; V1 determine all of V, future and past. and non-consecutively independent. This latter means that for each bi-infinite increasing integer sequence fn j g1 jD1 with no consecutive pair (always 1 C n j < n jC1 ), then the list of random variables : : : Vn 1 Vn 0 Vn 1 Vn 2 : : : is an independent process. Restricting the random variables to be countablyvalued, how much of the example survives? Joint work with Kalikow [39] produced a countably-valued stationary V which is non-consecutively independent as well as deterministic. (Strong determinism is ruled out, due to cardinality considerations.) A side-effect of the construction is that V’s time-reversal n 7! Vn is not deterministic. The Pinsker–Field and K-Automorphisms Consider the collection of zero-entropy sets, Z D Z T :D fA 2 X j E(T; (A; Ac )) D 0g :
(18)
Courtesy of Lemma 9b, Z is a T-invariant field, and 8Q Z : E(T; Q) D 0 :
(19)
The Pinsker field15 of T is this Z. It is maximal with respect to Eq. (19). Unsurprisingly, the Pinsker factor T Z has zero entropy, that is, E(T Z ) D 0. A transformation T is said to have completely-positive entropy if it has no (nontrivial) zero-entropy factors. That is, its Pinsker field Z T is the trivial field, ; :D f¿; Xg. 15 Traditionally, this called the Pinsker algebra where, in this context, “algebra” is understood to mean “ -algebra”.
Tail(T; P) :D
1 \
M _
Pj :
MD1 jD1
This T; P is a K-process if Tail(T; P) D ;. This turns out to be equivalent to what we might a call a strong form of “sensitive dependence on initial conditions”: For each fixed length L, the distant future _
Pj
j2(G:::GCL]
becomes more and more independent of _
Pj ;
j2(1:::0]
as the gap G ! 1. A transformation T is a Kolmogorov automorphism if it possesses a generating partition P for which T; P is a K-process. Here is a theorem that relates the “asymptotic forgetfulness” of Tail(T; P) D ;, to the lack of determinism implied by having no zero-entropy factors (See Walters, [23], p. 113. Related results appear in Berg [44]). Theorem 11 (Pinsker-Algebra Theorem) Suppose P is a generating partition for an ergodic T. Then Tail(T; P) equals Z T . Since Z T does not depend on P, this means that all generating partitions have the same tail field, and therefore K-ness of T can be detected from any generator. Another non-evident fact follows from the above. The future field of T; P is defined to be Tail (T 1 P). It is not obvious that if the present is independent of the distant past, then it is automatically independent of the distant future. (Indeed, the precise definitions are important; witness the Cautions on determinism section.) But since the entropy of a process, E(T; (A; Ac )), equals the entropy of the time-reversed process E(T 1 ; (A; Ac )), it follows that Z T equals Z T 1 . Ornstein Theory In 1970, Don Ornstein [46] solved the long-standing problem of showing that entropy was a complete isomorphism-
Entropy in Ergodic Theory
invariant of Bernoulli transformations; that is, that two independent processes with same entropy necessarily have the same underlying transformation. (Earlier, Sinai [52]) had shown that two such Bernoulli maps were weakly isomorphic, that is, each isomorphic to a factor of the other.) Ornstein introduced the notion of a process being finitely determined, see [46] for a definition, proved that a transformation T was Bernoulli IFF it had a finitely-determined generator IFF every partition was finitely-determined with respect to T, and showed that entropy completely classified the finitely-determined processes up to isomorphism. This seminal result led to a vast machinery for proving transformations to be Bernoulli, as well as classification and structure theorems [47,51,53]. Showing that the class of K-automorphisms far exceeds the Bernoulli maps, Ornstein and Shields produced in [48] an uncountable family of non-isomorphic K-automorphisms all with the same entropy.
And the topological entropy of T is Etop (T) :D supV Etop (T; V ) ;
(21)
taken over all open covers V. Thus Etop counts, in some sense, the growth rate in the number of T-orbits of length n. Evidently, topological entropy is an isomorphism invariant. Two continuous maps T : X ! X and S : Y ! Y are topologically conjugate (as isomorphism is called in this category) if there exists a homeomorphism : X ! Y with T D S . Lemma 12 (Subadditive Lemma) Consider a sequence s D (s l )1 1 [1; 1] satisfying s kCl s k C s l , for all k; l 2 Z. Then the following limit exists in [1; 1], and lim n!1 snn D inf n snn . Topological entropy, or “top ent” for short, satisfies many of the relations of measure-entropy. Lemma 13
Topological Entropy
(a) V 4 W implies
Adler, Konheim and McAndrew, in 1965, published the first definition of topological entropy in the eponymous article [33]. Here, T : X ! X is a continuous self-map of a compact topological space. The role of atoms is played by open sets. Instead of a finite partition, one uses a finite16 open-cover V D fU j gLjD1 , i. e. each patch U j is open, and S their union (V) D X. (Henceforth, ‘cover’ means “open cover”.) Let Card(V) be the minimum cardinality over all subcovers. [ ˚ (V0 ) D X ; Card(V) :D Min ] V0 j V0 V and and let H (V ) D Htop (V ) :D log(Card(V )) :
Analogous to the definitions for partitions, define V _ W :D fV \ W j V 2 V and W 2 W g ;
T V :D fT 1 (U) j U 2 Vg and V[0:::n) :D V0 _ V1 _ _ V n1 ; W < V ; if each W -patch is a subset of some V -patch:
The T; V-entropy is E T (V ) D E(T; V ) D Etop (T; V )
1 :D lim sup Htop V[0:::n) : n!1 n
(20)
16 Because we only work on a compact space, we can omit “finite”. Some generalizations of topological entropy to non-compact spaces require that only finite open-covers be used [37].
H (V ) H (W )
and E(T; V ) E(T; W ) :
(b) H (V _ W ) H (V) C H (W ). (c) H (T(V)) H (V), with equality if T is surjective. Also, E(T; V ) H (V). (d) In Eq. (20), the limn!1 n1 H V[0:::n) exists. (e) Suppose T is a homeomorphism. Then E(T 1 ; V ) D E(T; V ) ;
for each cover V. Consequently, Etop (T 1 ) D Etop (T). (f) Suppose C is a collection of covers such that: For each cover W , there exists a V 2 C with V < W . Then Etop (T) equals the supremum of Etop (T; V ), just taken over those V 2 C. (g) For all ` 2 N: Etop T ` D `Etop (T) : Proof (of (c)) Let C 4 V be a min-cardinality subcover. Then T C is a subcover of T V. So Card T V jT Cj D jCj. As for entropy, inequality (b) and the foregoing give H (V [0:::n) ) H (V )n. Proof (of (d)) Set s n :D H (V[0:::n) ). Then s kCl s k C H T k V[0:::l ) s k C s l ; by (b) and (c), and so the Subadditive Lemma 12, applies.
215
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Entropy in Ergodic Theory
Proof (of (g)) WLOG, ` D 3. Given V a cover, triple it to b V :D V \ T V \ T 2 V ; so _ j _ T3 b V D T i (V) : j2[0:::N)
i2[0:::3N)
Thus H T 3 ; b V ; N D H (T; V ; 3N), extending notation. Part (d) and sending N ! 1, gives E(T 3 ; b V) D 3H (T; V). Lastly, take covers such that
E T 3 ; C(k) ! Etop T 3
and
E T; D(k) ! Etop (T) ;
as k!1. Define V(k) :D C(k) _ D(k) . Apply the above to V (k) , then send k!1.
You Take the High Road and I’ll Take the Low Road There are several routes to computing top-ent, some via maximization, others, minimization. Our foregoing discussion computed Etop (T) by a family of sizes f k (n) D f kT (n), depending on a parameter k which specifies the fineness of scale. (In Sect. “Metric Preliminaries”, this k is an integer; in the original definition, an open cover.) Define two numbers: 1 b L f (k) :D lim sup log f k (n) and n!1 n 1 L f (k) :D lim inf log f k (n) : (22) n!1 n L f (k). If the limit exists Finally, let E f (T) :D sup k b f in Eq. (22) then agree to write L (k) for the common value. The A-K-M definition used the size fV (n) :D Card(V[0:::n) ), where Card(W ) :DMinimum cardinality of a subcover from W :
Using a Metric From now on, our space is a compact metric space (X; d). Dinaburg [36] and Bowen [34,35], gave alternative, equivalent, definitions of topological entropy, in the compact metric-space case, that are often easier to work with than covers. Bowen gave a definition also when X is not compact17 (see [35] and Chap. 7 in [23]). Metric Preliminaries An "-ball-cover comprises finitely many balls, all of radius ". Since our space is compact, every cover V has a Lebesgue number " > 0. I. e., for each z 2 X, the Bal(z; ") lies entirely inside at least one V-patch. (In particular, there is an "-ball-cover which refines V.) Let LEB(V) be the supremum of the Lebesgue numbers. Courtesy of Lemma 13f we can Fix a “universal” list V(1) 4 V(2) 4 : : : , with V(k) a 1k -ball-cover. For every T : X ! X, then, the lim k E(T; V(k) ) computes Etop (T). An "-Microscope Three notions are useful in examining a metric space (X; m) at scale ". Subset A X is an "separated-set, if m(z; z0 ) " for all distinct z; z0 2 A. Subset F X is "-spanning if 8x 2 X; 9z 2 F with m(x; z) < ". Lastly, a cover V is "-small if Diam(U) < ", for each U 2 V. 17 When X is not compact, the definitions need not coincide; e. g. [37]. And topologically-equivalent metrics, but which are not uniformly equivalent, may give the same T different entropies (see p. 171 in [23]).
Here are three metric-space sizes f" (n): Sep(n; ") :DMaximum cardinality of a dn –"-separated set: Spn(n; ") :DMinimum cardinality of a dn –"-spanning set: Cov(n; ") :DMinimum cardinality of a dn –"-small cover: These use a list (d n )1 nD1 of progressively finer metrics on X, where d N (x; y) :D Max j2[0:::N) d T j (x); T j (y) : Theorem 14 (All-Roads-Lead-to-Rome Theorem) Fix " and let W be any d-"-small cover. Then 8n : Cov(n; 2") Spn(n; ") Sep(n; ") Card(W [0:::n) ) : (ii) Take a cover V and a ı < LEB(V). Then 8n: Card(V[0:::n) ) Cov(n; ı) : (iii) The limit LCov (") D lim n n1 log(Cov(n; ")) exists in [0 : : : 1) : (iv)
(i)
ESep (T) D ESpn (T) D ECov (T) D ECard (T) by defn
D Etop (T) :
Proof (of (i)) Take F X, a min-cardinality dn -"-spanS ning set. So z2F Dz D X, where note
Dz :D dn -Bal(z; ") D
n1 \ jD0
T j Bal T j z; " :
Entropy in Ergodic Theory
This D :D fDz gz is a cover, and it is dn –2"–small. Thus Cov(n; 2") jDj D jFj. For any metric, a maximal "-separated-set is automatically "-spanning; adjoin a putative unspanned point to get a larger separated set. Let A be a max-cardinality dn -"-separated set. Take C, a min-cardinality subcover of W [0:::n) . For each z 2 A, pick a C-patch Cz 3 z. Could some pair x; y 2 A pick T j the same C? Well, write C D n1 jD0 T (Wj ), with each Wj 2 W . For every j 2 [0 : : : n), then, d(T j (x); T j (y)) Diam(Wj ) < " :
Proof (of (ii)) Choose a min-cardinality d n -ı-small cover C. For each C 2 C and j 2 [0 : : : n), the d-Diam(T j C) < ı. So there is a V-patch VC; j T j (C). Hence note
n1 \
T j VC; j C :
jD0
Thus V[0:::n) 4 C. So Card V[0:::n) Card(C) jCj D Cov(n; ı) :
Proof (of (iii)) To upper-bound Cov(k C l; ") let V and W be min-cardinality "-small covers, respectively, for metrics d k and d l . Then V \ T l (W ) is a "-small for d kCl . Consequently Cov(kCl; ") Cov(k; ") Cov(l; "). Thus n 7! log(Cov(n; ")) is subadditive. Proof (of (iv)) Pick a V from the list in Sect. “Metric Preliminaries”, choose some 2" < LEB(V) followed by an "-small W from Sect. “Metric Preliminaries”. Pushing n ! 1 gives LCard (V) LCov (2")
b LSep (") LSpn (") b LSpn (") LSep (")
1 log(Sep(n; ")) (24) n limit exists, in arguments that subsequently send "&0. Ditto for LSpn ("). LSep (") D lim
n!1
This will be used during the proof of the Variational Principle. But first, here are two entropy computations which illustrate the efficacy in having several characterizations of topological entropy. Etop (Isometry) D 0
Hence d n (x; y) < "; so x D y. Accordingly, the z 7! Cz map is injective, whence jAj jCj.
V [0:::n) 3
zero as "&0. Consequently, we can pretend that the
LCard (W ) : (23)
Now send V and W along the list in Sect. “Metric Preliminaries”. Pretension Topological entropy takes its values in [0; 1]. A useful corollary of Eq. (23) can be stated in terms of any Distance(; ) which topologizes [0; 1] as a compact interval. For each continuous T : X ! X on a compact metric-space, the Distance b LSep ("); LSep (") goes to
Suppose (T : X; d) is a distancepreserving map of a compact metric-space. Fixing ", a set is dn -"-separated IFF it is d-"-separated. Thus Sep(n; ") does not grow with n. So each b LSep (") is zero. Topological Markov Shifts Imagine ourselves back in the days when computer data is stored on large reels of fast-moving magnetic tape. One strategy to maximize the density of binary data stored is to not put timing-marks (which take up space) on the tape. This has the defect that when the tape-writer writes, say, 577 consecutive 1-bits, then the tape-reader may erroneously count 578 copies of 1. We sidestep this flaw by first encoding our data so : :1 word, then writing to tape. as to avoid the 11:577 Generalize this to a finite alphabet Q and a finite list F of disallowed Q-words. Extend each word to a common length K C 1; now F QKC1 . The resulting “K-step TMS (topological Markov shift)” is the shift on the set of doubly-1 Q-names having no substring in F . In the above magnetic-tape example, K D 576. Making it more realistic, suppose that some string of zeros, say 00:574 : :0, is also forbidden18 . Extending to length 577, we get 23 D8 new disallowed words of form 00:574 : :0b1 b2 b3 . We recode to a 1-step TMS (just called a TMS or a subshift of finite type) over the alphabet P :D QK . Each outlawed Q-word w0 w1 w K engenders a length-2 forbidden P-word (w0 ; : : : ; w K1 )(w1 ; : : : ; w K ). The resulting TMS is topologically conjugate to the original K-step. The allowed length-2 words can be viewed as the edges in a directed-graph and the set of points x 2 X is the set of doubly-1 paths through the graph. Once trivialities removed, this X is a Cantor set and the shift T : X ! X is a homeomorphism. The Golden Shift As the simplest example, suppose our magnetic-tape is constrained by the Markov graph, Fig. 5 that we studied measure-theoretically in Fig. 4. 18 Perhaps the ;-bad-length, 574, is shorter than the 1-bad-length because, say, ;s take less tape-space than 1s and so – being written more densely – cause ambiguity sooner.
217
218
Entropy in Ergodic Theory
Entropy in Ergodic Theory, Figure 5 Ignoring the labels on the edges, for the moment, the Golden shift, T, acts on the space of doubly-infinite paths through this graph. The space can be represented as a subset XGold fa; bgZ , namely, the set of sequences with no two consecutive b letters.
We want to store the text of The Declaration of Independence on our magnetic tape. Imagining that English is a stationary process, we’d like to encode English into this Golden TMS as efficiently as possible. We seek a shift-invariant measure on XGold of maximum entropy, should such exist. View PDfa; bg as the time-zero partition on X Gold ; that is, name xD: : : x1 x0 x1 x2 : : :, is in atom b IFF letter x0 is “b”. Any measure gives conditional probabilities (aja) D: s ; note
note
1 H (s; 1 s) 2s 1 slog(s) C (1 s)log(1 s) : D 2s
f (s) :D
(25)
Certainly f (0) D f (1) D 0, so f ’s maximum occurs at 0 s) the (it turns out) unique pointb ps where the derivative f (b equals zero. Thisb s D (1 C 5)/2. Plugging in, the maximum entropy supportable by the Golden Shift is p
p 1 C 5 p # 2 3 5 C log p : (26) 2 3 5 1 C 2
5
log
2
Exponentiating, the number of -typical n-names grows like Gn , where 2 G :D 4
2 1 C
p
p 1C p 5 5 5
5
1 ; 1 C jmj
(28)
0 . for the smallest jmj with x m ¤ x m
Lemma 15 Consider a subshift X. Then the 1 log(Names X (n)) n
exists in [0; 1], and equals Etop (X).
But recall, E(T) D H (P1 jP[1:::0) ) H (P1 jP0 ). So among all measures that make the conditional distribution Pja equal (s; c), the unique one maximizing entropy is the (s; c)-Markov-process. Its entropy, derived in Eq. (14), is
"
d(x; x 0 ) :D
n!1
(bjb) D 0 :
2 p MaxEnt D 5 5
Top-ent of the Golden Shift For a moment, let’s work more generally on an arbitrary subshift (a closed, shiftinvariant subset) X QZ , where Q is a finite alphabet. Here, the transformation is always the shift – but the space is varying – so agree to refer to the top-ent as Etop (X). Let ˚ Names X (n) be the number of distinct words in the set x [0:::n) j x 2 X . Note that a metric inducing the product-topology on QZ is
lim
(bja) D: c ;
(ajb) D 1 ;
This expression19 looks unpleasant to simplify – it isn’t even obviously an algebraic number – and yet topological entropy will reveal its familiar nature. This, because the Variational Principle (proved in the next section) says that the top-ent of a system is the supremum of measure-entropies supportable by the system.
32
p 3 3p5 2 5 5 54 5: p 3 5
(27)
Proof With " 2 (0; 1) fixed, two n-names are d n -"-separated IFF they are not the same name. Hence Sep(n; ") D Names X (n). To compute Etop (X Gold ), declare that a word is “golden” if it appears in some x 2 XGold . Each [n C 1]-golden word ending in a has form wa, where w is n-golden. An [n C 1]golden word ending in b, must end in ab and so has form wab, where w is [n 1]-golden. Summing up, NamesXGold (n C 1) D NamesXGold (n) C Names XGold (n 1) : This is the Fibonacci recurrence, and indeed, these are the Fibonacci numbers, since NamesXGold (0) D 1 and Names XGold (1) D 2. Consequently, we have that NamesXGold (n) Const n ; p
where D 1C2 5 is the Golden Ratio. So the sesquipedalian number G from Eq. (27) is simply , and Etop (X Gold ) D log(). Since log() 0:694, each thousand bits written on tape (subject to the “no bb substrings” constraint) can carry at most 694 bits of information. 19 A popular computer-algebra-system was not, at least under my inexpert tutelage, able to simplify this. However, once top-ent gave the correct answer, the software was able to detect the equality.
Entropy in Ergodic Theory
Top-ent of a General TMS A (finite) digraph G engenders a TMS T : XG ! XG , as well as a f0; 1g-valued adjacency matrix ADAG , where a i; j is the number of directed-edges from state i to j. (Here, each a i; j is 0 or 1.) The (i; j)-entry in power An is automatically the number of length-n paths from i to j. Employing the matrix-norm P kMk :D i; j jm i; j j, then, kAkn D NamesX (n) : Happily Gelfand’s formula (see 10.13 in [58] or Spectral_ radius in [60]) applies: For an arbitrary (square) complex matrix, 1
lim kAn k n D SpecRad(A) :
n!1
(29)
This right hand side, the spectral radius of A, means the maximum of the absolute values of A’s eigenvalues. So the top-ent of a TMS is thus the Etop (X G ) D SpecRad(AG )
˚ ˇ :D Max jej ˇ e is an eigenvalue of AG : (30)
The (a; b)-adjacency matrix of Fig. 5 is
1 1 ; 1 0 whose eigenvalues are and
1 .
Labeling Edges Interpret (s; c; 1) simply as edge-labels in Fig. 5. The set of doubly-1 paths can also be viewed as a subset YGold fs; c; 1gZ , and it too is a TMS. The shift on YGold is conjugate (topologically isomorphic) to the shift on XGold , so they a fortiori have the same topent, log(). The (s; c; 1)-adjacency matrix is 2 3 1 1 0 40 0 15 : 1 1 0 Its j j-largest eigenvalue is still , as it must. Now we make a new graph. We modify Fig. 5 by manufacturing a total of two s-edges, seven c-edges, and three edges 11 ; 12 ; 13 . Give these 2 C 7 C 3 edges twelve distinct labels. We could compute the resulting TMS-entropy from the corresponding 12 12 adjacency matrix. Alternatively, look at the (a; b)-adjacency matrix
2 7 A :D : 3 0 p The roots of its characteristic polynomial arep1 ˙ 22. Hence Etop of this 12-symbol TMS is log(1 C 22).
The Variational Principle Let M :D M (X; d) be the set of Borel probability measures, and M (T) :D M (T : X; d) the set of T-invariant 2 M . Assign EntSup(T) :D sup fE (T) j 2 M (T)g : Theorem 16 (Variational EntSup(T) D Etop (T).
principle
(Goodson))
This says that top-ent is the top entropy – if there is a measure which realizes the supremum. There doesn’t have to be. Choose a sequence of metric-systems (S k : Yk ; m k ) whose entropies strictly increase Etop (S k ) % L to some limit in (0; 1]. Let (S1 : Y1 ; m1 ) be the identity-map on a 1-point space. Define a new system (T : X; d), where F X :D k2[1:::1] Yk . Have T(x) :D S k (x), for the unique k with Yk 3 x. As for the metric, on Yk let d be a scaled version of m k , so that the d-Diam(Yk ) is less than 1/2 k . Finally, for points in distinct ˇcomponents, ˇ x 2 Yk and z 2 Y` , decree that d(x; z) :D ˇ2k 2` ˇ. Our T is continuous, and is a homeomorphism if each of the S k is. Certainly Etop (T) D L > Etop (S k ), for every k 2 [1 : : : 1]. If L is finite then there is no measure of maximal entropy; for must give mass to some Yk ; this pulls the entropy below L, since there are no compensatory components with entropy exceeding L. In contrast, when L D 1 then there is a maximalentropy measure (put mass 1/2 j on some component Yk j , where k j %1 swiftly); indeed, there are continuum-many maximal-entropy measures. But there is no 20 ergodic measure of maximal entropy. For a concrete L D 1 example, let S k be the shift on [1 : : : k]Z . Topology on M Let’s arrange our tools for establishing the Variational Principle. The argument will follow Misiurewicz’s proof, adapted from the presentations in [23] and [11]. Equip M with the weak- topology. 21 An A X is nice if its topological boundary @(A) is -null. And a partition is -nice if each atom is. 20 The ergodic measures are the extreme points of M(T); call them MErg (T). This M(T) is the set of barycenters obtained from Borel probability measures on MErg (T) (see Krein-Milman_ theorem, Choquet_theory in [60]). In this instance, what explains the failure to have an ergodic maximal-entropy measure? Let k be an invariant ergodic measure on Yk . These measures do converge to the one-point (ergodic) probability measure 1 on Y1 . But the map 7! E (T) is not continuousR at 1 . R 21 Measures ˛ ! IFF f d˛L ! f d, for each continuL ous f : X ! R. This metrizable topology makes M compact. Always, M(T) is a non-void compact subset (see Measure Preserving Systems.)
219
220
Entropy in Ergodic Theory
Proposition 17 If ˛L ! and A X is -nice, then ˛L (A) ! (A). Proof Define operator U (D) :D lim supL ˛L (D). It suffices to show that U (A) (A). For since Ac is -nice too, then U (Ac ) (Ac ). Thus limL ˛L (A) exists, and equals (A). Because C :D A is closed, the continuous functions f N & 1C pointwise, where f N (x) :D 1 Min(N d(x; C); 1) : By the Monotone Convergence theorem, then, Z N f N d ! (C) : And (C) D (A), since AR is nice. Fixing N, then, it suffices to establish U (A) f N d. But f N is continuous, so Z Z f N d D lim sup f N d˛L L!1 Z lim sup 1A d˛L D U (A) : L!1
Corollary 18 Suppose ˛L ! , and partition P is -nice. Then H˛L (P) ! H (P). The diameter of partition P is MaxA2P Diam(A). Proposition 19 Take 2 M and " > 0. Then there exists a -nice partition with Diam(P) < ". Proof Centered at an x, the uncountably many balls fBal(x; r) j r 2 (0; ")g have disjoint boundaries. So all but countably many are -nice; pick one and call it B x . Compactness gives a finite nice cover, say, fB1 ; : : : ; B7 g, at different centers. Then the partition P :D (A1 ; : : : ; A7 ) is S nice, 22 where A k :D B k X k1 jD1 B j . Here is a consequence of Jensen’s inequality. Lemma 20 (Distropy-Averaging Lemma) For ; 2 M , a partition R, and a number t 2 [0; 1], t H (R) C t c H (R) H tCt c (R) : Strategy for EntSup(T) Etop (T). Choose an " > 0. For L D 1; 2; 3; : : : , take a maximal (L; ")–separated–set FL X, then define 1 F D F" :D lim sup log(jFL j) : L!1 L 22 For any two sets B; B 0 X, the union @B [ @B 0 is a superset of the three boundaries @(B [ B0 ); @(B \ B0 ); @(B X B0 ).
Let 'L () be the equi-probable measure on FL ; each point has weight 1/jFL j. The desired invariant measure will come from the Cesàro averages 1 X T ` 'L ; ˛L :D L `2[0:::L)
which get more and more invariant. Lemma 21 Let be any weak- accumulation point of the above f˛L g1 1 . (Automatically, is T-invariant.) Then E (T) F. Indeed, if Q is any -nice partition with Diam(Q) < ", then E (T; Q) F. Tactics As usual, Q[0:::N) means Q0 _Q1 _: : :_Q N1 . Our goal is 8N :
?
F
1 H Q[0:::N) : N
(31)
Fix N and P :D Q[0:::N) , and a ı > 0. It suffices to verify: 8large L N, ? 1 1 log(jFL j) ı C H˛L (P) ; L N
(32)
since this and Corollary 17 will prove Eq. (31): Pushing L ! 1 along the sequence that produced essentially sends LhS(32) to F, courtesy Eq. (24). And RhS(32) goes to ı C N1 H (P), by Corollary 17, since P is -nice. Descending ı & 0, hands us the needed Eq. (31). Remark 22 The idea in the following proof is to mostly fill interval [0::L) with N-blocks, starting with a offset K 2 [0::N). Averaging over the offset will create a Cesàro average over each N-block. Averaging over the N-blocks will allow us to compute distropy with respect to the averaged measure, ˛L . Proof (of Eq. (32)) Since L is frozen, agree to use ' for the ' L probability measure. Our dL -"-separated set F L has at most one point in any given atom of Q[0:::L) , thereupon log(jFL j) D H' Q[0:::L) : Regardless of the “offset” K 2 [0 : : : N), we can always fit C :D b LN N c many N-blocks into [0 : : : L). Denote by G(K) :D [K : : : K C CN), this union of N-blocks, the good set of indices. Unsurprisingly, B(K) :D [0 : : : L) X G(K) is the bad index-set. Therefore, Bad(K)
‚ …„ _
H' Q[0:::L) H'
j2B(K)
ƒ
Good(K)
‚ …„ _
Q j C H'
ƒ Qj
:
j2G(K)
(33)
Entropy in Ergodic Theory
this last inequality, when n is large. The upshot: E (T) 2 C Etop (T). Applied to a power T ` , this asserts that E (T ` ) 2 C Etop (T ` ). Thus
Certainly Bad(K) 3Nlog(jQj). So 1 NL
X
Bad(K)
K2[0:::N)
3N log(jQj) : L
This is less than ı, since L is large. Applying to Eq. (28) now produces 1 1 X Good(K) : log(jFL j) ı C L NL
1 NL
P
K2[0:::N)
(34)
2 C Etop (T) ; ` using Lemma 9d and using Lemma 13g. Now coax ` ! 1. E (T)
K
Note _
j
T (Q) D
j2G(K)
Three Recent Results
_
:
c2[0:::C)T KCcN (P)
So Good(K)
X
Having given an survey of older results in measure-theoretic entropy and in topological entropy, let us end this survey with a brief discussion of a few recent results, chosen from many.
H' (T KCc N P) :
c
P
This latter, by definition, equals c H T KCcN(') (P). We conclude that 1 X 1 XX Good(K) H T KCcN ' (P) NL NL c K K 1 X H T ` ' (P) ; NL `2[0:::L)
by adjoining a few translates of P; 1 H˛L (P) ; N by the Distropy-averaging lemma 18; P since ˛L is the average L1 ` T ` '. Thus Eq. (34) implies Eq. (32), our goal. Proof (of EntSup(T) Etop (T)) Fix a T-invariant . For partition Q D (B1 ; : : : ; B K ), choose a compact set A k B k with (B k X A k ) small. (This can be done, since F is automatically a regular measure [58].) Letting c and P :D (D; A1 ; : : : ; A K ), we can have D :D i Ai made H (PjQ) as small as desired. Courtesy of Lemma 8b, then, we only need consider partitions of the form that P has. Open-cover V D (U1 ; : : : ; U K ) has patches U k :D D[ A k . What atoms of, say, P[0:::3) , can the intersection U9 \ T 1 (U2 ) \ T 2 (U5 ) touch? Only the eight atoms (D or A9 ) \ T 1 (D or A2 ) \ T 2 (D or A5 ) : Thus ] P[0:::n) 2n ] V[0:::n) . (Here, ] () counts the number of non-void atoms/patches.) So 1 1 H P[0:::n) 1 C log ] V [0:::n) n n 1 C 1 C Etop (T) ;
Ornstein–Weiss: Finitely-Observable Invariant In a landmark paper [10], Ornstein and Weiss show that all “finitely observable” properties of ergodic processes are secretly entropy; indeed, they are continuous functions of entropy. This was generalized by Gutman and Hochman [9]; some of the notation below is from their paper. Here is the setting. Consider an ergodic process, on a non-atomic space, taking on only finitely many values in N; let C be some family of such processes. An observation scheme is a metric space (˝; d) and a sequence of n functions S D (S n )1 1 , where Sn maps N : : : N into ˝. 1 On a point xE 2 N , the scheme converges if n 7! S n (x1 ; x2 ; : : : x n )
(35)
converges in ˝. And on a particular process X , say that S converges, if S converges on a.e. xE in X . A function J : C ! ˝ is isomorphism invariant if, whenever the underlying transformations of two processes X ; X 0 2 C are isomorphic, then J(X ) D J(X 0 ). Lastly, say that S “converges to J”, if for each X 2 C, scheme S converges to the value J(X ). The work of David Bailey [38], a student of Ornstein, produced an observation scheme for entropy. The Lempel-Ziv algorithm [43] was another entropy observer, with practical application. Ornstein and Weiss provided entropy schemes in [41] and [42]. Their recent paper “Entropy is the only finitelyobservable invariant” [10], gives a converse, a uniqueness result. Theorem 23 (Ornstein, Weiss) Suppose J is a finitely observable function, defined on all ergodic finite-valued processes. If J is an isomorphism invariant, then J is a continuous function of the entropy.
221
222
Entropy in Ergodic Theory
Gutman–Hochman: Finitely-Observable Extension Extending the Ornstein–Weiss result, Yonatan Gutman and Michael Hochman, in [9] proved that it holds even when the isomorphism invariant, J, is well-defined only on certain subclasses of the set of all ergodic processes. In particular they obtain the following result on three classes of zero-entropy transformations. Theorem 24 (Gutman, Hochman) Suppose J() is a finitely observable invariant on one of the following classes: (i)
The Kronecker systems; the class of systems with pure point spectrum. (ii) The zero-entropy mild mixing processes. (iii) The zero-entropy strong mixing processes. Then J() is constant.
Consider (G; G), a topological group and its Borel field (sigma-algebra). Let G X be the field on G X generated by the two coordinate-subfields. A map (36)
is measurable if 1
The paper introduces a new isomorphism invariant, the “f invariant”, and shows that, for Bernoulli actions, the f invariant agrees with entropy, that is, with the distropy of the independent generating partition. Exodos Ever since the pioneering work of Shannon, and of Kolmogorov and Sinai, entropy has been front and center as a major tool in Ergodic Theory. Simply mentioning all the substantial results in entropy theory would dwarf the length of this encyclopedia article many times over. And, as the above three results (cherry-picked out of many) show, Entropy shows no sign of fading away. . . Bibliography
Entropy of Actions of Free Groups
: GxX ! X
Theorem 25 (Lewis Bowen) Let G be a finite-rank free group. Then two Bernoulli G-actions are isomorphic IFF they have the same entropy.
(X ) G X :
Use g (x) for (g; x). This map in Eq. (36) is a (measure-preserving) group action if 8g; h 2 G: g ı h D g h , and each g : X ! X is measure preserving. This encyclopedia article has only discussed entropy for Z-actions, i. e., when G D Z. The ergodic theorem, our definition of entropy, and large parts of ergodic theory, involve taking averages (of some quantity of interest) over larger and larger “pieces of time”. In Z, we typically use the intervals I n :D [0 : : : n). When G is Z Z, we might average over squares I n I n . The amenable groups are those which possess, in a certain sense, larger and larger averaging sets. Parts of ergodic theory have been carried over to actions of amenable groups, e. g. [49] and [55]. Indeed, much of the Bernoulli theory was extended to certain amenable groups by Ornstein and Weiss, [50]. The stereotypical example of a non-amenable group, is a free group (on more than one generator). But recently, Lewis Bowen [8] succeeded in extending the definition of entropy to actions of finite-rank free groups.
Historical 1. Adler R, Weiss B (1967) Entropy, a complete metric invariant for automorphisms of the torus. Proc Natl Acad Sci USA 57:1573– 1576 2. Clausius R (1864) Abhandlungen ueber die mechanische Wärmetheorie, vol 1. Vieweg, Braunschweig 3. Clausius R (1867) Abhandlungen ueber die mechanische Wärmetheorie, vol 2. Vieweg, Braunschweig 4. Kolmogorov AN (1958) A new metric invariant of transitive automorphisms of Lebesgue spaces. Dokl Akad Nauk SSSR 119(5):861–864 5. McMillan B (1953) The basic theorems of information theory. Ann Math Stat 24:196–219 6. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423,623–656 7. Sinai Y (1959) On the Concept of Entropy of a Dynamical System, Dokl Akad Nauk SSSR 124:768–771
Recent Results 8. Bowen L (2008) A new measure-conjugacy invariant for actions of free groups. http://www.math.hawaii.edu/%7Elpbowen/ notes11.pdf 9. Gutman Y, Hochman M (2006) On processes which cannot be distinguished by finitary observation. http://arxiv.org/pdf/ math/0608310 10. Ornstein DS, Weiss B (2007) Entropy is the only finitely-observable invariant. J Mod Dyn 1:93–105; http://www.math.psu. edu/jmd
Ergodic Theory Books 11. Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge 12. Cornfeld I, Fomin S, Sinai Y (1982) Ergodic theory. Grundlehren der Mathematischen Wissenschaften, vol 245. Springer, New York
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13. Friedman NA (1970) Introduction to ergodic theory. Van Nostrand Reinhold, New York 14. Halmos PR (1956) Lectures on ergodic theory. The Mathematical Society of Japan, Tokyo 15. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. (With a supplementary chapter by Katok and Leonardo Mendoza). Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge 16. Keller G, Greven A, Warnecke G (eds) (2003) Entropy. Princeton Series in Applied Mathematics. Princeton University Press, Princeton 17. Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 18. Mañé R (1987) Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete, ser 3, vol 8. Springer, Berlin 19. Parry W (1969) Entropy and generators in ergodic theory. Benjamin, New York 20. Petersen K (1983) Ergodic theory. Cambridge University Press, Cambridge 21. Rudolph DJ (1990) Fundamentals of measurable dynamics. Clarendon Press, Oxford 22. Sinai Y (1994) Topics in ergodic theory. Princeton Mathematical Series, vol 44. Princeton University Press, Princeton 23. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, New York
Differentiable Entropy 24. Ledrappier F, Young L-S (1985) The metric entropy of diffeomorphisms. Ann Math 122:509–574 25. Pesin YB (1977) Characteristic Lyapunov exponents and smooth ergodic theory. Russ Math Surv 32:55–114 26. Young L-S (1982) Dimension, entropy and Lyapunov exponents. Ergod Theory Dyn Syst 2(1):109–124
Finite Rank 27. Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1):35– 65 28. King JLF (1988) Joining-rank and the structure of finite rank mixing transformations. J Anal Math 51:182–227
Maximal-Entropy Measures 29. Buzzi J, Ruette S (2006) Large entropy implies existence of a maximal entropy measure for interval maps. Discret Contin Dyn Syst 14(4):673–688 30. Denker M (1976) Measures with maximal entropy. In: Conze J-P, Keane MS (eds) Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974. Lecture Notes in Mathematics, vol 532. Springer, Berlin, pp 70–112 31. Misiurewicz M (1973) Diffeomorphism without any measure with maximal entropy. Bull Acad Polon Sci Sér Sci Math Astron Phys 21:903–910
Topological Entropy 32. Adler R, Marcus B (1979) Topological entropy and equivalence of dynamical systems. Mem Amer Math Soc 20(219)
33. Adler RL, Konheim AG, McAndrew MH (1965) Topological entropy. Trans Am Math Soc 114(2):309–319 34. Bowen R (1971) Entropy for group endomorphisms and homogeneous spaces. Trans Am Math Soc 153:401–414. Errata 181:509–510 (1973) 35. Bowen R (1973) Topological entropy for noncompact sets. Trans Am Math Soc 184:125–136 36. Dinaburg EI (1970) The relation between topological entropy and metric entropy. Sov Math Dokl 11:13–16 37. Hasselblatt B, Nitecki Z, Propp J (2005) Topological entropy for non-uniformly continuous maps. http://www.citebase.org/ abstract?id=oai:arXiv.org:math/0511495
Determinism and Zero-Entropy, and Entropy Observation 38. Bailey D (1976) Sequential schemes for classifying and predicting ergodic processes. Ph D Dissertation, Stanford University 39. Kalikow S, King JLF (1994) A countably-valued sleeping stockbroker process. J Theor Probab 7(4):703–708 40. King JLF (1992) Dilemma of the sleeping stockbroker. Am Math Monthly 99(4):335–338 41. Ornstein DS, Weiss B (1990) How sampling reveals a process. Ann Probab 18(3):905–930 42. Ornstein DS, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inf Theory 39(1):78–83 43. Ziv J, Lempel A (1977) A universal algorithm for sequential data compression. IEEE Trans Inf Theory 23(3):337–343
Bernoulli Transformations, K-Automorphisms, Amenable Groups 44. Berg KR (1975) Independence and additive entropy. Proc Am Math Soc 51(2):366–370; http://www.jstor.org/stable/2040323 45. Meshalkin LD (1959) A case of isomorphism of Bernoulli schemes. Dokl Akad Nauk SSSR 128:41–44 46. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 5:337–352 47. Ornstein DS (1974) Ergodic theory randomness and dynamical systems, Yale Math Monographs, vol 5. Yale University Press, New Haven 48. Ornstein DS, Shields P (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88 49. Ornstein DS, Weiss B (1983) The Shannon–McMillan–Briman theorem for a class of amenable groups. Isr J Math 44(3):53– 60 50. Ornstein DS, Weiss B (1987) Entropy and isomorphism theorems for actions of amenable groups. J Anal Math 48:1–141 51. Shields P (1973) The theory of Bernoulli shifts. University of Chicago Press, Chicago 52. Sinai YG (1962) A weak isomorphism of transformations having an invariant measure. Dokl Akad Nauk SSSR 147:797– 800 53. Thouvenot J-P (1975) Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli. Isr J Math 21:177– 207 54. Thouvenot J-P (1977) On the stability of the weak Pinsker property. Isr J Math 27:150–162
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Abramov Formula 55. Ward T, Zhang Q (1992) The Abramov–Rohlin entropy addition formula for amenable group actions. Monatshefte Math 114:317–329
Miscellaneous 56. Newhouse SE (1989) Continuity properties of entropy. Ann Math 129:215–235 57. http://www.isib.cnr.it/control/entropy/ 58. Rudin W (1973) Functional analysis. McGraw-Hill, New York 59. Tribus M, McIrvine EC (1971) Energy and information. Sci Am 224:178–184 60. Wikipedia, http://en.wikipedia.org/wiki/. pages: http://en.
wikipedia.org/wiki/Spectral_radius, wiki/Information_entropy
http://en.wikipedia.org/
Books and Reviews Boyle M, Downarowicz T (2004) The entropy theory of symbolic extensions. Invent Math 156(1):119–161 Downarowicz T, Serafin J (2003) Possible entropy functions. Isr J Math 135:221–250 Hassner M (1980) A non-probabilistic source and channel coding theory. Ph D Dissertation, UCLA Katok A, Sinai YG, Stepin AM (1977) Theory of dynamical systems and general transformation groups with an invariant measure. J Sov Math 7(6):974–1065
Ergodic Theorems
Ergodic Theorems ANDRÉS DEL JUNCO Department of Mathematics, University of Toronto, Toronto, Canada
Article Outline Glossary Definition of the Subject Introduction Ergodic Theorems for Measure-Preserving Maps Generalizations to Continuous Time and Higher–Dimensional Time Pointwise Ergodic Theorems for Operators Subadditive and Multiplicative Ergodic Theorems Entropy and the Shannon–McMillan–Breiman Theorem Amenable Groups Subsequence and Weighted Theorems Ergodic Theorems and Multiple Recurrence Rates of Convergence Ergodic Theorems for Non-amenable Groups Future Directions Bibliography
Glossary Dynamical system in its broadest sense, any set X, with a map T : X ! X. The classical example is: X is a set whose points are the states of some physical system and the state x is succeeded by the state Tx after one unit of time. Iteration repeated applications of the map T above to arrive at the state of the system after n units of time. Orbit of x the forward images x; Tx; T 2 X : : : of x 2 X under iteration of T. When T is invertible one may consider the forward, backward or two-sided orbit of x. Automorphism a dynamical system T : X ! X, where X is a measure space and T is an invertible map preserving measure. Ergodic average if f is a function on X let A n f (x) D P i n1 n1 iD0 f (T x); the average of the values of f over the first n points in the orbit of x. Ergodic theorem an assertion that ergodic averages converge in some sense. Mean ergodic theorem an assertion that ergodic averages converge with respect to some norm on a space of functions.
Pointwise ergodic theorem an assertion that ergodic averages A n f (x) converge for some or all x 2 X, usually for a.e. x. Stationary process a sequence (X1 ; X2 ; : : :) of random variables (real or complex-valued measurable functions) on a probability space whose joint distributions are invariant under shifting (X1 ; X2 ; : : :) to (X2 ; X3 ; : : :). Uniform distribution a sequence fx n g in [0; 1] is uniformly distributed if for each interval I [0; 1], the time it spends in I is asymptotically proportional to the length of I. Maximal inequality an inequality which allows one to bound the pointwise oscillation of a sequence of functions. An essential tool for proving pointwise ergodic theorems. Operator any linear operator U on a vector space of functions on X, for example one arising from a dynamical system T by setting U f (x) D f (Tx). More generally any linear transformation on a real or complex vector space. Positive contraction an operator T on a space of functions endowed with a norm k k such that T maps positive functions to positive functions and kT f k j f j. Definition of the Subject Ergodic theorems are assertions about the long-term statistical behavior of a dynamical system. The subject arose out of Boltzmann’s ergodic hypothesis which sought to equate the spatial average of a function over the set of states in a physical system having a fixed energy with the time average of the function observed by starting with a particular state and following its evolution over a long time period. Introduction Suppose that (X; B; ) is a measure space and T : (X; B; ) ! (X; B; ) is a measurable and measurepreserving transformation, that is (T 1 E) D (E) for all E 2 B. One important motivation for studying such maps is that a Hamiltonian physical system (see the article by Petersen in this collection) gives rise to a one-parameter group fTt : t 2 Rg of maps in the phase space of the system which preserve Lebesgue measure. The ergodic theorem of Birkhoff asserts that for f 2 L1 () n1 1X f (T i x) n
(1)
iD0
converges a.e. and that if RT is ergodic (to be defined shortly) then the limit is f d. This may be viewed
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as a justification for Boltzmann’s ergodic hypothesis that “space averages equal time averages”. See Zund [214] for some history of the ergodic hypothesis. For physicists, then, the problem is reduced to showing that a given physical system is ergodic, which can be very difficult. However ergodic systems arise in many natural ways in mathematics. One example is rotation z 7! z of the unit circle if is a complex number of modulus one which is not a root of unity. Another is the shift transformation on a sequence of i.i.d. random variables, for example a coin-tossing sequence. Another is an automorphism of a compact Abelian group. Often a transformation possesses an invariant measure which is not obvious at first sight. Knowledge of such a measure can be a very useful tool. See Petersen’s article for more examples. If (X; B; ) is a probability space and T is ergodic then Birkhoff’s ergodic theorem implies that if A is a measurable subset of X then for almost every x, the frequency with which x visits A is asymptotically equal to (A), a very satisfying justification of intuition. For example, applying this to the coin-tossing sequence one obtains the strong law of large numbers which asserts that almost every infinite sequence of coin tosses has tails occurring with asymptotic frequency 12 . One also obtains Borel’s theorem on normal numbers which asserts that for almost all x 2 [0; 1] each digit 0; 1; 2; : : : ; 9 occurs with limiting 1 frequency 10 . The so-called continued fraction transformation x 7! x 1 mod 1 on (0; 1) has a finite invariant dx . (Throughout this article x mod 1 denotes measure 1Cx the fractional part of x.) Applying Birkhoff’s theorem then gives precise information about the frequency of occurrence of any n 2 N in the continued fraction expansion of x, for a.e. x. See for example Billingsley [34]. These are the classical roots of the subject of ergodic theorems. The subject has evolved from these simple origins into a vast field in its own right, quite independent of physics or probability theory. Nonetheless it still has close ties to both these areas and has also forged new links with many other areas of mathematics. Our purpose here is to give a broad overview of the subject in a historical perspective. There are several excellent references, notably the books of Krengel [135] and Tempelman [194] which give a good picture of the state of the subject at the time they appeared. There has been tremendous progress since then. The time is ripe for a much more comprehensive survey of the field than is possible here. Many topics are necessarily absent and many are only glimpsed. For example this article will not touch on random ergodic theorems. See the articles [75,143] for some references on this topic.
I thank Mustafa Akcoglu, Ulrich Krengel, Michael Lin, Dan Rudolph and particularly Joe Rosenblatt and Vitaly Bergelson for many helpful comments and suggestions. I would like to dedicate this article to Mustafa Akcoglu who has been such an important contributor to the development of ergodic theorems over the past 40 years. He has also played a vital role in my mathematical development as well as of many other mathematicians. He remains a source of inspiration to me, and a valued friend. Ergodic Theorems for Measure-Preserving Maps Suppose (X; B; ) is a measure space. A (measure-preserving) endomorphism of (X; B; ) is a measurable mapping T : X ! X such that (T 1 E) D (E) for any measurable subset E X. If T has a measurable inverse then one says that it is an automorphism. In this article the unqualified terms “endomorphism” or “automorphism” will always mean measure-preserving. T is called ergodic if for all measurable E one has T 1 E D E ) (E) D 0
or (E c ) D 0 :
A very general class of examples comes from the notion of a stationary stochastic process in probability theory. A stochastic process is a sequence of measurable functions f1 ; f2 ; : : : on a probability space (X; B; ) taking values in a measurable space (Y; C ). The distribution of the process, a measure on Y N , is defined as the image of under the map ( f1 ; f2 ; : : :) : X ! Y N . captures all the essential information about the process f f i g. In effect one may view any process f f i g as a probability measure on Y N . The process is said to be stationary if is invariant under the left shift transformation S on Y N , S(y)(i) D y(i C 1), that is S is an endomorphism of the probability space (Y N ; ). From a probabilistic point of view the most natural examples of stationary stochastic processes are independent identically distributed processes, namely the case when is a product measure N for some measure on (Y; C ). More generally one can consider a stationary Markov process defined by transition probabilities on the state space Y and an invariant probability on Y. See for example Chap. 7 in [50], also Sect. “Pointwise Ergodic Theorems for Operators” below. The first and most fundamental result about endomorphisms is the celebrated recurrence theorem of Poincaré [173]). Theorem 1 Suppose is finite, A 2 B and (A) > 0. Then for a.e. x 2 A there is an n > 0 such that T n x 2 A, in fact there are infinitely many such n. It may be viewed as an ergodic theorem, in that it is a qualitative statement about how x behaves under iteration of T.
Ergodic Theorems
For a proof, observe that if E A is the measurable set of points which never return to A then for each n > 0 the set T n E is disjoint from E. Applying T m one gets also T (nCm) E \ T m E D ;. Thus E; T 1 E; T 2 E; : : : is an infinite sequence of disjoint sets all having measure (E) so (E) D 0 since (X) < 1. Much later Kac [120] formulated the following quantitative version of Poincaré’s theorem. Theorem 2 Suppose that (X) D 1, T is ergodic and let r A x denote the time of first return to A, that is r A (x) is the least n > 0 such that T n x 2 A. Then Z 1 1 r A d D ; (2) (A) A (A) that is, the expected value of the return time to A is (A)1 . Koopman [131] made the observation that associated to an automorphism T there is a unitary operator U D U T defined on the Hilbert space L2 () by the formula U f D f ı T. This led von Neumann [151] to prove his mean ergodic theorem. Theorem 3 Suppose H is a Hilbert space, U is a unitary operator on H and let P denote the orthogonal projection on the subspace of U-invariant vectors. Then for any x 2 H one has n1 1X U i x Px ! 0 as n ! 1 : (3) n iD0
Von Neumann’s theorem is usually quoted as above but to be historically accurate he dealt with unitary operators indexed by a continuous time parameter. Inspired by von Neumann’s theorem Birkhoff very soon proved his pointwise ergodic theorem [35]. In spite of the later publication by von Neumann his result did come first. See [29,214] for an interesting discussion of the history of the two theorems and of the interaction between Birkhoff and von Neumann. Theorem 4 Suppose (X; B; ) is a measure space and T is an endomorphism of (X; B; ). Then for any f 2 L1 D L1 () there is a T-invariant function g 2 L1 such that A n f (x) D
n1 1X f (T i x) ! g(x) n
a.e.
(4)
iD0
Moreover if is finite then the convergence alsoRholds with R respect to the L1 norm and one has E g d D E f d for all T-invariant subsets E.
Again this formulation of Birkhoff’s theorem is not historically accurate as he dealt with a smooth flow on a manifold. It was soon observed that the theorem, and its proof, remain valid for an abstract automorphism of a measure space although the realization that T need not be invertible seems to have taken a little longer. The notation A n f D A n (T) f as above will occur often in the sequel. Whenever T is an endomorphism one uses the notation T f D f ı T and with this notation P i A n (T) D n1 n1 iD0 T . When the scalars (R or C) for an L1 space are not specified the notation should be understood as referring to either possibility. In most of the theorems in this article the complex case follows easily from the real and any indications about proofs will refer to the real case. Although von Neumann originally used spectral theory to prove his result, there is a quick proof, attributed to Riesz by Hopf in his 1937 book [100], which uses only elementary properties of Hilbert space. Let I denote the (closed) subspace of U-invariant vectors and I 0 the (usually not closed) subspace of vectors of the form f U f . It is easy to check that any vector orthogonal to I 0 must be in I, whence the subspace I C I 0 is dense in H . For any vector of the form x D y C y 0 ; y 2 I; y 0 2 I 0 it is clear P i that A n x D n1 n1 iD0 U x converges to y D Px, since 0 A n y D y and if y D z Uz then the telescoping sum A n y 0 D n1 (z U n z) converges to 0. This establishes the desired convergence for x 2 I C I 0 and it is easy to extend it to the closure of I C I 0 since the operators An are contractions of H (kA n k 1). Lorch [147] used a soft argument in a similar spirit to extend non Neumann’s theorem from the case of a unitary operator on a Hilbert space to that of an arbitrary linear contraction on any reflexive Banach space. Sine [188] gave a necessary and sufficient condition for the strong convergence of the ergodic averages of a contraction on an arbitrary Banach space. Birkhoff’s theorem has the distinction of being one of the most reproved theorems of twentieth century mathematics. One approach to the pointwise convergence, parallel to the argument to argument just seen, is to find a dense subspace E of L1 so that the convergence holds for all f 2 E and then try to extend the convergence to all f 2 L1 by an approximation argument. The first step is not too hard. For simplicity assume that is a finite measure. As in the proof of von Neumann’s theorem the subspace E spanned by the T-invariant L1 functions together with functions of the form g T g; g 2 L1 , is dense in L1 (). This can be seen by using the Hahn–Banach theorem and the duality of L1 and L1 . (Here one needs finiteness of to know that L1 L1 .) The pointwise convergence of A n f for invariant f is trivial and for f D g T g it follows from telescoping of the sum and the fact that n1 T n g ! 0 a.e. This
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last can be shown by using the Borel–Cantelli lemma. The second step, extending pointwise convergence, as opposed to norm convergence, for f in a dense subspace to all f in L1 is a delicate matter, requiring a maximal inequality. Roughly speaking a maximal inequality is an inequality which bounds the pointwise oscillation of A n f in terms of the norm of f . The now standard maximal inequality in the context of Birkhoff’s theorem is the following, due to Kakutani and Yosida [209]. Birkhoff’s proof of his theorem includes a weaker version of this result. Let S n f D nA n f , the nth partial sum of the iterates of f . S Theorem 5 Given any real f 2 L1 let A D n1 fS n f 0g. Then Z f d 0 : (5) A
Moreover if one sets M f D supn1 A n f then for any ˛ > 0 fM f > ˛g
1 k f k1 ˛
(6)
A distributional inequality such as (6) will be referred to as a weak L1 inequality. Note that (6) follows easily from (5) by applying (5) to f ˛, at least in the case when is finite. With the maximal inequality in hand it is straightforward to complete the proof of Birkhoff’s theorem. For a realvalued function f let Osc f D lim sup A n f lim inf A n f :
(7)
Osc f D 0 a.e. if and only if A n f converges a.e.(to a possibly infinite limit). One has lim sup A n f M f Mj f j and by symmetry lim inf A n f Mj f j, so Osc f 2Mj f j. To establish the convergence of A n f for a real-valued f 2 L1 let > 0 and write f D g C h with g 2 E (the subspace where convergence has already been established), h 2 L1 and khk < . Then since Osc g D 0 one has Osc f D Osc h. Thus for any fixed ˛ > 0, using (6) n ˛o fOsc f > ˛g D fOsc h > ˛g M h > 2 2khk1 2 < : (8) ˛ ˛ Since > 0 was arbitrary one concludes that fOsc f > ˛g D 0 and since ˛ > 0 is arbitrary it follows that fOsc f > 0g D 0, establishing the a.e. convergence. Moreover a simple application of Fatou’s lemma shows that the limiting function is in L1 , hence finite a.e. There are many proofs of (5). Two of particular interest are Garsia’s [89], perhaps the shortest and most mysterious, and the proof via the filling scheme of Chacón
and Ornstein [61], perhaps the most intuitive, which goes like this. Given a function g 2 L1 write g C D max(g; 0), g D g C g and let Ug D T g C g . Interpretation: the region between the graph of g C and the X-axis is a sheaf of vertical spaghetti sticks, the intervals [0; g C (x)], x in X, and g is a hole. Now move the spaghetti (horizontally) by T and then let it drop (vertically) into the hole leaving a new hole and a new sheaf which are the negative and posS itive parts of Ug. Now let E 0 D n1 fU n f 0g, the set of points x at which the hole is eventually filled after finitely many iterations of U. The key point is that E D E 0 . Indeed if S n f (x) 0 for some n, then the total linear height of sticks over x; Tx; : : : T n1 x is greater than the total linear depth of holes at these points. The only way that spaghetti can escape from these points is by first filling the hole at x, which shows x 2 E 0 . Similar thinking shows that if x 2 E 0 and the hole at x is filled for the first time at time n then S n f (x) 0, so x 2 E, and that all the spaghetti that goes into the hole at x comes from points T i x which belong to E 0 . This shows that E D E 0 and that the part of the hole lying beneath E is eventually filled by spaghetti coming from E 0 D E. Thus the amount of spaghettiRover E is no less than the size of the hole under E, that is E f d 0. Most proofs of Birkhoff’s theorem use a maximal inequality in some form but a few avoid it altogether, for example [126,186]. It is also straightforward to deduce Birkhoff’s theorem directly from a maximal inequality, as Birkhoff does, without first establishing convergence on a dense subspace. However the technique of proving a pointwise convergence theorem by finding an appropriate dense subspace and a suitable maximal inequality has proved extremely useful, not only in ergodic theory. Indeed, in some sense maximal inequalities are unavoidable: this is the content of the following principle proved already in 1926 by Banach. The principle has many formulations; the following one is a slight simplification of the one to be found in [135]. Suppose B is a Banach space, (X; B; ) is a finite measure space and let E denote the space of -equivalence classes of measurable real-valued functions on X. A linear map T : B ! E is said to be continuous in measure if for each > 0 kx n xk ! 0 ) fjTx n Txj > g ! 0 : Suppose that T n is a sequence of linear maps from B to E which are continuous in measure and let Mx D sup n jTn xj. Of course if Tn x converges a.e. to a finite limit then Mx < 1 a.e. Theorem 6 (Banach principle) Suppose Mx < 1 a.e. for each x 2 X. Then there is a function C() such that
Ergodic Theorems
C() ! 0 as ! 1 such that for all x 2 B and > 0 one has fMx kxkg C() :
(9)
Moreover the set of x for which Tn x converges a.e. is closed in B. The first chapter of Garsia’s book [89] contains a nice introduction to the Banach principle. It should be noted that, for general integrable f , M f need not be in L1 . However if f 2 L p for p > 1 then M f does belong to Lp and one has the estimate kM f k p
p k f kp : p1
(10)
This can be derived from (6) using also the (obvious) fact that kM f k1 k f k1 . Any such estimate on the norm of a maximal function will be called a strong Lp inequality. It also follows from (6) that if (X) is finite and f 2 L log L, R that is j f j logC j f jd < 1, then M f 2 L1 . In fact Ornstein [160] has shown that the converse of this last statement holds provided T is ergodic. There is a special setting where one has uniform convergence in the ergodic theorem. Suppose T is a homeomorphism of a compact metric space X. By a theorem of Krylov and Bogoliouboff [138] there is at least one probability measure on the Borel -algebra of X which is invariant under T. T is said to be uniquely ergodic if there is only one Borel probability measure, say , invariant under T. It is easy to see that when this is the case then T is an ergodic automorphism of (X; B; ). As an example, if ˛ is an irrational number then the rotation z 7! e2 i˛ z is a uniquely ergodic transformation of the circle fjzj D 1g. Equivalently x 7! x C ˛ mod 1 is a uniquely ergodic map on [0; 1]. A quick way to see this is to show that the ˆ Fourier co-efficients (n) of any invariant probability are zero for n ¤ 0. The Jewett–Krieger theorem (see Jewett [109] and Krieger [137]) guarantees that unique ergodicity is ubiquitous in the sense that any automorphism of a probability space is measure-theoretically isomorphic to a uniquely ergodic homeomorphism. The following important result is due to Oxtoby [165]). Theorem 7 If T is uniquely ergodic, is its unique invariant probability measure and f 2 C(X) R then the ergodic averages A n ( f ) converge uniformly to f d. This result can be proved along the same lines as the proof given above of von Neumann’s theorem. In a nutshell, one uses the fact that the dual of CR (X) is the space of finite signed measures on X and the unique ergodicity to show
that functions of the form f f ı T together with the invariant functions (which are just constant functions) span a dense subspace of C(X). A sequence fx n g in the interval [0; 1]R is said to be uniP 1 formly distributed if n1 niD1 f (x n ) ! 0 f (x)dx for any f 2 C[0; 1] (equivalently for any Riemann integrable f or for any f D 1I where I is any subinterval of [0; 1]). As a simple application of Theorem 7 one obtains the linear case of the following result of Weyl [202]. Theorem 8 If ˛ is irrational and p(x) is any non-constant polynomial with integer coefficients then the sequence fp(n)˛mod1g is uniformly distributed. Furstenberg [84], see also [86], has shown that Weyl’s result, in full generality, can be deduced from the unique ergodicity of certain affine transformations of higher dimensional tori. For polynomials of degree k > 1 Weyl’s result is usually proved by inductively reducing to the case k D 1, using the following important lemma of van der Corput (see for example [139]). Theorem 9 Suppose that for each fixed h > 0 the sequence fx nCh x n
mod 1gn
is uniformly distributed. Then fx n g is uniformly distributed. When is infinite and T is ergodic the limiting function in Birkhoff’s theorem is 0 a.e. In 1937 Hopf [100] proved a generalization of Birkhoff’s theorem which is more meaningful in the case of an infinite invariant measure. It is a special case of a later theorem of Hurewicz [105], which we will discuss first. Suppose that (X; B; ) is a -finite measure space. If is another -finite measure on B write ( is absolutely continuous relative to ) if (E) D 0 implies (E) D 0 and one writes if and . Consider a non-singular automorphism : X ! X, meaning that is measurable with a measurable inverse and that (E) D 0 if and only if ( E) D 0. In other words D ı . By the Radon–Nikodym theorem there Ris a function 2 L1 () such that > 0 a.e. and (E) D E d for all measurable E. In order to obtain an associated operator T on L1 which is an (invertible) isometry one defines T f (x) D (x) f ( x) :
(11)
The dual operator on L1 is then given by T g D g ı 1 . If is a -finite measure equivalent to which is invariant under then the ergodic theory of can be reduced to the measure-preserving case using . The interesting case is when there is no such . It was an open
245
246
Ergodic Theorems
problem for some time whether there is always an equivalent invariant measure. In 1960 Ornstein [163] gave an example of a which does not have an equivalent invariant measure. It is curious that, with hindsight, such examples were already known in the fifties to people studying von Neumann algebras. Pn i For f 2 L1 let S n f D iD0 T f . is said to be conservative if there is no set E with (E) > 0 such that i E; i D 0; 1; : : : are pairwise disjoint, that is if the Poincaré recurrence theorem remains valid. For example, the shift on Z is not conservative. Hurewicz [105] proved the following ratio ergodic theorem. Theorem 10 Suppose is conservative, f ; g 2 L1 and a.e. to a -invarig(x) > 0 a.e. Then S n f /S n g converges R f d ant limit h. If is ergodic then h D R gd . In the case when is -invariant one has T f D f ı . If is invariant and finite, taking g D 1 one recovers Birkhoff’s theorem. If is invariant and -finite then Hurewicz’s theorem becomes the theorem of Hopf alluded to earlier. Wiener and Wintner [204] proved the following variant of Birkhoff’s theorem. Theorem 11 (Wiener–Wintner) Suppose T is an automorphism of a probability space (X; B; ). Then for any f 2 L1 there is a subset X 0 X of measure one such that for each x 2 X 0 and 2 C of modulus 1 the sequence 1 P n1 i i iD0 T f (x) converges. n It is an easy consequence of the ergodic theorem that one has a.e. convergence for a given f and but the point here is that the set on which the convergence occurs is independent of . Generalizations to Continuous Time and Higher-Dimensional Time A (measure-preserving) flow is a one-parameter group fTt ; t 2 Rg of automorphisms of (X; B; ), that is TtCs D Tt Ts , such that Tt x is measurable as a function of (t; x). It will always be implicitly assumed that the map (t; x) 7! Tt x from R X to X is measurable. Theorem 4 generalizes to flows by replacing sums with integrals and this generalization follows without difficulty from Theorem 4. (As already observed this observation reverses the historical record.) Theorem 4 may be viewed as a theorem about the “discrete flow” fTn D T n : n 2 Zg. Wiener was the first to generalize Birkhoff’s theorem to families of automorphisms fTg g indexed by groups more general than R or Z. A measure-preserving flow is an action of R while a single automorphism corresponds to an action of Z.
A (measure-preserving) action of a group G is a homomorphism T : g 7! Tg from G into the group of automorphisms of a measure space (X; B; ) (satisfying the appropriate joint measurability condition in case G is not discrete). Suppose now that G D R k or Z k and T is an action of G on (X; B; ). In the case of Z k an action amounts to an arbitrary choice of commuting maps T1 ; : : : Tk , Ti D T(e i ) where ei is the standard basis of Z k . In the case of R k one must specify k commuting flows. Let m denote counting measure on G in case G D Z k and Lebesgue measure in case G D R k . For any subset E of G with m(E) < 1 let Z 1 f (Tg x)dm(g) : (12) A E f (x) D m(E) E One may then ask whether A E f converges to a limit, either in the mean or pointwise, as E varies through some sequence of sets which “grow large” or, in case G D R k , “shrink to 0”. The second case is referred to as a local ergodic theorem. In the case of ergodic theorems at infinity the continuous and discrete theories are rather similar and often the continuous analogue of a discrete result can be deduced from the discrete result. In Wiener [203] proved the following result for actions of G D R k and ergodic averages over Euclidean balls Br D fx 2 R k : kxk2 rg. Theorem 12 Suppose T is an action of R k on (X; B; ) and f 2 L1 (). (a) For f 2 L1 limr!1 A B r f D g exists a.e. If is finite the convergence also holds with respect to the L1 -norm, R R g is T-invariant and I gd D I f d for every T-invariant set I. (b) limr!0 A B r f D f a.e. The local aspect of Wiener’s theorem is closely related to the Lebesgue differentiation theorem, see for example Proposition 3.5.4 of [132], which, in its simplest form, states Rthat for f 2 L1 (R k ; m) one has a.e. convergence of 1 m(B r ) B r f (x C t)dt to f (x) as r ! 0. The local ergodic theorem implies Lebesgue’s theorem, simply by considering the action of R k on itself by translation. In fact the local ergodic theorem can also be deduced from Lebesgue’s theorem by a simple application of Fubini’s theorem (see for example [135], Chap. 1, Theorem 2.4 in the case k D 1). The key point in Wiener’s proof is the following weak L1 maximal inequality, similar to (6). Theorem 13 Let M f D supr>0 jA B r f j. Then one has fM f > ˛g
C k f k1 ; ˛
(13)
Ergodic Theorems
where C is a constant depending only on the dimension d. (In fact one may take C D 3d .) In the case when T is the action of Rk on itself by translation (13) is the well-known maximal inequality for the Hardy–Littlewood maximal function ([132], Lemma 3.5.3). Wiener proves (13) by way of the following covering lemma. If B is a ball in R k let B0 denote the concentric ball with three times the radius. Rk
Theorem 14 Suppose a compact subset K of is covered by a (finite) collection U of open balls. Then there exist pairS wise disjoint B i 2 U; i D 1; : : : k such that i B0i covers K. To find the Bi it suffices to let B1 be the largest ball in U, then B2 the largest ball which does not intersect B1 and in general Bn the largest ball which does not intersect Sn1 0 iD1 B i . Then it is not hard to argue that the B i cover K. In general, a covering lemma is, roughly speaking, an assertion that, given a cover U of a set K in some space (often a group), one may find a subcollection U0 U which covers a substantial part of K and is in some sense efficient P in that U 2U0 1U C, C some absolute constant. Covering lemmas play an important role in the proofs of many maximal inequalities. See Sect. 3.5 of [132] for a discussion of several of the best-known classical covering lemmas. As Wiener was likely aware, the same kind of covering argument easily leads to maximal inequalities and ergodic theorems for averages over “sufficiently regular” sets. For example, in the case of G D Z k use the standard total order on G and for n 2 N k let S n D fm 2 Z k : 0 m ng. Let e(n) denote the maximum value of n i /n j . Then one has the following result.
vergence if f 2 L p for some p > 1, provided is finite. Let T1 ; : : : ; Tk be any k (possibly non-commuting!) automorphisms of a finite measure space (X; B; ). For n n D (n1 ; n2 ; : : : ; n k ) let Tn D T1n 1 : : : Tk k . (This is not an action of Z k unless the T i commute with each other.) As before when F Z k is a finite subset write A F f D 1 P n2F Tn f . Finally let P j f D lim n A n (T j ) f . jFj Theorem 16 For f 2 L p lim A S n f D P1 : : : Pk f
n!1
both a.e. and in Lp . The proof of Theorem 16 uses repeated applications of Birkhoff’s theorem and hinges on (10). Somewhat surprisingly Hurewicz’s theorem was not generalized to higher dimensions until the very recent work of Feldman [79]. In fact the theorem fails if one considers averages over the cube [0; n 1]d in Zd . However Feldman was able to prove a suitably formulated generalization of Hurewicz’s theorem for averages over symmetric cubes. For f 2 L1 (R) the classical Hilbert transform is defined for a.e. t by Z f (t s) 1 lim ds (16) H f (t) D !0C jsj> s R f (ts) Let H f (t) D 1 sup >0 j jsj> s dsj, the corresponding maximal function. The proof that the limit (16) exists a.e. is based on the maximal inequality mfH f > g
Theorem 15 For any e > 0 and any integrable f lim
n!1;e(n)<e
AS n f
exists a.e. and the limit is characterized by the same properties as in Theorem 12. Moreover there is a maximal inequality ( ) C sup jA S n f j > ˛ k f k1 ; (14) ˛ e(n)<e
(15)
C k f k1 ;
(17)
where C is an absolute constant. See Loomis [146] (1946) for a proof of (17) using real-variable methods. Cotlar [65] proved the existence of the following ergodic analogue of the Hilbert transform (actually for the n-dimensional Hilbert transform). Theorem 17 Suppose fTt g is a measure-preserving flow on a probability space (X; B; ) and f 2 L1 . Then Z f (Tt x) dt (18) lim !0 g
g
0 a.e. 1 kD0 T p is infinite a.e. on C and finite a.e. on D. These results are the cornerstone of much of the subsequent work on L1 contractions.
Ergodic Theorems
Theorem 19 contains Birkhoff’s theorem for an automorphism of (X; B; ), simply by defining T f D f ı . In fact one can also deduce Theorem 19 from Birkhoff’s theorem (if one assumes only that X is standard). Indeed the hypotheses of Hopf’s theorem imply that the kernel P associated to T is stochastic (T 1 D 1) and and that is P-invariant (T1 D 1). Hopf’s theorem then follows by applying Birkhoff’s theorem to the shift on the stationary Markov process associated to P and . In fact Kakutani [123] (see also Doob [71]) had already made essentially the same observation, except that his result assumes the stationary Markov process to be already given. In 1955 Dunford and Schwartz [73] made essential use of Hopf’s work to prove the following result. Theorem 20 Suppose (X) D 1 and T is a (not necessarily positive) contraction with respect to both the L1 and L1 norms. Then the conclusion of Hopf’s theorem remains valid. Note that the assumption that T contracts the L1 -norm is meaningful, as L1 L1 . The proof of the result is reduced to the positive case by defining a positive contraction jTj analogous to the total variation of a complex measure. Then in 1960 Chacón and Ornstein [61] proved a definitive ratio ergodic theorem for all positive contractions of L1 which generalizes both Hopf’s theorem and the Hurewicz theorem. Theorem 21 Suppose T is a positive contraction of L1 (), where is -finite, f ; g 2 L1 and g 0. Then Pn Ti f PiD0 n i iD0 T g converges a.e. on the set f
(22) P1
iD0
T i g > 0g.
In 1963 Chacón [58] proved a very general theorem for non-positive operators which includes the Chacón–Ornstein theorem as well as the Dunford–Schwartz theorem. Theorem 22 Suppose T is a contraction of L1 and p n 0 is a sequence of measurable functions with the property that g 2 L1 ; jgj p n ) jT gj p nC1 :
(23)
Then Pn Ti f PiD0 n iD0 p i converges a.e. to a finite limit on the set f
(24) P1
iD0
p i > 0g.
If T is an L1 ; L1 -contraction and p n D 1 for all n then the hypotheses of this theorem are satisfied, so Theorem 22
reduces to the result of Dunford and Schwartz. See [59] for a concise overview of all of the above theorems in this section and the relations between them. The identification of the limit in the Chacón–Ornstein theorem on the conservative part C of X is a difficult problem. It was solved by Neveu [152] in case C D X, and in general by Chacón [57]. Chacón [60] has shown that there is a non-singular automorphism of (X; B; ) such that for the associated invertible isometry T of L1 () given by (11) there is an f 2 L1 () such that lim sup A n f D 1 and lim inf A n f D 1. In 1975 Akcoglu [3] solved a major open problem when he proved the following celebrated theorem. Theorem 23 Suppose T : L p 7! L p is a positive contracP i tion. Then A n f D n1 n1 iD0 T f converges a.e. Moreover one has the strong Lp inequality sup jA n f j n
p
p k f kp : p1
(25)
As usual, the maximal inequality (25) is the key to the convergence. Note that it is identical in form to (10) the classical strong Lp inequality for automorphisms. (25) was proved by A. Ionesco–Tulcea (now Bellow) [106] in the case of positive invertible isometries of Lp . It is a result of Banach [16], see also [140], that in this case T arises from a non-singular automorphism of (X; B; ) in the form T f D 1/p f ı . By a series of reductions Bellow was able to show that in this case (25) can be deduced from (10). Akcoglu’s brilliant idea was to consider a dilation S of T which is a positive invertible isometry on a larger Lp space L˜ p D L p (Y; C ; ). What this means is that there is a positive isometric injection D : L p ! L˜ p and a positive projection P on L˜ p whose range is D(L p ) such that DT n D PS n D for all n 0. Given the existence of such an S it is not hard to deduce (25) for T from (25) for S. In fact Akcoglu constructs a dilation only in the case when Lp is finite dimensional and shows how to reduce the proof of (25) to this case. In the finite dimensional case the construction is very concrete and P is a conditional expectation operator. Later Akcoglu and Kopp [7] gave a construction in the general case. It is noteworthy that the proof of Akcoglu’s theorem consists ultimately of a long string of reductions to the classical strong Lp inequality (10), which in turn is a consequence of (5). Subadditive and Multiplicative Ergodic Theorems Consider a family fX n;m g of real-valued random variables on a probability space indexed by the set of pairs (n; m) 2
249
250
Ergodic Theorems
Z2 such that 0 n < m. fX(n;m) g is called a (stationary) subadditive process if (a) the joint distribution of fX n;m g is the same as that of fX nC1;mC1 g (b) X n;m X n;l C X l ;m whenever n < l < m. Denoting the index set by fn < mg Z2 the distribution of the process is a measure on Rfn<mg which is invariant under the shift Tx(n; m) D x(n C 1; m C 1). Thus there is no loss of generality in assuming that there is an underlying endomorphism T such that X n;m ı T D X nC1;mC1 . In 1968 Kingman [129] proved the following generalizaR tion of Birkhoff’s theorem. D inf n1 X 0;n d is called the time constant of the process. Theorem 24 If the X n;m are integrable and > 1 then 1 n X 0;n converges a.e. and R in L1 -norm to a T -invariant limit ¯ D . X¯ 2 L1 () satisfying Xd It is easy to deduce from the above that if one assumes only C that X 0;1 is integrable then n1 X0;n still converges a.e. to a T-invariant limit X¯ taking values in [1; 1). Subadditive processes first arose in the work of Hammersley and Welsh [99] on percolation theory. Here is an example. Let G be the graph with vertex set Z2 and with edges joining every pair of nearest neighbors. Let E denote the edge set and let fTe : e 2 Eg be non-negative integrable i.i.d. random variables. To each finite path P in G P associate the “travel time” T(P) D e2E Te . For integers m > n 0 let X n;m be the infimum of T(P) over all paths P joining (0; R n) to (0; m). This is a subadditive process with 0 < Te d and it is not hard to see that the underlying endomorphism is ergodic. Thus Kingman’s theorem yields the result that n1 X0;n ! a.e. Suppose now that T is an ergodic automorphism of a probability space (X; B; ) and P is a function on X taking values in the space of d d real matrices. Define Pn;m D P(T m1 x)P(T m2 x) : : : P(T n x) and let Pn;m (i; j) denote the i; j entry of Pn;m . Then X n;m D log(kPn;m k) (use any matrix norm) is a subadditive process so one obtains the first part of the following result of Furstenberg and Kesten [87] (1960) originally proved by more elaborate methods. The second part can also be deduced from the subadditive theorem with a little more work. See Kingman [130] for details and for some other applications of subadditive processes. Theorem 25 R (a) Suppose logC (jPj)d < 1. Then kP0;n k1/n converges a.e. to a finite limit. (b) Suppose that for each i; j P(i; j) is a strictly positive function such that log P(i; j) is integrable. Then the
1
limit p D lim(P0;n (i; j)) n exists a.e. and is independent of i and j. Partial results generalizing Kingman’s theorem to the multiparameter case were obtained by Smythe [189] and Nguyen [157]. In 1981 Akcoglu and Krengel [8] obtained a definitive multi-parameter subadditive theorem. They d consider an action fTm g of the semigroup G D Z0 by endomorphisms of a measure space (X; B; ). Using the standard total ordering of G an interval in G is any set of the form fk 2 G : m k ng for any m n 2 G. Let I denote the set of non-empty intervals. Reversing the direction of the inequality, they define a superadditive process as a collection of integrable functions FI ; I 2 I , such that (a) FI ı Tm D FICm , (b) FI FI 1 C : : : C FI k whenever I is the disjoint union of I1 ; : : : ; I k and R (c) D supI2I jIj1 FI d < 1. A sequence fI n g of sets in I is called regular if there is an increasing sequence I 0n such that I n I 0n and jI 0n j CjI n j for some constant C. Theorem 26 (Akcoglu–Krengel) Suppose F I is a superadditive process and fI n g is regular. Then jI1n j FI n converges a.e. fFI g is additive if the inequality in (b) is replaced by equalP ity. In this case FI D n2I f ı Tn where f is an integrable function. Thus in the additive case the Akcoglu–Krengel result is a theorem about ordinary multi-dimensional ergodic averages, which is in fact a special case of an earlier result of Tempelman [196] (see Sect. “Amenable Groups” below). Kingman’s proof of Theorem 24 hinged on the existence of a certain (typically non-unique) decomposition for subadditive processes. Akcoglu and Krengel’s proof of the multi-parameter result does not depend on a Kingman-type decomposition, in fact they show that there is no such decomposition in general. They prove a weak maximal inequality fsup jI n j1 FI n > g
0
then the i are constants, 1 < 0 and r > 0. Raghunathan [174] gave a much shorter proof of Oseledec’s theorem, valid for matrices with entries in a locally compact normed field. He showed that it could be reduced to the Furstenberg–Kesten theorem by considering the exterior powers of P. Ruelle [180] extended Oseledec’s theorem to the case where P takes values in the set of bounded operators on a Hilbert space. Walters [199] has given a proof (under slightly stronger hypotheses) which avoids the matrix calculations and tools from multilinear algebra used in other proofs.
(27)
log((A)) may be viewed as a quantitative measure of the amount of information contained in the statement that a randomly chosen x 2 X happens to belong to A. So h(P) is the expected information if one is about to observe which atom of P a randomly chosen point falls in. See Billingsley [34] for more motivation of this concept. See also the article in this collection on entropy by J. King or any introductory book on ergodic theory, e. g. Petersen [171]. W If P and Q are partitions P Q denotes the common refinement which consists of all sets p\q, p 2 P; q 2 Q. It W is intuitive and not hard to show that h(P Q) h(P) C W n i h(Q). Now let P0n D n1 iD0 T P and h n D h(P0 ). The subadditivity of entropy implies that h nCm h n C h m , so by a well-known elementary lemma the limit h(P; T) D lim n
(c) The functions m i and i are T-invariant.
(p) log((p))
p2P
hn hn D inf 0 n n n
(28)
exists. If one thinks of P(x) as a measurement performed on the space X and Tx as the state succeeding x after one second has elapsed then h(P; T) is the expected information per second obtained by repeating the experiment every second for a very long time. See [177] for an alternative and very useful approach to h(P; T) via name-counting. The following result, which is known as the Shannon– McMillan–Breiman theorem, has proved to be of fundamental importance in ergodic theory, notably, for example, in the proof of Ornstein’s celebrated isomorphism theorem for Bernoulli shifts [159]. Theorem 28 If T is ergodic then 1 lim log((P0n1 (x)) D h(P; T) n
n!1
a.e. and in L1 -norm.
(29)
251
252
Ergodic Theorems
In other words, the actual information obtained per second by observing x over time converges to the constant h(P; T), namely the limiting expected information per second. Shannon [185] formulated Theorem 28 and proved convergence in probability. McMillan [149] proved L1 convergence and Breiman [49] obtained the a.e. convergence. The original proofs of a.e. convergence used the martingale convergence theorem, were not very intuitive and did not generalize to Z n -actions, where the martingale theorem is not available. Ornstein and Weiss [161] (1983) found a beautiful and more natural argument which bypasses the martingale theorem and allows generalization to a class of groups which includes Z n . Amenable Groups Let G be any countable group and T D fTg g an action of G by automorphisms of a probability space. Suppose is a complex measure on G, that is, f(g)g g2G is an absolutely summable sequence. Let T g act on functions via Tg f D f ı Tg , so T g is an isometry of Lp for every P 1 p 1. Let (T) D g2G (g)Tg . A very general framework for formulating ergodic theorems is to consider a sequence fn g and ask whether n (T) f converges, a.e. or in mean, for f in some Lp space. When n (T) f converges for all actions T and all f in Lp in p-norm or a.e. then one says that n is mean or pointwise good in Lp . When the n are probability measures it is natural to call such results weighted ergodic theorems and this terminology is retained for complex as well. Birkhoff’s theorem says that if G D Z and n is the normalized counting measure on f0; 1; : : : n 1g then fn g is pointwise good in L1 . This section will be concerned only with sequences fn g such that n is normalized counting measure on a finite subset Fn G so one speaks of mean or pointwise good sequences fFn g. A natural condition to require of fFn g, which will ensure that the limiting function is invariant, is that it be asymptotically (left) invariant in the sense that
jgFn Fn j !0 jFn j
8g 2 G :
(30)
Such a sequence is called a Følner sequence and a group G is amenable if it has a Følner sequence. As in most of this article G is restricted to be a discrete countable group for simplicity but most of the results to be seen actually hold for a general locally compact group. Amenability of G is equivalent to the existence of a finitely additive left invariant probability measure on G. It is not hard to see that any Abelian, and more generally
any solvable, group is amenable. On the other hand the free group F 2 on two generators is not amenable. See Paterson [167] for more information on amenable groups. The Følner property by itself is enough to give a mean ergodic theorem. Theorem 29 Any Følner sequence is mean good in Lp for 1 p < 1. The proof of this result is rather similar to the proof of Theorem 3. In fact Theorem 29 is only a special case of quite general results concerning amenable semi-groups acting on abstract Banach spaces. See the book of Paterson [167] for more on this. Turning to pointwise theorems, the Følner condition alone does not yield a pointwise theorem, even when G D Z and the F n are intervals. For example Akcoglu and del Junco [6] have shown that when G D Z and p Fn D [n; n C n] \ Z the pointwise ergodic theorem fails for any aperiodic T and for some characteristic function f . See also del Junco and Rosenblatt [119]. The following pointwise result of Tempelman [196] is often quoted. A Følner sequence fFn g is called regular if there is a constant C such that jFn1 Fn j CjFn j and there is an increasing sequence Fn0 such that Fn Fn0 and jFn0 j CjFn j. Theorem 30 Any regular Følner sequence is pointwise good in L1 . In case the F n are intervals in Z n this result can be proved by a variant of Wiener’s covering argument and in the general case by an abstraction thereof. The condition jFn1 Fn j CjFn j captures the property of rectangles which is needed for the covering argument. Emerson [78] independently proved a very similar result. The work on ergodic theorems for abstract locally compact groups was pioneered by Calderón [53] who built on Wiener’s methods. The main result in this paper is somewhat technical but it already contains the germ of Tempelman’s theorem. Other ergodic theorems for amenable groups, whose main interest lies in the case of continuous groups, include Tempelman [195], Renaud [175], Greenleaf [93] and Greenleaf and Emerson[94]. The discrete versions of these results are all rather close to Tempelman’s theorem. Among pointwise theorems for discrete groups Tempelman’s result was essentially the best available for a long time. It was not known whether every amenable group had a Følner sequence which is pointwise good for some Lp . In 1988 Shulman [187] introduced the notion of a tempered
Ergodic Theorems
Følner sequence fFn g, namely one for which ˇ ˇ ˇ[ ˇ ˇ 1 ˇ F i Fn ˇ < CjFn j ; ˇ ˇ ˇ
(31)
i 0 and (@E) D 0. Let S D fi > 0 : T i g 2 Eg. Then S is pointwise good in L1 . Krengel [133] constructed the first example of a sequence S N which is pointwise universally bad, in the strong sense that for any aperiodic T the a.e. convergence of S;n (T) f fails for some characteristic function f . Bellow [17] proved that any lacunary sequence (meaning a nC1 > ca n for some c > 1) is pointwise universally bad in L1 . Later Akcoglu at al. [1] were able to show that for lacunary sequences fS;n g is even strongly sweeping out. A sequence fn g of probability measures on Z is said to be strongly sweeping out if for any ergodic TRand for all ı > 0 there is a characteristic function f with f d < ı such that lim sup n (T) f D 1 a.e. It is not difficult to show that if fn g is strongly sweeping out then there are characteristic functions f such that lim inf n (T) f D 0 and lim sup n (T) f D 1. Thus for lacunary sequences the ergodic theorem fails in the worst possible way. Bellow and Losert [21] gave the first example of a sequence S Z of density 0 which is universally good for pointwise convergence, answering a question posed by Furstenberg. They construct an S which is pointwise good in L1 . This paper also contains a good overview of the progress on weighted and subsequence ergodic theorems at that time. Weyl’s theorem on uniform distribution (Theorem 9) suggests the possibility of an ergodic theorem for the sequence fn2 g. It is not hard to see that fn2 g is mean good in L2 . In fact the spectral theorem and the dominated convergence theorem show that it is enough to prove that theL1 P n 2 on the unit bounded sequence of functions n1 n1 iD0 z circle converges at each point z of the unit circle. When z is not a root of unity the sequence converges to 0 by Weyl’s result and when z is a root of unity the convergence is 2 trivial because fz n g is periodic. In 1987 Bourgain [39,43] proved his celebrated pointwise ergodic theorem for polynomial subsequences. Theorem 33 If p is any polynomial with rational coefficients taking integer values on the integers then S D fp(n)g is pointwise good in L2 . The first step in Bourgain’s argument is to reduce the problem of proving a maximal inequality to the case of the shift map on the integers, via Calderón’s transfer princi-
253
254
Ergodic Theorems
ple. Then the problem is transferred to the circle by using Fourier transforms. At this point the problem becomes a very delicate question about exponential sums and a whole arsenal of tools is brought to bear. See Rosenblatt and Wierdl [176] and Quas and Wierdl [28](Appendix B) for nice expositions of Bourgain’s methods. Bourgain subsequently improved this to all Lp , p > 1 and also extended it to sequences f[q(n)]g where now q is an arbitrary real polynomial and [] denotes the greatest integer function. He also announced that his methods can be used to show that the sequence of primes is pointwise p 1C 3 good in Lp for any p > 2 . Wierdl [205] (1988) soon extended the result for primes to all p > 1. Theorem 34 The primes are pointwise good in Lp for p >1. It has remained a major open question for quite some time whether any of these results hold for p D 1. In 2005 there appeared a preprint of Mauldin and Buczolich [148], which remains unpublished, showing that polynomial sequences are L1 -universally bad. Another major result of Bourgain’s is the so-called return times theorem [44]. A simplification of Bourgain’s original proof was published jointly with Furstenberg, Katznelson and Ornstein as an appendix to an article [47] of Bourgain. To state it let us agree to say that a a sequence of complex numbers fa(n)gn0 has property P if the seP quence of complex measures n D n1 n1 iD0 a(i)ı i has property P, where ı i denotes the point mass at i. Theorem 35 (Bourgain) Suppose T is an automorphism of a probability space (X; B; ), 1 p; q 1 are conjugate exponents and f 2 L p (). Then for almost all x the sequence f f (T n x)g is pointwise good in Lq . Applying this to characteristic functions f D 1E one sees that the return time sequence fi > 0 : S i x 2 Eg is good for pointwise convergence in L1 . Theorem 32 is a very special case. It is also easy to see that Theorem 35 contains the Wiener–Wintner theorem. In 1998 Rudolph [179] proved a far-reaching generalization of the return times theorem using the technique of joinings. For an introduction to joinings see the article by de la Rue in this collection and also Thouvenot [198], Glasner [90] (2003) and Rudolph’s book [177]. Rudolph’s result concerns the convergence of multiple averages N1 k 1 XY f j (T jn x) N nD0
(32)
jD1
where each T j is an automorphism of a probability space (X j ; B j ; j ) and the f j are L1 functions. The point is that the convergence occurs whenever each x j 2 X 0j , sets of
measure one which may be chosen sequentially for j D 1; : : : ; k without knowing what T i or f i are for any i > j. He actually proves something stronger, namely he identifies an intrinsic property of a sequence fa i g, which he calls fully generic, such that the following hold. (a) The constant sequence {1} is fully generic. (b) If fa i g is fully generic then for any T and f 2 L1 the sequence a i f (T i x) is fully generic for almost all x. (c) Fully generic implies pointwise good in L1 . The definition of fully generic will not be quoted here as it is somewhat technical. For a proof of the basic return times theorem using joinings see Rudolph [178]. Assani, Lesigne and Rudolph [13] took a first step towards the multiple theorem, a Wiener–Wintner version of the return times theorem. Also Assani [11] independently gave a proof of Rudolph’s result in the case when all the T j are weakly mixing. Ornstein and Weiss [162] have proved the following version of the return times theorem for abstract discrete groups. As with Z, let us say that a sequence fa g g g2G of complex numbers has property P for fFn g if the sequence P n D jF1 j g2Fn a(g)ı g of complex measures has propn erty P. Theorem 36 Suppose that the increasing Følner sequence fFn g satisfies the Tempelman condition supn jFn1 Fn j/ S jFn j < 1 and Fn D G. If b 2 L1 then for a.a. x the sequence fb(Tg x)g is pointwise good in L1 for fFn g. Recently Demeter, Lacey, Tao and Thiele [67] have proved that the return times theorem remains valid for any 1 < p 1 and q 2. On the other hand Assani, Buczolich and Mauldin [14] (2005) showed that it fails for p D q D 1. Bellow, Jones and Rosenblatt have a series of papers [22,23,24,25] studying general weighted averages associated to a sequence n of probability measures on Z, and, in some cases, more general groups. The following are a few of their results. [23] is concerned with Z-actions and moving block averages given by n D m I n , where the I n are finite intervals and mI denotes normalized counting measure on I. They resolve the problem completely, obtaining a checkable necessary and sufficient condition for such a sequence to be pointwise good in L1 . [24] gives sufficient conditions on a sequence n for it to be pointwise good in Lp , p > 1, via properties of the Fourier transforms ˆ n . A particular consequence is P that if lim n!1 k2Z jn (k) n (k 1)j D 0 then fn g has a subsequence which is pointwise good in Lp ,
Ergodic Theorems
p > 1. In [25] they obtain convergence results for sequences n D n , the convolution powers of a probability measure . A consequence of one of their main results P is that if the expectation k2Z k(k) is zero, the second P moment k2Z k 2 (k) is finite and is aperiodic (its support is not contained in any proper coset in Z) then n is pointwise good in Lp for p > 1. Bellow and Calderón [19] later showed that this last result is valid also for p D 1. This is a consequence of the following sufficient condition for a sequence T to satisfy a weak L1 inequality. Given an automorphism of a probability space (X; B; ) let M f D sup jn (T) f (x)j be the maximal operator associated to fn g. Theorem 37 (Bellow and Calderón) Suppose there is an ˛ 2 (0; 1] and C > 0 such that for each n > 1 one has jn (x C y) n (x)j C
jyj˛ jxj1C˛
for all x; y 2 Z
Theorem 39 Suppose T is an automorphism of a probability space (X; B; ), (B) > 0 and k 1. Then there is T an n > 0 such that ( kiD1 T in B) > 0. In 1977, Furstenberg [85] proved the following ergodic theorem which implies the multiple recurrence theorem. He also established a general correspondence principle which puts the shaky analogy between the multiple recurrence theorem and Szemerédi’s theorem on a firm footing and allows each to be deduced from the other. Thus he obtained an ergodic theoretic proof of Szemerédi’s combinatorial result. Theorem 40 Suppose T is an automorphism of a probabilR ity space (X; B; ), f 2 L1 , f 0, f d > 0 and k 1. Then lim inf N
N1 Z k 1 X Y in T f d > 0 : N nD0
(34)
iD1
such that 0 < 2jyj jxj Then there is a constant D such that fM f > g
D k f k1
f 2 L1 ()
for all T;
and > 0 :
Ergodic Theorems and Multiple Recurrence Suppose S N. The upper density of S is ¯ D lim sup jS \ [1; n]j : d(S) n n
(33)
and the density d(S) is the limit of the same quantity, if it exists. In 1975 Szemerédi [190] proved the following celebrated theorem, answering an old question of Erd˝os and Turán. Theorem 38 Any subset of N with positive upper density contains an arithmetic progression of length k for each k 1. This result has a distinctly ergodic-theoretic flavor. Letting T denote the shift map on Z, it says that for each k there T is an n such that S 0 D kiD1 T i n S is non-empty. In fact ¯ 0 ) > 0. the result gives more: there is an n for which d(S In this light Szemerédi’s theorem becomes a multiple recurrence theorem for the shift map on N, equipped with ¯ Of course d¯ is not the invariant “measure-like” quantity d. even finitely additive so it is not a measure. d, however, is at least finitely additive, when defined, and d(N) D 1. This point of view suggests the following multiple recurrence theorem.
Furstenberg’s result opened the door to the study of socalled ergodic Ramsey theory which has yielded a vast array of deep results in combinatorics, many of which have no non-ergodic proof as yet. The focus of this article is not on this direction but the reader is referred to Furstenberg’s book [86] for an excellent introduction and to Bergelson [27,28] for surveys of later developments. There is also the article by Frantzikinakis and McCutcheon in this collection. Furstenberg’s proof relies on a deep structure theorem for a general automorphism which was also developed independently by Zimmer [213], [212] in a more general context. A factor of T is any sub- -algebra F B such that T(F ) D F . (It is more accurate to think of the factor as the action of T on the measure space (X; F ; jF ).) The structure theorem asserts that there is a transfinite increasing sequence of factors fF˛ g of T such that the following conditions hold. (a) F˛C1 is compact relative to F˛ . W (b) F˛ D ˇ 0 the set fn 2 Z : kT n f f k2 < g has bounded gaps and (34) follows easily. In the case when T is weakly mixing (34) is a consequence of the following theorem which Furstenberg proves in [85] (as a warm-up for it’s much harder relative version). Theorem 41 If T is weakly mixing and f1 ; f2 ; : : : ; f k are L1 functions then lim N
N1 Z k k Z Y 1 X Y in fi T f i d D N nD0 iD1
(35)
iD1
Later Bergelson [26] showed that the result can be obtained easily by an induction argument using the following Hilbert space generalization of van der Corput’s lemma. Theorem 42 (Bergelson) Suppose fx n g is a bounded sequence of vectors in Hilbert space such that for each h > 0 P N1 one has N1 nD0 hx nCh ; x n i ! 0 as N ! 1. Then P N1 1 k N nD0 x n k ! 0. Ryzhikov has also given a beautiful short proof of Theorem 41 using joinings ([182]). Bergelson’s van der Corput lemma and variants of it have been a key tool in subsequent developments in ergodic Ramsey theory and in the convergence results to be be discussed in this section. Bergelson [26] used it to prove the following mean ergodic theorem for weakly mixing automorphisms. Theorem 43 Suppose T is weakly mixing, f1 ; : : : ; f k are L1 functions and p1 ; : : : ; p k are polynomials with rational coefficients taking integer values on the integers such that no p i p j is constant for i ¤ j. Then N1 k k Z 1 XY Y p i (n) f i d D 0 : (36) T fi lim N!1 N nD0 iD1
iD1
Theorems 40 and 41 immediately raise the question of P N1 Q k in convergence of the multiple averages N1 nD0 iD1 T f i for a general T. Several authors obtained partial results on the question of mean convergence. It was finally resolved only recently by Host and Kra [104], who proved the following landmark theorem.
Theorem 44 Suppose f1 ; f2 ; : : : ; f k 2 L1 . Then there is a g 2 L1 such that N1 k 1 XY T in f i g D 0 : lim N nD0 iD1
(37)
2
Independently and somewhat later Ziegler [211]obtained the same result by somewhat different methods. Furstenberg had already established Theorem 44 for k D 2 in [85]. It was proved for k D 3 in the case of a totally ergodic T by Conze and Lesigne [63]and in general by Host and Kra [102]. It can also be obtained using the methods developed by Furstenberg and Weiss [88]. In this paper Furstenberg and Weiss proved a result for polynomial powers of T in the case k D 2. They also formalized the key notion of a characteristic factor. A factor C of T is said to be characteristic for the averages (37) if, roughly speaking, the L2 limiting behavior of the averages is unchanged when any one of the f i ’s is replaced by its conditional expectation on C . This means that the question of convergence of these averages may be reduced to the case when f i are all C -measurable. So the problem is to find the right (smallest) characteristic factor and prove convergence for that factor. The importance of characteristic factors was already apparent in Furstenberg’s original paper [85], where he showed that the maximal distal factor is characteristic for the averages (37). In fact he showed that for a given k a k-step distal factor is characteristic. (An automorphism is k-step distal if it is the top rung in a k-step ladder of factors as in the Furstenberg–Zimmer structure-theorem.) It turns out, though, that the right characteristic factor for (37) is considerably smaller. In their seminal paper [63] Conze and Lesigne identified the characteristic factor for k D 3, now called the Conze–Lesigne factor. As shown in [104], and [211], the characteristic factor for a general k is (isomorphic to) an inverse limit of k-step nilflows. A k-step nilflow is a compact homogeneous space N/ of a k-step nilpotent Lie group N, endowed with its unique left-invariant probability measure, on which T acts via left translation by an element of N. Ergodic properties of nilflows have been studied for some time in ergodic theory, for example in Parry [166]. In this way the problem of L2 -convergence of (37) is reduced to the case when T is a nilflow. In this case one has more: the averages converge pointwise by a result of Leibman [141] (See also Ziegler [210]). There have already been a good many generalizations of (37). Host and Kra [103], Frantzikinakis and Kra [82, 83], and Leibman [141] have proved results which replace
Ergodic Theorems
linear powers of T by polynomial powers. In increasing degrees of generality Conze and Lesigne [63], Frankzikinakis and Kra [81] and Tao [192] have obtained results which replace the maps T; T 2 ; : : : ; T k in (37) by commuting maps T1 ; : : : ; Tk . Bergelson and Leibman [30,31] have obtained results, both positive and negative, in the case of two noncommuting maps. In the direction of pointwise convergence the only general result is the following theorem of Bourgain [48] which asserts pointwise convergence in the case k D 2. Theorem 45 Suppose S and T are powers of a single autoPN f (T n x)g(S n x) morphism R and f ; g 2 L1 . Then N1 nD1 converges a.e. When T is a K-automorphism Derrien and Lesigne [68] have proved that the averages (35) converge pointwise to the product of the integrals, even with polynomial powers of T replacing the linear powers. Gowers [91] has given a new proof of Szemerédi’s theorem by purely finite methods using only harmonic analysis on Z n . His results give better quantitative estimates in the statement of finite versions of Szemerédi’s theorem. Although his proof contains no ergodic theory it is to some extent guided by Furstenberg’s approach. This section would be incomplete without mentioning the spectacular recent result of Green and Tao [92] on primes in arithmetic progression and the subsequent extensions of the Green-Tao theorem due to Tao [191] and Tao and Ziegler [193]. Rates of Convergence There are many results which say in various ways that, in general, there are no estimates for the rate of convergence of the averages A n f in Birkhoff’s theorem. For example there is the following result of Krengel [134]. Theorem 46 Suppose limn!1 c n D 1 and T is any ergodic automorphism of a probability space. Then there R is a bounded measurable f with f d D 0 such that lim sup c n A n f D 1 a.e. See Part 1 of Derriennic [69] for a selection of other results in this direction. In spite of these negative results one can obtain quantitative estimates by reformulating the ergodic theorem in various ways. Bishop [37] proved the following result which is purely finite and constructive in nature and evidently implies the a.e. convergence in Birkhoff’s theorem. If y D (y1 ; : : : ; y n ) is a finite sequence of real numbers and a < b, an upcrossing of y over [a; b] is a minimal integer interval [k; l] [1; n] satisfying y k < a and y l > b.
Theorem 47 Suppose T is an automorphism of the probability space (X; B; ). Let U(n; a; b; f ; x) be the number of upcrossings of the sequence A0 f (x); : : : ; A n f (x) over [a; b]. Then for every n fx : U(n; a; b; f ; x) > kg
k f k1 : (b a)k
(38)
Ivanov [107] has obtained the following stronger upcrossing inequality for an arbitrary positive measurable f , which also implies Birkhoff’s theorem. Theorem 48 For any positive measurable f and 0 < a kg
a k b
:
(39)
Note the exponential decay and the remarkable fact that the estimate does not depend on f . Ivanov has also obtained the following result (Theorem 23 in [121]) about fluctuations of A n f . An -fluctuation of a real sequence y D (y1 : : : y n ) is a minimal integer interval [k; l] satisfying jy k y l j . If f 2 L1 (R) let F( ; f ; x) be the number of -fluctuations of the sequence fA n f (x)g1 nD0 . Theorem 49 1
fx : F( ; f ; x) kg C
(log k) 2 k
(40)
where C is a constant depending only on k f k1 / . If f 2 L1 then fx : F( ; f ; x) kg AeBk ;
(41)
where A and B are constants depending only on k f k1 / . See Kachurovskii’s survey [121] of results on rates of convergence in the ergodic theorem and also the article of Jones et al. [115] for more results on oscillation type inequalities. Under special assumptions on f and T it is possible to give more precise results on speed of convergence. If the sequence T i f is independent then there is a vast literature in probability theory giving very precise results, for example the central limit theorem and the law of the iterated logarithm (see, for example, [50]). See the surveys of Derrienic [69] (2006) and of Merlevède, Peligrad and Utev [150] for results on the central limit theorem for dynamical systems.
257
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Ergodic Theorems for Non-amenable Groups Guivarch [95] was the first to prove an ergodic theorem for a general pair of non-commuting unitary operators. Much work has been done in the last 15 years on mean and pointwise theorems for non-amenable groups. See the extensive survey by Nevo [155]. Here is just one result of Nevo [154] as a sample. Suppose G is a discrete group and fn g is a sequence of probability measures on G. Say that fn g is mean ergodic if for any unitary representation of G on a Hilbert space H and any x 2 H one P has g2G n (g)(g)x converges in norm to the projection of x on the subspace of -invariant vectors. Say that fn g is pointwise ergodic if for any measure preserving action T of G on a probability space and f 2 L2 one has P g2G n (g)Tg f converges a.e. to the projection of f on the subspace of T-invariant functions. Theorem 50 Let G be the free group on k generators, k 1. Let n be the normalized counting measure on the set of elements whose word length (in terms of the generators and their inverses) is n. Let n D (n C nC1 )/2 and P n0 D n1 niD1 i . Then n and n0 are mean and pointwise ergodic but n is not mean ergodic. Future Directions This section will be devoted to a few open problems and some questions. Many of these have been suggested by my colleagues acknowledged in the introduction. The topic of convergence of Furstenberg’s multiple averages has seen some remarkable achievements in the last few years and will likely continue to be vital for some time to come. The question of pointwise convergence of multiple averages is completely open, beyond Bourgain’s result (Theorem 45). Even extending Bourgain’s result to three different powers of R or to any two commuting automorphisms S and T would be a very significant achievement. Another natural question is whether one has pointwise convergence of the averages of the sequence 2 f (T n (x))g(T n (x)), which are mean convergent by the result of Furstenberg and Weiss [88] (which is now subsumed in much more general results). A long-standing open problem relating to Akcoglu’s ergodic theorem (Theorem 23) for positive contractions of Lp is whether it can be extended to power-bounded operators T (this means that kTkn is bounded). It is also an open question whether it extends to non-positive contractions, excepting the case p D 2 where Burkholder [52] has shown that it fails. There are many natural questions in the area of subsequence and weighted ergodic theorems. For example,
which of Bourgain’s several pointwise theorems can be extended to L1 ? Are there other natural subsequences of an arithmetic character which have density 0 and for which an ergodic theorem is valid, either mean or pointwise, and in what Lp spaces might such theorems be valid? Are there higher dimensional analogues? Since lacunary sequences are bad, to have any sort of an pointwise theorem l n D log a nC1 must get close to 0 and for simplicity let us assume that lim l n D 0. How fast must the convergence be in order to get an ergodic theorem? Jones and Wierdl [113] have shown that if l n > 1 (log n) 2 C then the pointwise ergodic theorem fails in L2 while Jones, Lacey and Wierdl [116] have shown that an only slightly faster rate permits a sequence which is pointwise good in L2 . How well or badly does the ergodic theorem succeed or fail depending on the rate of convergence of ln ? In particular is there a (slow) rate which still guarantees strong sweeping out? [116] contains some interesting conjectures in this direction. There are also interesting questions concerning the mean and pointwise ergodic theorems for subsequences which are chosen randomly in some sense. See Bourgain [42] and [116] for some results in this direction. Again [116] contains some interesting conjectures along these lines. In a recent paper [32] Bergelson and Leibman prove some very interesting and surprising results about the distribution of generalized polynomials. A generalized polynomial is any function which can be built starting with polynomials in R[x] using the operations of addition, multiplication and taking the greatest integer. As a consequence they derive a generalization of von Neumann’s mean ergodic theorem to averages along generalized polynomial sequences. The following is a special case. Theorem 51 Suppose p is a generalized polynomial taking integer values on the integers and U is a unitary operator P p(i) x is norm convergent for all x 2 on H . Then n1 n1 iD0 U H. This begs the question: does one have pointwise convergence? If so, this would be a far-reaching generalization of Bourgain’s polynomial ergodic theorem. There are also lots of questions concerning the nature of Følner sequences fFn g in an amenable group which give a pointwise theorem. For example Lindenstrauss [145] has shown that in the lamplighter group, a semi-direct product L of Z with i2Z Z/2Z on which Z acts by the shift, there is no sequence satisfying Tempelman’s condition and that any fFn g satisfying the Shulman condition must grow super-exponentially. So, it is natural to ask for slower rates of growth. In particular, in any amenable group is there al-
Ergodic Theorems
ways a sequence fFn g which is pointwise good and grows at most exponentially? Can one do better either in general or in particular groups? Lindenstrauss’s theorem at least guarantees the existence of Følner sequences which are pointwise good in L1 but in particular groups there are often natural sequences L which one hopes might be good. For example in 1 iD1 Z one may take F n to be a cube based at 0 of sidelength ln and dimension dn (that is, all but the first dn co-ordinates are zero), where both sequences increase to 1. What conditions on ln and dn will give a good sequence? Note that no such sequence is regular in Tempelman’s sense. If d n D n then fl n g must be superexponential to ensure Shulman’s condition. Can one do better? What about l n D d n D n? Bibliography 1. Akcoglu M, Bellow A, Jones RL, Losert V, Reinhold–Larsson K, Wierdl M (1996) The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergodic Theory Dynam Systems 16(2):207–253 2. Akcoglu M, Jones RL, Rosenblatt JM (2000) The worst sums in ergodic theory. Michigan Math J 47(2):265–285 3. Akcoglu MA (1975) A pointwise ergodic theorem in Lp-spaces. Canad J Math 27(5):1075–1082 4. Akcoglu MA, Chacon RV (1965) A convexity theorem for positive operators. Z Wahrsch Verw Gebiete 3:328–332 (1965) 5. Akcoglu MA, Chacon RV (1970) A local ratio theorem. Canad J Math 22:545–552 6. Akcoglu MA, del Junco A (1975) Convergence of averages of point transformations. Proc Amer Math Soc 49:265–266 7. Akcoglu MA, Kopp PE (1977) Construction of dilations of positive Lp -contractions. Math Z 155(2):119–127 8. Akcoglu MA, Krengel U (1981) Ergodic theorems for superadditive processes. J Reine Angew Math 323:53–67 9. Akcoglu MA, Sucheston L (1978) A ratio ergodic theorem for superadditive processes. Z Wahrsch Verw Gebiete 44(4):269– 278 10. Alaoglu L, Birkhoff G (1939) General ergodic theorems. Proc Nat Acad Sci USA 25:628–630 11. Assani I (2000) Multiple return times theorems for weakly mixing systems. Ann Inst H Poincaré Probab Statist 36(2):153–165 12. Assani I (2003) Wiener–Wintner ergodic theorems. World Scientific Publishing Co. Inc., River Edge, NJ 13. Assani I, Lesigne E, Rudolph D (1995) Wiener–Wintner returntimes ergodic theorem. Israel J Math 92(1–3):375–395 14. Assani I, Buczolich Z, Mauldin RD (2005) An L1 counting problem in ergodic theory. J Anal Math 95:221–241 15. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. With the assistance of Markus L, Massey W and an appendix by Greenberg L Annals of Mathematics Studies, No 53. Princeton University Press, Princeton, NJ 16. Banach S (1993) Théorie des opérations linéaires. Éditions Jacques Gabay, Sceaux, reprint of the 1932 original 17. Bellow A (1983) On “bad universal” sequences in ergodic theory II. In: Belley JM, Dubois J, Morales P (eds) Measure theory and its applications. Lecture Notes in Math, vol 1033. Springer, Berlin, pp 74–78
18. Bellow A (1999) Transference principles in ergodic theory. In: Christ M, Kenig CE, Sadowsky C (eds) Harmonic analysis and partial differential equations, Chicago Lectures in Math. Univ Chicago Press, Chicago, pp 27–39 19. Bellow A, Calderón A (1999) A weak-type inequality for convolution products. In: Christ M, Kenig CE, Sadowsky C (eds) Harmonic analysis and partial differential equations, Chicago Lectures in Math. Univ Chicago Press, Chicago, pp 41–48 20. Bellow A, Jones R (eds) (1991) Almost everywhere convergence, II. Academic Press, Boston 21. Bellow A, Losert V (1985) The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences. Trans Amer Math Soc 288(1):307–345 22. Bellow A, Jones R, Rosenblatt J (1989) Almost everywhere convergence of powers. In: Edgar GA, Sucheston L (eds) Almost everywhere convergence. Academic, Boston, pp 99– 120 23. Bellow A, Jones R, Rosenblatt J (1990) Convergence for moving averages. Ergodic Theory Dynam Systems 10(1):43–62 24. Bellow A, Jones RL, Rosenblatt J (1992) Almost everywhere convergence of weighted averages. Math Ann 293(3):399– 426 25. Bellow A, Jones R, Rosenblatt J (1994) Almost everywhere convergence of convolution powers. Ergodic Theory Dynam Systems 14(3):415–432 26. Bergelson V (1987) Weakly mixing PET. Ergodic Theory Dynam Systems 7(3):337–349 27. Bergelson V (1996) Ergodic Ramsey theory—an update. In: Pollicott M, Schmidt K (eds) Ergodic theory of Z d actions. London Math Soc Lecture Note Ser, vol 228. Cambridge Univ Press, Cambridge, pp 1–61 28. Bergelson V (2006) Combinatorial and Diophantine applications of ergodic theory. In: Hasselblatt B, Katok A (eds) Handbook of dynamical systems, vol 1B, Appendix A by Leibman A, Appendix B by Quas A, Wierdl M. Elsevier, Amsterdam, pp 745–869 29. Bergelson V (2007) Some historical remarks and modern questions around the ergodic theorem. Internat Math Nachrichten 205:1–10 30. Bergelson V, Leibman A (2002) A nilpotent Roth theorem. Invent Math 147(2):429–470 31. Bergelson V, Leibman A (2004) Failure of the Roth theorem for solvable groups of exponential growth. Ergodic Theory Dynam Systems 24(1):45–53 32. Bergelson V, Leibman A (2007) Distribution of values of bounded generalized polynomials. Acta Math 198(2):155– 230 33. Berkson E, Bourgain J, Gillespie TA (1991) On the almost everywhere convergence of ergodic averages for powerbounded operators on Lp -subspaces. Integral Equ Operator Theory 14(5):678–715 34. Billingsley P (1965) Ergodic theory and information. Wiley, New York 35. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci USA 17:656–660 36. Bishop E (1966) An upcrossing inequality with applications. Michigan Math J 13:1–13 37. Bishop E (1967/1968) A constructive ergodic theorem. J Math Mech 17:631–639 38. Blum JR, Hanson DL (1960) On the mean ergodic theorem for subsequences. Bull Amer Math Soc 66:308–311
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Ergodic Theory: Fractal Geometry JÖRG SCHMELING Center for Mathematical Sciences, Lund University, Lund, Sweden Article Outline Glossary Definition of the Subject Introduction Preliminaries Brief Tour Through Some Examples Dimension Theory of Low-Dimensional Dynamical Systems – Young’s Dimension Formula Dimension Theory of Higher-Dimensional Dynamical Systems Hyperbolic Measures General Theory Endomorphisms Multifractal Analysis Future Directions Bibliography Glossary Dynamical system A (discrete time) dynamical system describes the time evolution of a point in phase space. More precisely a space X is given and the time evolution is given by a map T : X ! X. The main interest is to describe the asymptotic behavior of the trajectories (orbits) T n (x), i. e. the evolution of an initial point x 2 X under the iterates of the map T. More generally one is interested in obtaining information on the geometrically complicated invariant sets or measures which describe the asymptotic behavior. Fractal geometry Many objects of interest (invariant sets, invariant measures etc.) exhibit a complicated structure that is far from being smooth or regular. The aim of fractal geometry is to study those objects. One of the main tools is the fractal dimension theory that helps to extract important properties of geometrically “irregular” sets. Definition of the Subject The connection between fractal geometry and dynamical system theory is very diverse. There is no unified approach and many of the ideas arose from significant examples. Also the dynamical system theory has been shown to have a strong impact on classical fractal geometry. In this article there are first presented some examples showing nontrivial results coming from the application of dimension the-
ory. Some of these examples require a deeper knowledge of the theory of smooth dynamical systems then can be provided here. Nevertheless, the flavor of these examples can be understood. Then there is a brief overview of some of the most developed parts of the application of fractal geometry to dynamical system theory. Of course a rigorous and complete treatment of the theory cannot be given. The cautious reader may wish to check the original papers. Finally, there is an outlook over the most recent developments. This article is by no means meant to be complete. It is intended to give some of the ideas and results from this field. Introduction In this section some of the aspects of fractal geometry in dynamical systems are pointed out. Some notions that are used will be defined later on and I intend only to give a flavor of the applications. The nonfamiliar reader will find the definitions in the corresponding sections and can return to this section later. The geometry of many invariant sets or invariant measures of dynamical systems (including attractors, measures defining the statistical properties) look very complicated at all scales, and their geometry is impossible to describe using standard geometric tools. For some important classes of dynamical systems, these complicated structures are intensively studied using notions of dimension. In many cases it becomes possible to relate these notions of dimension to other fundamental dynamical characteristics, such as Lyapunov exponents, entropies, pressure, etc. On the other hand tools from dynamical systems, especially from ergodic theory and thermodynamic formalism, are extremely useful to explore the fractal properties of the objects in question. This includes dimensions of limit sets of geometric constructions (the standard Cantor set being the most famous example), which a priori, are not related to dynamical systems [46,100]. Many dimension formulas for asymptotic sets of dynamical systems are obtained by means of Bowen-type formulas, i. e. as roots of some functionals arising from thermodynamic formalism. The dimension of a set is a subtle characteristic which measures the geometric complexity of the set at arbitrarily fine scales. There are many notions of dimension, and most definitions involve a measurement of geometric complexity at scale " (which ignores the irregularities of the set at size less than ") and then considers the limiting measurement as " ! 0. A priori (and in general) these different notions can be different. An important result is the affirmative solution of the Eckmann–Ruelle conjecture by Barreira, Pesin and Schmeling [17], which says that for smooth nonuniformly hyperbolic systems, the pointwise
Ergodic Theory: Fractal Geometry
dimension is almost everywhere constant with respect to a hyperbolic measure. This result implies that many dimension characteristics for the measure coincide. The deep connection between dynamical systems and dimension theory seems to have been first discovered by Billingsley [21] through several problems in number theory. Another link between dynamical systems and dimension theory is through Pesin’s theory of dimension-like characteristics. This general theory is a unification of many notions of dimension along with many fundamental quantities in dynamical system such as entropies, pressure, etc. However, there are numerous examples of dynamical systems exhibiting pathological behavior with respect to fractal geometrical characteristics. In particular higher-dimensional systems seem to be as complicated as general objects considered in geometric measure theory. Therefore, a clean and unified theory is still not available. The study of characteristic notions like entropy, exponents or dimensions is an essential issue in the theory of dynamical systems. In many cases it helps to classify or to understand the dynamics. Most of these characteristics were introduced for different questions and concepts. For example, entropy was introduced to distinguish nonisomorphic systems and appeared to be a complete invariant for Bernoulli systems (Ornstein). Later, the thermodynamic formalism (see [107]) introduced new quantities like the pressure. Bowen [27] and also Ruelle discovered a remarkable connection between the thermodynamic formalism and the dimension theory for invariant sets. Since then many efforts were taken to find the relations between all these different quantities. It occurred that the dimension of invariant sets or measures carries lots of information about the system, combining its combinatorial complexity with its geometric complexity. Unfortunately it is extremely difficult to compute the dimension in general. The general flavor is that local divergence of orbits and global recurrence cause complicated global behavior (chaos). It is impossible to study the exact (infinite) trajectory of all orbits. One way out is to study the statistical properties of “typical” orbits by means of an invariant measure. Although the underlying system might be smooth the invariant measures may often be singular. Preliminaries Throughout the article the following situation is considered. Let M be a compact Riemannian manifold without boundary. On M is acting a dynamical system generated by a C 1C˛ diffeomorphism T : M ! M. The presence of a dynamical system provides several important additional tools and methods for the theory of fractal dimensions.
Also the theory of fractal dimensions allows one to draw deep conclusions about the dynamical system. The importance and relevance of the study of fractal dimension will be explained in later sections. In the next sections some of the most important tools in the fractal theory of dynamical systems are considered. The definitions given here are not necessarily the original definitions but rather the ones which are closer to contemporary use. More details can be found in [95]. Some Ergodic Theory Ergodic theory is a powerful method to analyze statistical properties of dynamical systems. All the following facts can be found in standard books on ergodic theory like [103,124]. The main idea in ergodic theory is to relate global quantities to observations along single orbits. Let us consider an invariant measure: ( f 1 A) D (A) for all measurable sets A. Such a measure “selects” typical trajectories. It is important to note that the properties vary with the invariant measures. Any such invariant measure can be decomposed into elementary parts (ergodic components). An invariant measure is called ergodic if for any invariant set A D T 1 A one has (A)(M n A) D 0 (with the agreement 0 1 D 0), i. e. from the measure-theoretic point of view there are no nontrivial invariant subsets. The importance of ergodic probability measures (i. e. (M) D 1) lies in the following theorem of Birkhoff Theorem 1 (Birkhoff) Let be an ergodic probability measure and ' 2 L1 (). Then Z n1 1X k '(T x) D ' d a.e : lim n!1 n M kD0
Hausdorff Dimension With Z R N and s 0 one defines
X diam(B i )s : mH (s; Z) D lim inf ı!0 fB i g
i
sup diam(B i ) < " and i
[
Bi Z :
i
Note that this limit exists. mH (s; Z) is called the s-dimensional outer Hausdorff measure of Z. It is immediate that there exists a unique value s , called the Hausdorff dimension of Z, at which mH (s; Z) jumps from 1 to 0. In general it is very hard to find optimal coverings and hence it is often impossible to compute the Hausdorff dimension of a set. Therefore a simpler notion – the lower
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and upper Box dimension – was introduced. The difference to the Hausdorff dimension is that the covering balls are assumed to have the same radius ". Since then the limit as " ! 0 does not have to exist one arrives at the notion of the upper and lower dimension. Dimension of a Measure Definition 1 Let Z R N and let be a probability measure supported on Z. Define the Hausdorff dimension of the measure by dimH ()
inf
K Z : (K)D1
dimH (K):
Pointwise Dimension Most invariant sets or measures are not strongly self-similar, i. e. the local geometry at arbitrarily fine scales might look different from point to point. Therefore, the notion of pointwise dimension with respect to a Borel probability measure is defined. Let be a Borel probability measure. By B(x; ") the ball with center x and radius " is denoted. The pointwise dimension of the measure at the point x is defined as d (x) :D lim
"!0
log (B(x; ")) log "
if the above limit exists. If d (x) D d, then for small " the measure of small balls scales as (B(x; ")) "d . Proposition 1 Suppose is a probability measure supported on Z R N . Then d (x) D d for – almost all x 2 Z implies dimH () D d. One should not take the existence of a local dimension (even for good measures) for granted. Later on it will be seen that the spectrum of the pointwise dimension (dimension spectrum) is a main object of study in classical multifractal analysis. The dimension of a measure or a set of measures is its geometric complexity. However, under the presence of a dynamical system one also wants to measure the dynamical (combinatorial) complexity of the system. This leads to the notion of entropy. Dimension-Like Characteristics and Topological Entropy Pesin’s theory of dimension-like characteristics provides a unified treatment of dimensions and important dynamical quantities like entropies and pressure. The topological entropy of a continuous map f with respect to a subset Z
in a metric space (X; ) (in particular X D M – a Riemannian manifold) can be defined as a dimension-like characteristic. For each n 2 N and " > 0, define the Bowen ball B n (x; ") D fy 2 X : (T i (x); T i (y)) " for 0 i ng. Then let m h (Z; ˛; e; n) :D ( ) X [ ˛n i e : n i > n; B n i (x; e) Z : lim inf n!1
i
i
This gives rise to an outer measure that jumps from 1 to 0 at some value ˛ . This threshold value ˛ is called the topological entropy of Z (at scale e). However, in many situations this value does not depend on e. The topological entropy is denoted by htop (TjZ). If Z is f – invariant and compact, this definition of topological entropy coincides with the usual definition of topological entropy [124]. ˚The entropy h of a measure is defined as h D inf htop (TjZ) : (Z) D 1 . For ergodic measures this definition coincides with the Kolmogorov–Sinai entropy (see [95]). One has to note that in the definition of entropy metric (“round”) balls are substituted by Bowen balls and the metric diameter by the “depth” of the Bowen ball. Therefore, the relation between entropy and dimension is determined by the relation of “round” balls to “oval” dynamical Bowen balls. If one understands how metric balls can be efficiently used to cover dynamical balls one can use the dynamical and relatively easy relation to compute notion of entropy to determine the dimension. However, in higher dimensions this relation is by far nontrivial. A heuristic argument for comparing “round” balls with dynamical balls is given in Subsect. “The Kaplan–Yorke Conjecture”. The Pressure Functional A useful tool in the dimension analysis of dynamical systems is the pressure functional. It was originally defined by means of statistical physics (thermodynamic formalism) as the free energy (or pressure) of a potential ' (see for example [107]). However, in this article a dimension-like definition (see [95]) is more suitable. Again an outer measure using Bowen balls will be used. Let ' : M ! R be a continuous function and ( X exp ˛n i m P (Z; ˛; "; n; ') D lim inf n!1
C
i
sup
n X
x2B n i (x;") kD0
) '(T x) : k
Ergodic Theory: Fractal Geometry
This defines an outer measure that jumps from 1 to 0 as ˛ increases.The threshold value ˛ is called the topological pressure of the potential ' denoted by P('). In many situations it does not depend on ". There is also a third way of defining the pressure in terms of a variational principle (see [124]): Z ' d P(') D sup h C invariant
M
Brief Tour Through Some Examples Before describing the fractal theory of dynamical systems in more detail some ideas about its role are presented. The application of dimension theory has many different aspects. At this point some (but by far not all) important examples are considered that should give the reader some feeling about the importance and wide use of dimension theory in dynamical system theory. Dimension of Conformal Repellers: Ruelle’s Pressure Formula Computing or estimating a dimension via a pressure formula is a fundamental technique. Explicit properties of pressure help to analyze subtle characteristics. For example, the smooth dependence of the Hausdorff dimension of basic sets for Axiom- A surface diffeomorphisms on the derivative of the map follows from smoothness of pressure. Ruelle proved the following pressure formula for the Hausdorff dimension of a conformal repeller. A conformal repeller J is an invariant set T(J) D J D fx 2 M : f n x 2 V8n 2 N and some neighborhood V of J} such that for any x 2 J the differential D x T D a(x) Isox where a(x) is a scalar with ja(x)j > 1 and Iso x an isometry of the tangent space Tx M. Theorem 2 ([108]) Let T : J ! J be a conformal repeller, and consider the function t ! P(t log jD x Tj), where P denotes pressure. Then P(s log jD x Tj) D 0
()
s D dimH (J)
Iterated Function Systems In fractal geometry one of the most-studied objects are iterated function systems, where there are given n contractions F1 ; ; Fn of Rd . The unique compact set J fulfilling S J D i F i (J) is called the attractor of the iterated function system (see [42]). One question is to evaluate its Hausdorff dimension. Often (for example under the open set condition) the attractor J can be represented as the repeller of a piecewise expanding map T : Rd ! Rd where the F i are
the inverse branches of the map T. In general it is by far not trivial to determine the dimension of J or even the dimension of a measure sitting on J. The following example explains some of those difficulties. Let 1/2 < < 1 and consider the maps F i : [0; 1] ! [0; 1] given by F1 (x) D x and F2 (x) D x C (1 ). Then the images of F1 ; F2 have an essential overlap and J D [0; 1]. If one randomizes this construction in the way that one applies both maps each with probability 1/2 a probability measure is induced on J. This measure might be absolute continuous with respect to Lebesgue measure or not. Already Erdös realized that for some special values of (for example for the inverse of the golden mean) the induced measure is singular. In a breakthrough paper B. Solomyak ([119]) proved that for a.e. the induced measure is absolutely continuous. A main ingredient in the proof is a transversality condition in the parameter space: the images of arbitrary two random samples of the (infinite) applications of the maps F i have to cross with nonzero speed when the parameter changes. This is a general mechanism which allows one to handle more general situations. Homoclinic Bifurcations for Dissipative Surface Diffeomorphisms Homoclinic tangencies and their bifurcations play a fundamental role in the theory of dynamical systems [87,88,89]. Systems with homoclinic tangencies have a complicated and subtle quasi-local behavior. Newhouse showed that homoclinic tangencies can persist under small perturbations, and that horseshoes may co-exist with infinitely many sinks in a neighborhood of the homoclinic orbit and hence the system is not hyperbolic (Newhouse phenomenon). Let T : M 2 ! M 2 be a smooth parameter family of surface diffeomorphisms that exhibits for D 0 an invariant hyperbolic set 0 (horseshoe) and undergoes a homoclinic bifurcation. The Hausdorff dimension of the hyperbolic set 0 for T 0 determines whether hyperbolicity is the typical dynamical phenomenon near T 0 or not. If dimH 0 < 1, then hyperbolicity is the prevalent dynamical phenomenon near f 0 . This is not the case if dimH 0 > 1. More precisely, let N W denote the set of nonwandering points of T in an open neighborhood of 0 after the homoclinic bifurcation. Let ` denote Lebesgue measure. Theorem 3 ([87,88]) If dimH 0 < 1, then lim
0 !0
`f 2 [0; 0 ] : N W is hyperbolicg D1: 0
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The hyperbolicity co-exists with the Newhouse phenomena for a residual set of parameter values. Theorem 4 (Palis–Yoccoz) If dimH 0 > 1, then `f 2 [0; 0 ] : N W is hyperbolicg < 1: 0 !0 0 lim
Some Applications to Number Theory Sometimes dimension problems in number theory can be transferred to dynamical systems and attacked using tools from dynamics. Example Diadic expansion of numbers: Consider a real number expanded in base 2, i. e. x D P1 n nD1 x n /2 . Let ( Xp D
) n1 1X x : lim xk D p : n!1 n kD0
1
xD
1
a1 C
1
a2 C a3 C
1 : a4 C : :
D [a1 ; a2 ; a3 ; a4 ; ] ; and the approximants p n (x)/q n (x) D [a1 ; a2 ; : : : ; a n ] (i. e. the approximation given by the finite continued fraction after step n). The set of numbers which admit a faster approximation by rational numbers is defined as follows. For 2, let F ˇ ˇ
ˇ pˇ 1 F D x 2 [0; 1] : ˇˇ x ˇˇ q q for infinitely many p/q :
Borel showed that `(X 1/2 ) D 1, where ` denotes Lebesgue measure. This result is an easy consequence of the Birkhoff ergodic theorem applied to the characteristic function of the digit 1 (which in this simple case is the Strong Law of Large Numbers for i.i.d. processes).
It is well known that this set has zero measure for each 2. Jarnik [58] computed the Hausdorff dimension of F and showed that dimH (F ) D 2/. However, nowadays there are methods from dynamical systems which not only allow unified proofs but also can handle more subtle subsets of the reals defined by some properties of their continued fraction expansion (see for example [3,105]).
One can ask how large is the set X p in general. Eggleston [39] discovered the following wonderful dimension formula, which Billingsley [21] interpreted in terms of dynamics and reproved using tools from ergodic theory.
Infinite Iterated Function Systems and Parabolic Systems
Theorem 5 ([39]) The Hausdorff dimension of X p is given by dimH (X p ) D
1 p log p (1 p) log(1 p) : log 2
The underlying dynamical system is E2 (x) D 2x mod 1 and the dimension is the dimension of the Bernoulli p measure. Other cases included that by Rényi who proposed generalization of the base d expansion from integer base d to noninteger base ˇ. In this case the underlying dynamical system is the (in general non-Markovian) beta shift ˇ(x) D ˇx mod 1 studied in [110]. There are many investigations concerning the approximation of real numbers by rationals by using dynamical methods. The underlying dynamical system in this case is the Gauss map T(x) D (1/x) mod 1. This map is uniformly expanding, but has infinitely many branches. Example Continued fraction approximation of numbers Consider the continued fraction expansion of x 2 [0; 1], i. e.,
In the previous section a system with infinitely many branches appeared. This can be regarded as an iterated function system with infinitely many maps F i . This situation is quite general. If one considers a (one-dimensional) system with a parabolic (indifferent) fixed point, i. e. there is a fixed point where the derivative has absolute value equal to 1, one often uses an induced system. For this one chooses nearby the parabolic point a higher iterate of the map in order to achieve uniform expansion away from the parabolic point. This leads to infinitely many branches since the number of iterates has to be increased the closer the parabolic point is. The main difference to the finite iterated function system is that the setting is no longer compact and many properties of the pressure functional are lost. Mauldin and Urba´nski and others (see for example [3,75,76,78]) developed a thermodynamic formalism adapted to the pressure functional for infinite iterated function systems. Besides noncompactness one of the main problems is the loss of analyticity (phase transitions) and convexity of the pressure functional for infinite iterated function systems.
Ergodic Theory: Fractal Geometry
Complex Dynamics Let T : C ! C be a polynomial and J its Julia set (repeller of this system). If this set is hyperbolic, i. e. the derivative at each point has absolute value larger than 1 the study of the dimension can be related to the study of a finite iterated function system. However, in the presence of a parabolic fixed point this leads to an infinite iterated function system. If one considers the coefficients of the polynomial as parameters one often sees a qualitative change in the asymptotic behavior. For example, the classical Mandelbrodt set for polynomials z2 C c is the locus of values c 2 C for which the orbit of the origin stays bounded. This set is well known to be fractal. However, its complete description is still not available. Embedology and Computational Aspects of Dimension Tools from dynamical systems are becoming increasingly important to study the time evolution of deterministic systems in engineering and the physical and biological sciences. One of the main ideas is to model a “real world” system by a smooth dynamical system which possesses a strange attractor with a natural ergodic invariant measure. When studying a complicated real world system, one can measure only a very small number of variables. The challenge is to reconstruct the underlying attractor from the time measurement of a scaler quantity. An idealized measurement is considered as a function h : M n ! R. The main tool researchers currently use to reconstruct the model system is called attractor reconstruction (see papers and references in [86]). This method is based on embedding with time delays (see the influential paper [33], where the authors attribute the idea of delay coordinates to Ruelle), where one attempts to reconstruct the attractor for the model using a single long trajectory. Then one considers the points in R pC1 defined by (x k ; x kC ; x kC2 ; : : : ; x kCp ). Takens [122] showed that for a smooth T : M n ! M n and for typical smooth h, the mapping ' : M n ! R2nC1 defined by x ! (h(x); h( f (x)); ; h( f 2n (x)) is an embedding. Since the box dimension of the attractor may be much less than the dimension of the ambient manifold n, an interesting mathematical question is whether there exists p < 2n C 1 such that the mapping on the attractor ' : ! R p defined by x ! (h(x); h( f (x)); : : : ; h( f p (x)) is 1 1? It is known that for a typical smooth h the mapping ' is 1 1 for p > 2 dimB () [29]. Denjoy Systems This section will give some ideas indicating the principle difficulties that arise in systems with low complexity.
Contrary to hyperbolic systems (each vector in the tangent space is either contracted or expanded) finer mechanisms determine the local behavior of the scaling of balls. While in hyperbolic systems the dynamical scaling of small balls is exponential in a low-complexity system this scaling is subexponential and hence the linearization error is of the same magnitude. Up to now there is no general dimension theory for low complexity systems. A specific example presented here is considered in [67]. Poincaré showed that to each orientation preserving homeomorphism of the circle S 1 D R/Z is associated a unique real parameter ˛ 2 [0; 1), called the rotation number, so that the ordered orbit structure of T is the same as that of the rigid rotation R˛ , where R˛ (t) D (t C ˛) mod 1, provided that ˛ is irrational. Half a century later, Denjoy [35] constructed examples of C1 diffeomorphisms that are not conjugate (via a homeomorphism) to rotations. This was improved later on by Herman [55]. In these examples, the minimal set of T is necessarily a Cantor set ˝. The arithmetic properties of the rotation number have a strong effect on the properties of T. One area that has been well understood is the relation between the differentiability of T, the differentiability of the conjugation and the arithmetic properties of the rotation number. (See, for example, Herman [55]) Without stating any precise theorem, the results differ sharply for Diophantine and for Liouville rotation numbers (definition follows). Roughly speaking the conjugating map is always regular for Diophantine rotation numbers while it might be not smooth at all for Liuoville rotation numbers. Definition 2 An irrational number ˛ is of Diophantine class D (˛) 2 RC if kq˛k
where k k denotes the distance to the nearest integer. In [67] the effect of the rotation number on the dimension of ˝ is studied. There the main result is Theorem 6 Assume that 0 < ı < 1 and that ˛ 2 (0; 1) is of Diophantine class 2 (0; 1). Then an orientation preserving C 1Cı diffeomorphism of the circle with rotation number ˛ and minimal set ˝˛ı satisfies dimH ˝˛ı
ı :
Furthermore, these results are sharp, i. e. the standard Denjoy examples attain the minimum.
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Return Times and Dimension Recently an interesting connection between the pointwise dimensions, multifractal analysis, and recurrence behavior of trajectories was discovered [1,11,23]. Roughly speaking, given an ergodic probability measure the return time asymptotics (as the neighborhood of the point shrinks) of -a.e point is determined by the pointwise dimension of at this point. The deeper understanding of this relation would help to get a unified approach to dimensions, exponents, entropies, recurrence times and correlation decay. Dimension Theory of Low-Dimensional Dynamical Systems – Young’s Dimension Formula In this section a remarkable extension of Ruelle’s dimension formula by Young [128] for the dimension of a measure is discussed. Theorem 7 Let T : M 2 ! M 2 be a C2 surface diffeomorphism and let be an ergodic measure. Then 1 1 dimH () D h ( f ) ; 1 2 where 1 2 are the two Lyapunov exponents for .
2) Number theoretic properties of some scaling rates [104, 106] enter into dimension calculations in ways they do not in low dimensions (see Subsect. “Iterated Function Systems”). 3) The dimension theory of sets is often reduced to the theory of invariant measures. However, there is no invariant measure of full dimension in general and measure-theoretic considerations do not apply [79]. 4) The stable and unstable foliations for higher dimensional systems are typically not C1 [51,109]. Hence, to split the system into an expanding and a contracting part is far more subtle. Dimension Theory of Higher-Dimensional Dynamical Systems Here an example of a hyperbolic attractor in dimension 3 is considered to highlight some of the difficulties. Let 4 denote the unit disc in R2 . Let f : S 1 4 ! S 1 4 be of the form T(t; x; y) D ('(t);
Theorem 8 Let be a basic set for a C2 Axiom-A surface diffeomorphism T : M 2 ! M 2 . Then dimH () D s1 C s2 , where s1 and s2 satisfy P(s log kDTx jE xu k) D 0 P(s log kDTx jE xs k) D 0 : where Es and Eu are the stable and unstable directions, respectively. Some Remarks on Dimension Theory for Low-Dimensional versus High-Dimensional Dynamical Systems Unlike lower dimensions (one, two, or conformal repellers), for higher-dimensional dynamical systems there are no general dimension formulas (besides the Ledrappier–Young formula), and in general dimension theory is much more difficult. This is due to several problems: 1) The geometry of the Bowen balls differs in a substantial way from round balls.
(t; x);
2
(t; y)) ;
with 0 < max
S 1 4
In [79], Manning and McCluskey prove the following dimension formula for a basic set (horseshoe) of an Axiom-A surface diffeomorphism which is a set-version of Young’s formula.
1
@ @x
1
(t; x) < min
S 1 4
@ @y
2
(t; y) < < 1 :
The limit set \ T n (S 1 4) :D n2N
is called the attractor or the solenoid. It is an example of a structurally stable basic set and is one of the fundamental examples of a uniformly hyperbolic attractor. The following result can be proved. Theorem 9 ([24,52]) For all t, the thermodynamic pressure ˇ ˇ ˇ@ 2 ˇ s ˇ (t; y)ˇˇ D 0 : P dimH t log ˇ @y In particular, the stable dimension is independent of the stable section. In this particular case the invariant axes for strong and weak contraction split the system smoothly and the difficulty is to show that the strong contraction is dominated by the weaker. In particular one has to ensure that effects as described in Subsect. “Iterated Function Systems” do not appear. In the general situation this is not the case and one lacks a similar theorem in the general situation. In particular, the unstable foliation is not better than Hölder and does not provide a “nice” coordinate.
Ergodic Theory: Fractal Geometry
Hyperbolic Measures
General Theory
Given x 2 M and a vector v 2 Tx M the Lyapunov exponent is defined as
In this section the dimension theory of higher-dimensional dynamical systems is investigated. Most developed is this theory for invariant measures. There is an important connection between Lyapunov exponents and the measure theoretic entropy and dimensions of measures that will be presented here. Let be an ergodic invariant measure. The Oseledec and Pesin Theory guarantee that local stable manifolds exist at -a.e. point. As for the Kaplan–Yorke formula the idea is to consider the contributions to the entropy and to the dimension in the directions of i . Historically the first connections between exponents and entropy were discovered by Margulis and Ruelle. They proved that
(x; v) D lim
n!1
1 log kD x T n vk : n
provided this limit exists. For fixed x, the numbers (x; ) attain only finitely many values. By ergodicity they are a.e. constant, and the corresponding values are denoted by 1 p ; where p D dim M. Denote by s (s stands for stable) the largest index such that s < 0. Definition 3 The invariant ergodic measure is said to be hyperbolic if i ¤ 0 for every i D 1, : : :, p. The Kaplan–Yorke Conjecture In this section a heuristic argument for the dimension of an invariant ergodic measure will be given. This argument uses a specific cover of a dynamical ball by “round” balls. These ideas are essentially developed by Kaplan and Yorke [61] for the estimation of the dimension of an invariant set. Their estimates always provide an upper bound for the dimension. Kaplan and Yorke conjectured that in typical situations these estimates also provide a lower bound for the dimension of an attractor. Ledrappier and Young [72,73] showed that in case this holds for an invariant measure, then this measure has to be very special: an SBR-measure (after Sinai, Ruelle, and Bowen) (an SRBmeasure is a measure that describes the asymptotic behavior for a set of initial points of positive Lebesgue measure and has absolutely continuous conditional measures on unstable manifolds). Consider a small ball B in the phase space. The image T n B is almost an ellipsoid with axes of length e1 n ; ; e p n : For 1 i s, cover T n B by balls of radius e i n . Then approximately exp[ p n] exp[ iC1 n] ::: : exp[ i n] exp[ i n] balls are needed for covering. The dimension can be estimated from above by P j>i j (1) C (p i) :D dimLi : dimB j i j This is the Kaplan–Yorke formula.
h ( f )
s X
i
iD1
for a C1 diffeomorphism T. Pesin [91] showed that this inequality is actually an equality if the measure is essentially the Riemannian volume on unstable manifolds. Ledrappier and Young [72] showed that this is indeed a necessary condition. They also provided an exact formula: Theorem 10 (Ledrappier–Young [72,73]) With d 0 D 0 for a C2 diffeomorphism holds h ( f ) D
s X
i d i d i1
iD1
where di are the dimensions of the (conditional) measure on the ith unstable leaves. The proof of this theorem is difficult and uses the theory of nonuniform hyperbolic systems (Pesin theory). In dimension 1 and 2 the above theorem resembles Ruelle’s and Young’s theorems. The reader should note that the above theorem includes also the existence of the pointwise dimension along the stable and unstable direction. Here the question arises whether this implies the existence of the pointwise dimension itself. The Existence of the Pointwise Dimension for Hyperbolic Measure – the Eckmann–Ruelle Conjecture In [17], Barreira, Pesin, and Schmeling prove that every hyperbolic measure has an almost local product structure, i. e., the measure of a small ball can be approximated by the product of the stable conditional measure of the stable component and the unstable conditional measure of
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the unstable component, up to small exponential error. This was used to prove the existence of the pointwise dimension of every hyperbolic measure almost everywhere. Moreover, the pointwise dimension is the sum of the contributions from its stable and unstable part. This implies that most dimension-type characteristics of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide, and provides a rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures. The existence of the pointwise dimension for hyperbolic measures had been conjectured long before by Eckmann and Ruelle. The hypotheses of this theorem are sharp. Ledrappier and Misiurewicz [70] constructed an example of a nonhyperbolic measure for which the pointwise dimension is not constant a.e. In [101], Pesin and Weiss present an example of a Hölder homeomorphism with Hölder constant arbitrarily close to one, where the pointwise dimension for the unique measure of maximal entropy does not exist a.e. There is also a one-dimensional example by Cutler [34].
Endomorphisms The previous section indicates that the dimensional properties of hyperbolic measures under invariant conditions for a diffeomorphism are understood. However, partial differential equations often generate only semi-flows and the corresponding dynamical system is noninvertible. Also, Poincaré sections sometimes introduce singularities. For such dynamical systems the theory of diffeomorphisms does not apply. However, the next theorem allows under some conditions application of this theory. It essentially rules out similar situations as considered in Subsect. “Iterated Function Systems”.
The Dynamical Characteristic View of Multifractal Analysis The aim of multifractal analysis is an attempt to understand the fine structure of the level sets of the fundamental asymptotic quantities in ergodic theory (e. g., Birkhoff averages, local entropy, Lyapunov exponents). For ergodic measures these quantities are a.e. constant, however may depend on the underlying ergodic measure. Important elements of multifractal analysis entail determining the range of values these characteristics attain, an analysis of the dimension of the level sets, and an understanding of the sets where the limits do not exist. A general concept of multifractal analysis was proposed by Barreira, Schmeling and Pesin [16]. An important field of applications of multifractal analysis is to describe sets of real numbers that have constraints on their digits or continued fraction expansion. General Multifractal Formalism In this section the abstract theory of multifractal analysis is described. Let X; Y be two measurable spaces and g : X n B ! Y be any measurable function where B is a measurable (possibly empty) subset of X (in the standard applications Y D R or Y D C). The associated multifractal decomposition of X is defined as [ g K˛ X DB[ ˛2Y
where g
K˛ :D fx 2 X : g(x) D ˛g For a given set function G : 2 X ! R the multifractal spectrum is defined by g
F (˛) :D G(K˛ ) :
Definition 4 A system (possibly with singularities) is almost surely invertible if it is invertible on a set of full measure. This implies that a full measure set of points has unique forward and backward trajectories.
At this point some classical and concrete examples of this general framework are considered.
Theorem 11 (Schmeling–Troubetzkoy [114]) A two-dimensional system with singularities is almost surely invertible (w.r.t. an SRB–measure) if and only if Young’s formula holds.
Let be an ergodic invariant measure for T : X ! X. If one sets
The Entropy Spectrum
g E (x) :D h (x) 2 Y D R and
Multifractal Analysis A group of physicists [50] suggested the idea of a multifractal analysis.
G E (Z) D htop (TjZ) the associated multifractal spectrum fE (˛) is called the entropy spectrum.
Ergodic Theory: Fractal Geometry
The Dimension Spectrum This is the classical multifractal spectrum. Let be an invariant ergodic measure on a complete separable metric space X. Set g D (x) :D d (x) 2 Y D R and G D (Z) D dimH Z : The associated multifractal spectrum fD (˛) is called the dimension spectrum. The Lyapunov Spectrum Let n1 1X g L (x) :D (x) D lim (T k x) 2 Y D R n!1 n kD0
where
(x) D log jD x Tj and
G D (Z) D dimH Z
or G E (Z) D htop (jZ) :
The associated multifractal spectra f LD (˛) and f LE (˛) are called Lyapunov spectra. It was observed by H. Weiss [126] and Barreira, Pesin and Schmeling [16] that for conformal repellers log 2 f LD (˛) D fDh (2) ˛ where the dimension spectrum on the right-hand side is with respect to the measure of maximal entropy, and f LE (˛) D fED (dimH ˛)
(3)
where the entropy spectrum on the right-hand side is with respect to the measure of maximal dimension. The following list summarizes the state of the art for the dynamical characteristic multifractal analysis of dynamical systems. The precise statements can be found in the original papers. [102,126] For conformal repellers and Axiom-A surface diffeomorphisms, a complete multifractal analysis exists for the Lyapunov exponent. [16,102,126] For mixing a subshift of finite type, a complete multifractal analysis exists for the Birkhoff average for a Hölder continuous potential and for the local entropy for a Gibbs measure with Hölder continuous potential.
[12,97] There is a complete multifractal analysis for hyperbolic flows. [123] There is a generalization of the multifractal analysis on subshifts with specification and continuous potentials. [13,19] There is an analysis of “mixed” spectra like the dimension spectrum of local entropies and also an analysis of joint level sets determined by more than one (measurable) function. [105] For the Gauss map (and a class of nonuniformly hyperbolic maps) a complete multifractal analysis exists for the Lyapunov exponent. [57] A general approach to multifractal analysis for repellers with countably many branches is developed. It shows in contrary to finitely many branches features of nonanalytic behavior. In the first three statements the multifractal spectra are analytic concave functions that can be computed by means of the Legendre transform of the pressure functional with respect to a suitable chosen family of potentials. In the remaining items this is no longer the case. Analyticity and convexity properties of the pressure functional are lost. However, the authors succeeded to provide a satisfactory theory in these cases. Multifractal Analysis and Large Deviation Theory There are deep connections between large deviation theory and multifractal analysis. The variational formula for pressure is an important tool in the analysis, and can be viewed (and proven) as a large deviation result [41]. Some authors use large deviation theory as a tool to effect multifractal analysis. Future Directions The dimension theory is fast developing and of great importance in the theory of dynamical systems. In the most ideal situations (low dimensions and hyperbolicity) a generally far reaching and powerful theory has been developed. It uses ideas from statistical physics, fractal geometry, probability theory and other fields. Unfortunately, the richness of this theory does not carry over to higher-dimensional systems. However, recent developments have shown that it is possible to obtain a general theory for the dimension of measures. Part of this theory is the development of the analytic tools of nonuniformly hyperbolic systems. Therefore, the dimension theory of dynamical systems is far from complete. In particular, it is usually difficult to apply the general theory to concrete examples, for instance
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if one really wants to compute the dimension. The general theory does not provide a way to compute the dimension but gives rather connections to other characteristics. Moreover, in the presence of neutral directions (zero Lyapunov exponents) one encounters all the difficulties arising in low-complexity systems. Another important open problem is to understand the dimension theory of invariant sets in higher-dimensional spaces. One way would be to relate the dimension of sets to the dimension of measures. Such a connection is not clear. The reason is that most systems do not exhibit a measure whose dimension coincides with the dimension of its support (invariant set). But there are some reasons to conjecture that any compact invariant set of an expanding map in any dimension carries a measure of maximal dimension (see [48,63]). If this conjecture is true one obtains an invariant measure whose unstable dimension coincides with the unstable dimension of the invariant set. There is also a measure of maximal stable dimension. Combining these two measures one could establish an analogous theory for invariant sets as for invariant measures. Last but not least one has to mention the impact of the dimension theory of dynamical systems on other fields. This new point of view makes in many cases the posed problems more tractable. This is illustrated in examples from number theory, geometric limit constructions and others. The applications of the dimension theory of dynamical systems to other questions seem to be unlimited.
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101. Pesin Y, Weiss H (1997) A Multifractal Analysis of Equilibrium Measures For Conformal Expanding Maps and Moran-like Geometric Constructions. J Stat Phys 86:233–275 102. Pesin Y, Weiss H (1997) The Multifractal Analysis of Gibbs Measures: Motivation. Mathematical Foundation and Examples. Chaos 7:89–106 103. Petersen K (1983) Ergodic theory. Cambridge studies in advanced mathematics 2. Cambridge Univ Press, Cambridge 104. Pollicott M, Weiss H (1994) The Dimensions Of Some Self Affine Limit Sets In The Plane And Hyperbolic Sets. J Stat Phys 77:841–866 105. Pollicott M, Weiss H (1999) Multifractal Analysis for the Continued Fraction and Manneville-Pomeau Transformations and Applications to Diophantine Approximation. Comm Math Phys 207(1):145–171 ´ 106. Przytycki F, Urbanski M (1989) On Hausdorff Dimension of Some Fractal Sets. Studia Math 93:155–167 107. Ruelle D (1978) Thermodynamic Formalism. Addison-Wesley 108. Ruelle D (1982) Repellers For Real Analytic Maps. Erg Th Dyn Syst 2:99–107 109. Schmeling J (1994) Hölder Continuity of the Holonomy Maps for Hyperbolic basic Sets II. Math Nachr 170:211–225 110. Schmeling J (1997) Symbolic Dynamics for Beta-shifts and Self-Normal Numbers. Erg Th Dyn Syst 17:675–694 111. Schmeling J (1998) A dimension formula for endomorphisms – the Belykh family. Erg Th Dyn Syst 18:1283–1309 112. Schmeling J (1999) On the Completeness of Multifractal Spectra. Erg Th Dyn Syst 19:1–22 113. Schmeling J (2001) Entropy Preservation under Markov Codings. J Stat Phys 104(3–4):799–815 114. Schmeling J, Troubetzkoy S (1998) Dimension and invertibility of hyperbolic endomorphisms with singularities. Erg Th Dyn Syst 18:1257–1282 115. Schmeling J, Weiss H (2001) Dimension theory and dynamics. AMS Proceedings of Symposia in Pure Mathematics series 69:429–488 116. Series C (1985) The Modular Surface and Continued Fractions. J Lond Math Soc 31:69–80 117. Simon K (1997) The Hausdorff dimension of the general Smale–Williams solenoidwith different contraction coefficients. Proc Am Math Soc 125:1221–1228 118. Simpelaere D (1994) Dimension Spectrum of Axiom-A Diffeomorphisms, II. Gibbs Measures. J Stat Phys 76:1359–1375 P 119. Solomyak B (1995) On the random series ˙n (an Erdös problem). Ann of Math (2) 142(3):611–625 120. Solomyak B (2004) Notes on Bernoulli convolutions. In: Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Proc Sympos Pure Math, 72, Part 1. Amer Math Soc, Providence, pp 207–230 121. Stratmann B (1995) Fractal Dimensions for Jarnik Limit Sets of Geometrically Finite Kleinian Groups; The Semi-Classical Approach. Ark Mat 33:385–403 122. Takens F (1981) Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics, vol 898. Springer 123. Takens F, Verbitzki E (1999) Multifractal analysis of local entropies for expansive homeomorphisms with specification. Comm Math Phys 203:593–612 124. Walters P (1982) Introduction to Ergodic Theory. Springer 125. Weiss H (1992) Some Variational Formulas for Hausdorff Di-
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Ergodic Theory: Non-singular Transformations ALEXANDRE I. DANILENKO1, CESAR E. SILVA 2 1 Institute for Low Temperature Physics & Engineering, Ukrainian National Academy of Sciences, Kharkov, Ukraine 2 Department of Mathematics, Williams College, Williamstown, USA Article Outline Glossary Definition of the Subject Basic Results Panorama of Examples Mixing Notions and multiple recurrence Topological Group Aut(X, ) Orbit Theory Smooth Nonsingular Transformations Spectral Theory for Nonsingular Systems Entropy and Other Invariants Nonsingular Joinings and Factors Applications. Connections with Other Fields Concluding Remarks Bibliography Glossary Nonsingular dynamical system Let (X; B; ) be a standard Borel space equipped with a -finite measure. A Borel map T : X ! X is a nonsingular transformation of X if for any N 2 B, (T 1 N) D 0 if and only if (N) D 0. In this case the measure is called quasi-invariant for T; and the quadruple (X; B; ; T) is called a nonsingular dynamical system. If (A) D (T 1 A) for all A 2 B then is said to be invariant under T or, equivalently, T is measure-preserving. Conservativeness T is conservative if for all sets A of positive measure there exists an integer n > 0 such that (A \ T n A) > 0. Ergodicity T is ergodic if every measurable subset A of X that is invariant under T (i. e., T 1 A D A) is either null or -conull. Equivalently, every Borel function f : X ! R such that f ı T D f is constant a. e. Types II, II1 , II1 and III Suppose that is non-atomic and T ergodic (and hence conservative). If there exists a -finite measure on B which is equivalent to and invariant under T then T is said to be of type II. It is easy to see that is unique up to scaling. If is finite then T is of type II 1 . If is infinite then T is of type II1 . If T is not of type II then T is said to be of type III.
Definition of the Subject An abstract measurable dynamical system consists of a set X (phase space) with a transformation T : X ! X (evolution law or time) and a finite or -finite measure on X that specifies a class of negligible subsets. Nonsingular ergodic theory studies systems where T respects in a weak sense: the transformation preserves only the class of negligible subsets but it may not preserve . This survey is about dynamics and invariants of nonsingular systems. Such systems model ‘non-equilibrium’ situations in which events that are impossible at some time remain impossible at any other time. Of course, the first question that arises is whether it is possible to find an equivalent invariant measure, i. e. pass to a hidden equilibrium without changing the negligible subsets? It turns out that there exist systems which do not admit an equivalent invariant finite or even -finite measure. They are of our primary interest here. In a way (Baire category) most of systems are like that. Nonsingular dynamical systems arise naturally in various fields of mathematics: topological and smooth dynamics, probability theory, random walks, theory of numbers, von Neumann algebras, unitary representations of groups, mathematical physics and so on. They also can appear in the study of probability preserving systems: some criteria of mild mixing and distality, a problem of Furstenberg on disjointness, etc. We briefly discuss this in Sect. “Applications. Connections with Other Fields”. Nonsingular ergodic theory studies all of them from a general point of view:
What is the qualitative nature of the dynamics? What are the orbits? Which properties are typical withing a class of systems? How do we find computable invariants to compare or distinguish various systems?
Typically there are two kinds of results: some are extensions to nonsingular systems of theorems for finite measure-preserving transformations (for instance, the entire Sect. “Basic Results”) and the other are about new properly ‘nonsingular’ phenomena (see Sect. “Mixing Notions and Multiple Recurrence” or Sect. “Orbit Theory”). Philosophically speaking, the dynamics of nonsingular systems is more diverse comparatively with their finite measurepreserving counterparts. That is why it is usually easier to construct counterexamples than to develop a general theory. Because of shortage of space we concentrate only on invertible transformations, and we have not included as many references as we had wished. Nonsingular endomorphisms and general group or semigroup actions are practically not considered here (with some exceptions in
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Sect. “Applications. Connections with Other Fields” devoted to applications). A number of open problems are scattered through the entire text. We thank J. Aaronson, J.R. Choksi, V.Ya. Golodets, M. Lema´nczyk, F. Parreau, E. Roy for useful remarks. Basic Results This section includes the basic results involving conservativeness and ergodicity as well as some direct nonsingular counterparts of the basic machinery from classic ergodic theory: mean and pointwise ergodic theorems, Rokhlin lemma, ergodic decomposition, generators, Glimm–Effros theorem and special representation of nonsingular flows. The historically first example of a transformation of type III (due to Ornstein) is also given here with full proof. Nonsingular Transformations In this paper we will consider only invertible nonsingular transformations, i. e. those which are bijections when restricted to an invariant Borel subset of full measure. Thus when we refer to a nonsingular dynamical system (X; B; ; T) we shall assume that T is an invertible nonsingular transformation. Of course, each measure on B which is equivalent to , i. e. and have the same null sets, is also quasi-invariant under T. In particular, since is -finite, T admits an equivalent quasi-invariant probability measure. For each i 2 Z, we denote by ! i or ! i the Radon–Nikodym derivative d( ı T i )/d 2 L1 (X; ). The derivatives satisfy the cocycle equation ! iC j (x) D ! i (x) ! j (T i x) for a. e. x and all i; j 2 Z. Basic Properties of Conservativeness and Ergodicity A measurable set W is said to be wandering if for all i; j 0 with i ¤ j, T i W \ T j W D ;. Clearly, if T has a wandering set of positive measure then it cannot be conservative. A nonsingular transformation T is incompressible if whenever T 1 C C, then (C n T 1 C) D 0. A set W of positive measure is said to be weakly wandering if there is a sequence n i ! 1 such that T n i W \ T n j W D ; for all i ¤ j. Clearly, a finite measure-preserving transformation cannot have a weakly wandering set. Hajian and Kakutani [83] showed that a nonsingular transformation T admits an equivalent finite invariant measure if and only if T does not have a weakly wandering set. Proposition 1 (see e. g. [123]) Let (X; B; ; T) be a nonsingular dynamical system. The following are equivalent: (i) T is conservative. S n A) D 0. (ii) For every measurable set A, (A n 1 nD1 T
(iii) T is incompressible. (iv) Every wandering set for T is null. Since any finite measure-preserving transformation is incompressible, we deduce that it is conservative. This is the statement of the classical Poincaré recurrence lemma. If T is a conservative nonsingular transformation of (X; B; ) and A 2 B a subset of positive measure, we can define an induced transformation T A of the space (A; B \ A; A) by setting TA x :D T n x if n D n(x) is the smallest natural number such that T n x 2 A. T A is also conservative. As shownRin [179], if (X) D 1 and T is conservative and erP !(x) d(x) D 1, which is a nonsingular godic, A n(x)1 iD0 version of the well-known Kaçs formula. Theorem 2 (Hopf Decomposition, see e. g. [3]) Let T be a nonsingular transformation. Then there exist disjoint invariant sets C; D 2 B such that X D C t D, T reF1 n stricted to C is conservative, and D D nD1 T W, 1 where W is a wandering set. If f 2 L (X; ), f > 0, Pn1 i then C D fx : iD0 f (T x) ! i (x) D 1 a. e.g and D D P n1 i fx : iD0 f (T x) ! i (x) < 1 a. e.g. The set C is called the conservative part of T and D is called the dissipative part of T. If T is ergodic and is non-atomic then T is automatically conservative. The translation by 1 on the group Z furnished with the counting measure is an example of an ergodic non-conservative (infinite measure-preserving) transformation. Proposition 4 Let (X; B; ; T) be a nonsingular dynamical system. The following are equivalent: (i) T is conservative and ergodic. (ii) For every set A of positive measure, (X n S1 n A) D 0. (In this case we will say A sweeps nD1 T out.) (iii) For every measurable set A of positive measure and for a. e. x 2 X there exists an integer n > 0 such that T n x 2 A. (iv) For all sets A and B of positive measure there exists an integer n > 0 such that (T n A \ B) > 0. (v) If A is such that T 1 A A, then (A) D 0 or (Ac ) D 0. This survey is mainly about systems of type III. For some time it was not quite obvious whether such systems exist at all. The historically first example was constructed by Ornstein in 1960. Example 5 (Ornstein [149]) Let A n D f0; 1; : : : ; ng, n (0) D 0:5 and n (i) D 1/(2n) for 0 < i n and all n 2 N. Denote by (X; ) the infinite product probability
Ergodic Theory: Non-singular Transformations
N space 1 nD1 (A n ; n ). Of course, is non-atomic. A point of X is an infinite sequence x D (x n )1 nD1 with x n 2 A n for all n. Given a1 2 A1 ; : : : ; a n 2 A n , we denote the cylinder fx D (x i )1 iD1 2 X : x1 D a1 ; : : : ; x n D a n g by [a1 ; : : : ; a n ]. Define a Borel map T : X ! X by setting 8 if i < l(x) ˆ l(x) ; xi ; where l(x) is the smallest number l such that x l ¤ l. It is easy to verify that T is a nonsingular transformation of (X; ) and 1 Y n ((Tx)n ) d ı T (x) D d n (x n ) nD1 8 < (l(x) 1)! ; if x l (x) D 0 l(x) D : (l(x) 1)! ; if x l (x) ¤ 0 :
We prove that T is of type III by contradiction. Suppose that there exists a T-invariant -finite measure equivalent to . Let ' :D d/d. Then
! i (x) D '(x) '(T i x)1 for a. a. x 2 X and all i 2 Z: (2) Fix a real C > 1 such that the set EC :D ' 1 ([C 1 ; C])
X is of positive measure. By a standard approximation argument, for each sufficiently large n, there is a cylinder [a1 ; : : : ;a n ] such that (EC \ [a1 ; : : : ; a n ]) > 0:9([a1 ; : : : ; a n ]). Since nC1 (0) D 0:5, it follows that (EC \ [a1 ; : : : ; a n ; 0]) > 0:8([a1 ; : : : ; a n ; 0]). Moreover, by the pigeon hole principle there is 0 < i n C 1 with (EC \ [a1 ; : : : ; a n ; i]) > 0:8([a1 ; : : : ; a n ; i]). Find N n > 0 such that T N n [a1 ; : : : ; a n ; 0] D [a1 ; : : : ; a n ; i]. Since ! Nn is constant on [a1 ; : : : ; a n ; 0], there is a subset E0 EC \ [a1 ; : : : ; a n ; 0] of positive measure such that T N n E0 EC \[a1 ; : : : ; a n ; i]. Moreover, ! Nn (x) D nC1 (i)/nC1 (0) D (n C 1)1 for a. a. x 2 [a1 ; : : : ; a n ; 0]. On the other hand, we deduce from (2) that ! N n (x) C 2 for all x 2 E0 , a contradiction. Mean and Pointwise Ergodic Theorems. Rokhlin Lemma Let (X; B; ; T) be a nonsingular dynamical system. Define a unitary operator U T of L2 (X; ) by setting s d( ı T) f ıT: (3) U T f :D d We note that U T preserves the cone of positive functions L2C (X; ). Conversely, every positive unitary operator in
L2 (X; ) that preserves L2C (X; ) equals U T for a -nonsingular transformation T. Theorem 6 (von Neumann mean Ergodic Theorem, see e. g. [3]) If T has no -absolutely continuous T-invariant P i probability, then n1 n1 iD0 U T ! 0 in the strong operator topology. Denote by I the sub--algebra of T-invariant sets. Let E [:jI ] stand for the conditional expectation with Rrespect to I . Note that if T is ergodic, then E [ f jI ] D f d. Now we state a nonsingular analogue of Birkhoff’s pointwise ergodic theorem, due to Hurewicz [105] and in the form stated by Halmos [84]. Theorem 7 (Hurewicz pointwise Ergodic Theorem) If T is conservative, (X) D 1, f ; g 2 L1 (X; ) and g > 0, then n1 X iD0 n1 X
f (T i x) ! i (x) ! g(T i x) ! i (x)
E [ f jI ] as n ! 1 for a. e. x : E [gjI ]
iD0
A transformation T is aperiodic if the T-orbit of a. e. point from X is infinite. The following classical statement can be deduced easily from Proposition 1. Lemma 8 (Rokhlin’s lemma [161]) Let T be an aperiodic nonsingular transformation. For each " > 0 and integer N > 1 there exists a measurable set A such that the sets A; TA; : : : ; T N1 A are disjoint and (A [ TA [ [ T N1 A) > 1 ". This lemma was refined later (for ergodic transformations) by Lehrer and Weiss as follows. Theorem 9 ("-free Rokhlin lemma [132]) Let T be ergodic and non-atomic. Then for a subset B X and any N S k N (X n B) D X, there is a set A such for which 1 kD0 T that the sets A; TA; : : : ; T N1 A are disjoint and A [ TA [ [ T N1 A B. S k N (X n B) D X holds of course The condition 1 kD0 T for each B ¤ X if T is totally ergodic, i. e. T p is ergodic for any p, or if N is prime. Ergodic Decomposition A proof of the following theorem may be found in [3]. Theorem 10 (Ergodic Decomposition Theorem) Let T be a conservative nonsingular transformation on a standard probability space (X; B; ). There there exists a standard probability space (Y; ; A) and a family of probability measures y on (X; B), for y 2 Y, such that
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(i)
For each A 2 B the map y 7! y (A) is Borel and for each A 2 B Z (A) D y (A) d(y) :
(ii) For y; y 0 2 Y the measures y and y 0 are mutually singular. (iii) For each y 2 Y the transformation T is nonsingular and conservative, ergodic on (X; B; y ). (iv) For each y 2 Y d y ı T d ı T y -a. e : D d d y (v) (Uniqueness) If there exists another probability space (Y 0 ; 0 ; A0 ) and a family of probability measures 0y 0 on (X; B), for y 0 2 Y 0 , satisfying (i)–(iv), then there exists a measure-preserving isomorphism : Y ! Y 0 such that y D 0 y for -a. e. y. It follows that if T preserves an equivalent -finite measure then the system (X; B; y ; T) is of type II for a. a. y. The space (Y; ; A) is called the space of T-ergodic components. Generators It was shown in [157,162] that a nonsingular transformation T on a standard probability space (X; B; ) has a countable generator, i. e. a countable partition P so that W1 n nD1 T P generates the measurable sets. It was refined by Krengel [126]: if T is of type II1 or III then there exists a generator P consisting of two sets only. Moreover, given a sub--algebra F B such that F
S T F and k>0 T k F D B, the set fA 2 F j (A; X n A) is a generator of Tg is dense in F . It follows, in particular, that T is isomorphic to the shift on f0; 1gZ equipped with a quasi-invariant probability measure. The Glimm–Effros Theorem The classical Bogoliouboff–Krylov theorem states that each homeomorphism of a compact space admits an ergodic invariant probability measure [33]. The following statement by Glimm [76] and Effros [61] is a “nonsingular” analogue of that theorem. (We consider here only a particular case of Z-actions.) Theorem 11 Let X be a Polish space and T : X ! X an aperiodic homeomorphism. Then the following are equivalent: T has a recurrent point x, i. e. x D lim n!1 T n i x for a sequence n1 < n2 < : (ii) There is an orbit of T which is not locally closed.
(i)
(iii) There is no a Borel set which intersects each orbit of T exactly once. (iv) There is a continuous probability Borel measure on X such that (X; ; T) is an ergodic nonsingular system. A natural question arises: under the conditions of the theorem how many such can exists? It turns out that there is a wealth of such measures. To state a corresponding result we first write an important definition. Definition 12 Two nonsingular systems (X; B; ; T) and (X; B0 ; 0 ; T 0 ) are called orbit equivalent if there is a oneto-one bi-measurable map ' : X ! X with 0 ı ' and such that ' maps the T-orbit of x onto the T 0 -orbit of '(x) for a. a. x 2 X. The following theorem was proved in [116,128] and [174]. Theorem 13 Let (X; T) be as in Theorem 11. Then for each ergodic dynamical system (Y; C ; ; S) of type II1 or III, there exist uncountably many mutually disjoint Borel measures on X such that (X; T; B; ) is orbit equivalent to (Y; C ; ; S). On the other hand, T may not have any finite invariant measure. Indeed, let T be an irrational rotation on the circle T and X a non-empty T-invariant Gı subset of T of full Lebesgue measure. Let (X; T) contain a recurrent point. Then the unique ergodicity of (T ; T) implies that (X; T) has no finite invariant measures. Let T be an aperiodic Borel transformation of a standard Borel space X. Denote by M(T) the set of all ergodic T-nonsingular continuous measures on X. Given 2 M(T), let N() denote the family of all Borel -null subsets. Shelah and Weiss showed [178] T that 2M(T) N() coincides with the collection of all Borel T-wandering sets. Special Representations of Ergodic Flows Nonsingular flows (D R-actions) appear naturally in the study of orbit equivalence for systems of type III (see Sect. “Orbit Theory”). Here we record some basic notions related to nonsingular flows. Let (X; B; ) be a standard Borel space with a -finite measure on B. A nonsingular flow on (X; ) is a Borel map S : XR 3 (x; t) 7! S t x 2 X such that S t Ss D S tCs for all s; t 2 R and each St is a nonsingular transformation of (X; ). Conservativeness and ergodicity for flows are defined in a similar way as for transformations. A very useful example of a flow is a flow built under a function. Let (X; B; ; T) be a nonsingular dynamical system and f a positive Borel function on X such that P1 P1 i i iD0 f (T x) D iD0 f (T x) D 1 for all x 2 X. Set
Ergodic Theory: Non-singular Transformations
X f :D f(x; s) : x 2 X; 0 s < f (x)g. Define f to be the restriction of the product measure Leb on X R to X f and define, for t 0, ! n1 X f n i S t (x; s) :D T x; s C t f (T x) ; iD0
where n is the unique integer that satisfies n1 X
f (T i x) < s C t
iD0
n X
If x ¤ (0; 0; : : : ), we let l(x) be the smallest number l such that the lth coordinate of x is not m l 1. We define a Borel map T : X ! X by (1) if x ¤ (m1 ; m2 ; : : : ) and put Tx :D (0; 0; : : : ) if x D (m1 ; m2 ; : : : ). Of course, T is isomorphic to a rotation on a compact monothetic totally disconnected Abelian group. It is easy to check that T is -nonsingular and 1 Y d ı T n ((Tx)n ) (x) D d n (x n ) nD1
f (T i x) :
iD0
A similar definition applies when t < 0. In particular, f when 0 < s C t < '(x), S t (x; s) D (x; s C t), so that the flow moves the point (x; s) up t units, and when it reaches (x; '(x)) it is sent to (Tx; 0). It can be shown that f S f D (S t ) t2R is a free f -nonsingular flow and that it pref serves if and only if T preserves [148]. It is called the flow built under the function ' with the base transformation T. Of course, S f is conservative or ergodic if and only if so is T. Two flows S D (S t ) t2R on (X; B; ) and V D (Vt ) t2R on (Y; C ; ) are said to be isomorphic if there exist invariant co-null sets X 0 X and Y 0 Y and an invertible nonsingular map : X 0 ! Y 0 that interwines the actions of the flows: ı S t D Vt ı on X 0 for all t. The following nonsingular version of Ambrose–Kakutani representation theorem was proved by Krengel [120] and Kubo [130]. Theorem 14 Let S be a free nonsingular flow. Then it is isomorphic to a flow built under a function. Rudolph showed that in the Ambrose–Kakutani theorem one can choose the function ' to take two values. Krengel [122] showed that this can also be assumed in the nonsingular case. Panorama of Examples This section is devoted entirely to examples of nonsingular systems. We describe here the most popular (and simple) constructions of nonsingular systems: odometers, nonsingular Markov odometers, tower transformations, rankone and finite rank systems and nonsingular Bernoulli shifts. Nonsingular Odometers Given a sequence mn of natural numbers, we let A n :Df0; 1;: : : ;m n 1g. Let n be a probability on An and n (a) > 0 for all a 2 A n . Consider now the infinite product N1 probability space (X; ) :D nD1 (A n ; n ). Assume that Q1 maxf (a) j a 2 A g D 0. Then is non-atomic. n n nD1 Given a1 2 A1 ; : : : ; a n 2 A n , we denote by [a1 ; : : : ; a n ] the cylinder x D (x i ) i>0 j x1 D a1 ; : : : ; x n D a n .
D
l (x)1 n (0) l (x) (x l (x) C 1) Y l (x) (x l (x) ) n (m n 1) nD1
for a. a. x D (x n )n>0 2 X. It is also easy to verify that T is ergodic. It is called the nonsingular odometer associated to (m n ; n )1 nD1 . We note that Ornstein’s transformation (Example 5) is a nonsingular odometer. Markov Odometers We define Markov odometers as in [54]. An ordered Bratteli diagram B [102] consists of (i)
a vertex set V which is a disjoint union of finite sets V (n) , n 0, V 0 is a singleton; (ii) an edge set E which is a disjoint union of finite sets E (n) , n > 0; (iii) source mappings s n : E (n) ! V (n1) and range mappings r n : E (n) ! V (n) such that s 1 n (v) ¤ ; for all (v) ¤ ; for all v 2 V (n) , n > 0; v 2 V (n1) and r1 n 0 (iv) a partial order on E so that e; e 2 E are comparable if and only if e; e 0 2 E (n) for some n and r n (e) D r n (e 0 ).
A Bratteli compactum X B of the diagram B is the space of infinite paths ˚ x D (x n )n>0 j x n 2 E (n) and r(x n ) D s(x nC1 ) on B. X B is equipped with the natural topology induced Q by the product topology on n>0 E (n) . We will assume always that the diagram is essentially simple, i. e. there is only one infinite path xmax D (x n )n>0 with xn maximal for all n and only one xmin D (x n )n>0 with xn minimal for all n. The Bratteli–Vershik map TB : X B ! X B is defined as follows: Txmax D xmin . If x D (x n )n>0 ¤ xmax then let k be the smallest number such that xk is not maximal. Let yk be a successor of xk . Let (y1 ; : : : ; y k ) be the unique path such that y1 ; : : : ; y k1 are all minimal. Then we let TB x :D (y1 ; : : : ; y k ; x kC1 ; x kC2 ; : : : ). It is easy to see that T B is a homeomorphism of X B . Suppose that we are given (n) a sequence P(n) D (P(v;e)2V n1 E (n) ) of stochastic matrices, i. e. (n) (i) Pv;e > 0 if and only if v D s n (e) and P (n) (n1) . (ii) f e2E (n) js n (e)Dv g Pv;e D 1 for each v 2 V
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For e1 2 E (1) ; : : : ; e n 2 E (n) , let [e1 ; : : : ; e n ] denote the cylinder fx D (x j ) j>0 j x1 D e1 ; : : : ; x n D e n g. Then we define a Markov measure on X B by setting P ([e1 ; : : : ; e n ]) D Ps11 (e 1 );e 1 Ps22 (e 2 );e 2 Psnn (e n );e n for each cylinder [e1 ; : : : ; e n ]. The dynamical system (X B ; P ; TB ) is called a Markov odometer. It is easy to see that every nonsingular odometer is a Markov odometer where the corresponding V (n) are all singletons. Tower Transformations This construction is a discrete analogue of flow under a function. Given a nonsingular dynamical system (X; ; T) and a measurable map f : X ! N, we define a new dynamical system (X f ; f ; T f ) by setting X f :D f(x; i) 2 X ZC j 0 i < f (x)g ; d f (x; i) :D d(x) and ( (x; i C 1); if i C 1 < f (x) f T (x; i) :D (Tx; 0); otherwise : Then T f is f -nonsingular and (d f ı T f /d f )(x; i) D (d ı T/d)(x) for a. a. (x; i) 2 X f . This transformation is called the (Kakutani) tower over T with height function f . It is easy to check that T f is conservative if and only if T is conservative; T f is ergodic if and only if T is ergodic; T f is of type III if and only if T is of type III. Moreover, the induced transformation (T f ) Xf0g is isomorphic to T. Given a subset A X of positive measure, T is the tower over the induced transformation T A with the first return time to A as the height function. Rank-One Transformations. Chacón Maps. Finite Rank The definition uses the process of “cutting and stacking”. We construct by induction a sequence of columns Cn . A column Cn consists of a finite sequence of bounded intervals (left-closed, right-open) C n D fI n;0 ; : : : ; I n;h n 1 g of height hn . A column Cn determines a column map TC n that sends each interval I n;i to the interval above it I n;iC1 by the unique orientation-preserving affine map between the intervals. TC n remains undefined on the top interval I n;h n 1 . Set C0 D f[0; 1)g and let fr n > 2g be a sequence of positive integers, let fs n g be a sequence of functions s n : f0; : : : ; r n 1g ! N0 , and let fw n g be a sequence of probability vectors on f0; : : : ; r n 1g. If Cn has been defined, column C nC1 is defined as follows. First “cut” (i. e., subdivide) each interval I n;i in Cn into rn subintervals I n;i [ j]; j D 0; : : : ; r n 1, whose lengths are in the proportions w n (0) : w n (1) : : w n (r n 1). Next place, for each
j D 0; : : : ; r n 1, s n ( j) new subintervals above I n;h n 1 [ j], all of the same length as I n;h n 1 [ j]. Denote these intervals, called spacers, by S n;0 [ j]; : : : ; S n;s n ( j)1 [ j]. This yields, for each j 2 f0; : : : ; r n 1g, rn subcolumns each consisting of the subintervals I n;0 [ j]; : : : ; I n;h n 1 [ j] followed by the spacers S n;0 [ j]; : : : ; S n;s n ( j)1 [ j] : Finally each subcolumn is stacked from left to right so that the top subinterval in subcolumn j is sent to the bottom subinterval in subcolumn j C 1, for j D 0; : : : ; r n 2 (by the unique orientation-preserving affine map between the intervals). For example, S n;s n (0)1 [0] is sent to I n;0 [1]. This defines a new column C nC1 and new column map TC nC1 , which remains undefined on its top subinterval. Let X be the union of all intervals in all columns and let be Lebesgue measure restricted to X. We assume that as n ! 1 the maximal length of the intervals in Cn converges to 0, so we may define a transformation T of (X; ) by Tx :D lim n!1 TC n x. One can verify that T is welldefined a. e. and that it is nonsingular and ergodic. T is said to be the rank-one transformation associated with (r n ; w n ; s n )1 nD1 . If all the probability vectors wn are uniform the resulting transformation is measure-preserving. The measure is infinite (-finite) if and only if the total mass of the spacers is infinite. In the case r n D 3 and s n (0) D s n (2) D 0, s n (1) D 1 for all n 0, the associated rank-one transformation is called a nonsingular Chacón map. It is easy to see that every nonsingular odometer is of rank-one (the corresponding maps sn are all trivial). Each rank-one map T is a tower over a nonsingular odometer (to obtain such an odometer reduce T to a column Cn ). A rank N transformation is defined in a similar way. A nonsingular transformation T is said to be of rank N or less if at each stage of its construction there exits N disjoint columns, the levels of the columns generate the -algebra and the Radon–Nikodym derivative of T is constant on each non-top level of every column. T is said to be of rank N if it is of rank N or less and not of rank N 1 or less. A rank N transformation, N 2, need not be ergodic. Nonsingular Bernoulli Transformations – Hamachi’s Example A nonsingular Bernoulli transformation is a transformation T such that there exists a countable generator P (see Subsect. “Generators”) such that the partitions T n P , n 2 Z, are mutually independent and such that the Radon– Nikodym derivative ! 1 is measurable with respect to the W sub--algebra 0nD1 T n P .
Ergodic Theory: Non-singular Transformations
In [87], Hamachi constructed examples of conservative nonsingular Bernoulli transformations, hence ergodic (see Subsect. “Weak Mixing, Mixing, K-Property”), with a 2-set generating partition that are of type III. Krengel [121] asked if there are of type II1 examples of nonsingular Bernoulli automorphisms and the question remains open. Hamachi’s construction is the left-shift on Q the space X D 1 nD1 f0; 1g. The measure is a product Q1 D nD1 n where n D (1/2; 1/2) for n 0 and for n < 0 n is chosen carefully alternating on large blocks between the uniform measure and different non-uniform measures. Kakutani’s criterion for equivalence of infinite product measures is used to verify that is nonsingular. Mixing Notions and multiple recurrence The study of mixing and multiple recurrence are central topics in classical ergodic theory [33,70]. Unfortunately, these notions are considerably less ‘smooth’ in the world of nonsingular systems. The very concepts of any kind of mixing and multiple recurrence are not well understood in view of their ambiguity. Below we discuss nonsingular systems possessing a surprising diversity of such properties that seem equivalent but are different indeed. Weak Mixing, Mixing, K-Property Let T be an ergodic conservative nonsingular transformation. A number 2 C is an L1 -eigenvalue for T if there exists a nonzero f 2 L1 so that f ı T D f a. e. It follows that jj D 1 and f has constant modulus, which we assume to be 1. Denote by e(T) the set of all L1 -eigenvalues of T. T is said to be weakly mixing if e(T) D f1g. We refer to Theorem 2.7.1 in [3] for proof of the following Keane’s ergodic multiplier theorem: given an ergodic probability preserving transformation S, the product transformation T S is ergodic if and only if S (e(T)) D 0, where S denotes the measure of (reduced) maximal spectral type of the unitary U S (see (3)). It follows that T is weakly mixing if and only T S is ergodic for every ergodic probability preserving S. While in the finite measure-preserving case this implies that T T is ergodic, it was shown in [5] that there exits a weakly mixing nonsingular T with T T not conservative, hence not ergodic. In [11], a weakly mixing T was constructed with T T conservative but not ergodic. A nonsingular transformation T is said to be doubly ergodic if for all sets of positive measure A and B there exists an integer n > 0 such that (A \ T n A) > 0 and (A \ T n B) > 0. Furstenberg [70] showed that for finite measure-preserving transformations double ergodicity is equivalent to weak mixing. In [20] it is shown that for nonsingular transformations weak mixing does not imply
double ergodicity and double ergodicity does not imply that T T is ergodic. T is said to have ergodic index k if the Cartesian product of k copies of T is ergodic but the product of k C 1 copies of T is not ergodic. If all finite Cartesian products of T are ergodic then T is said to have infinite ergodic index. Parry and Kakutani [113] constructed for each k 2 N [ f1g, an infinite Markov shift of ergodic index k. A stronger property is power weak mixing, which requires that for all nonzero integers k1 ; : : : ; kr the product T k 1 T k r is ergodic [47]. The following examples were constructed in [12,36,38]: (i) power weakly mixing rank-one transformations, (ii) non-power weakly mixing rank-one transformations with infinite ergodic index, (iii) non-power weakly mixing rank-one transformations with infinite ergodic index and such that T k 1 T k r are all conservative, k1 ; : : : ; kr 2 Z, of types II1 and III (and various subtypes of III, see Sect. “Orbit Theory”). Thus we have the following scale of properties (equivalent to weak mixing in the probability preserving case), where every next property is strictly stronger than the previous ones: T is weakly mixing ( T is doubly ergodic ( T T is ergodic ( T T T is ergodic ( ( T has infinite ergodic index ( T is power weakly mixing : We also mention a recent example of a power weakly mixing transformation of type II1 which embeds into a flow [46]. We now consider several attempts to generalize the notion of (strong) mixing. Given a sequence of measurable sets fA n g let k (fA n g) denote the -algebra generated by A k ; A kC1 ; : : : . A sequence fA n g is said to be remotely trivT ial if 1 kD0 k (fA n g) D f;; Xg mod , and it is semi-remotely trivial if every subsequence contains a subsequence that is remotely trivial. Krengel and Sucheston [124] define a nonsingular transformation T of a -finite measure space to be mixing if for every set A of finite measure the sequence fT n Ag is semi-remotely trivial, and completely mixing if fT n Ag is semi-remotely trivial for all measurable sets A. They show that T is completely mixing if and only if it is type II1 and mixing for the equivalent finite invariant measure. Thus there are no type III and II1 completely mixing nonsingular transformations on probability spaces. We note that this definition of mixing in infinite
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measure spaces depends on the choice of measure inside the equivalence class (but it is independent if we replace the measure by an equivalent measure with the same collection of sets of finite measure). Hajian and Kakutani showed [83] that an ergodic infinite measure-preserving transformation T is either of zero type: limn!1 (T n A \ A) D 0 for all sets A of finite measure, or of positive type: lim supn!1 (T n A \ A) > 0 for all sets A of finite positive measure. T is mixing if and only if it is of zero type [124]. For 0 ˛ 1 Kakutani suggested a related definition of ˛-type: an infinite measure preserving transformation is of ˛-type if lim supn!1 (A \ T n A) D ˛(A) for every subset A of finite measure. In [153] examples of ergodic transformations of any ˛-type and a transformation of not any type were constructed. It may seem that mixing is stronger than any kind of nonsingular weak mixing considered above. However, it is not the case: if T is a weakly mixing infinite measure preserving transformation of zero type and S is an ergodic probability preserving transformation then T S is ergodic and of zero type. On the other hand, the L1 -spectrum e(T S) is nontrivial, i. e. T S is not weakly mixing, whenever S is not weakly mixing. We also note that there exist rank-one infinite measure-preserving transformations T of zero type such that T T is not conservative (hence not ergodic) [11]. In contrast to that, if T is of positive type all of its finite Cartesian products are conservative [7]. Another result that suggests that there is no good definition of mixing in the nonsingular case was proved recently in [110]. It is shown there that while the mixing finite measure-preserving transformations are measurably sensitive, there exists no infinite measure-preserving system that is measurably sensitive. (Measurable sensitivity is a measurable version of the strong sensitive dependence on initial conditions – a concept from topological theory of chaos.) A nonsingular transformation T of (X; B; ) is called K-automorphism [180] if there exists a sub--algebra F
T WC1 B such that T 1 F F , k0 T k F D f;; Xg, kD0 T k F D B and the Radon–Nikodym derivative d ı T/ d is F -measurable (see also [156] for the case when T is of type II1 ; the authors in [180] required T to be conservative). Evidently, a nonsingular Bernoulli transformation (see Subsect. “Nonsingular Bernoulli Transformations – Hamachi’s Example”) is a K-automorphism. Parry [156] showed that a type II1 K-automorphism is either dissipative or ergodic. Krengel [121] proved the same for a class of Bernoulli nonsingular transformations, and finally Silva and Thieullen extended this result to nonsingular K-automorphisms [180]. It is also shown in [180] that if T is
a nonsingular K-automorphism, for any ergodic nonsingular transformation S, if S T is conservative, then it is ergodic. It follows that a conservative nonsingular K-automorphism is weakly mixing. However, it does not necessarily have infinite ergodic index [113]. Krengel and Sucheston [124] showed that an infinite measure-preserving conservative K-automorphism is mixing. Multiple and Polynomial Recurrence Let p be a positive integer. A nonsingular transformation T is called p-recurrent if for every subset B of positive measure there exists a positive integer k such that (B \ T k B \ \ T k p B) > 0 : If T is p-recurrent for any p > 0, then it is called multiply recurrent. It is easy to see that T is 1-recurrent if and only if it is conservative. T is called rigid if T n k ! Id for a sequence n k ! 1. Clearly, if T is rigid then it is multiply recurrent. Furstenberg showed [70] that every finite measure-preserving transformation is multiply recurrent. In contrast to that Eigen, Hajian and Halverson [64] constructed for any p 2 N [ f1g, a nonsingular odometer of type II1 which is p-recurrent but not (p C 1)-recurrent. Aaronson and Nakada showed in [7] that an infinite measure preserving Markov shift T is p-recurrent if and only if the product T T (p times) is conservative. It follows from this and [5] that in the class of ergodic Markov shifts infinite ergodic index implies multiple recurrence. However, in general this is not true. It was shown in [12,45] and [82] that for each p 2 N [ f1g there exist (i) power weakly mixing rank-one transformations and (ii) non-power weakly mixing rank-one transformations with infinite ergodic index which are p-recurrent but not (p C 1)-recurrent (the latter holds when p ¤ 1, of course). A subset A is called p-wandering if (A \ T k A \ pk \T A) D 0 for each k. Aaronson and Nakada established in [7] a p-analogue of Hopf decomposition (see Theorem 2). Proposition 15 If (X; B; ; T) is conservative aperiodic nonsingular dynamical system and p 2 N then X D C p [ D p , where Cp and Dp are T-invariant disjoint subsets, Dp is a countable union of p-wandering sets, T C p is p-reP k B \ \T d k B) D 1 for current and 1 kD1 (B \ T every B C p . Let T be an infinite measure-preserving transformation and let F be a -finite factor (i. e., invariant subalgebra)
Ergodic Theory: Non-singular Transformations
of T. Inoue [106] showed that for each p > 0, if T F is p-recurrent then so is T provided that the extension T ! T F is isometric. It is unknown yet whether the latter assumption can be dropped. However, partial progress was recently achieved in [140]: if T F is multiply recurrent then so is T. Let P :D fq 2 Q[t] j q(Z) Z and q(0) D 0g. An ergodic conservative nonsingular transformation T is called p-polynomially recurrent if for every q1 ; : : : ; q p 2 P and every subset B of positive measure there exists k 2 N with (B \ T q 1 (k) B \ \ T q p (k) B) > 0 : If T is p-polynomially recurrent for every p 2 N then it is called polynomially recurrent. Furstenberg’s theorem on multiple recurrence was significantly strengthened in [17], where it was shown that every finite measure-preserving transformation is polynomially recurrent. However, Danilenko and Silva [45] constructed (i)
nonsingular transformations T which are p-polynomially recurrent but not (p C 1)-polynomially recurrent (for each fixed p 2 N), (ii) polynomially recurrent transformations T of type II1 , (iii) rigid (and hence multiply recurrent) transformations T which are not polynomially recurrent. Moreover, such T can be chosen inside the class of rankone transformations with infinite ergodic index. Topological Group Aut(X, ) Let (X; B; ) be a standard probability space and let Aut(X; ) denote the group of all nonsingular transformations of X. Let be a finite or -finite measure equivalent to ; the subgroup of the -preserving transformations is denoted by Aut0 (X; ). Then Aut(X; ) is a simple group [62] and it has no outer automorphisms [63]. Ryzhikov showed [169] that every element of this group is a product of three involutions (i. e. transformations of order 2). Moreover, a nonsingular transformation is a product of two involutions if and only if it is conjugate to its inverse by an involution. Inspired by [85], Ionescu Tulcea [107] and Chacon and Friedman [21] introduced the weak and the uniform topologies respectively on Aut(X; ). The weak one – we denote it by dw – is induced from the weak operator topology on the group of unitary operators in L2 (X; ) by the embedding T 7! U T (see Subsect. “Mean and Pointwise Ergodic Theorems. Rokhlin Lemma”). Then (Aut(X; ); dw ) is a Polish topological group and Aut0 (X; ) is a closed subgroup of Aut(X; ).
This topology will not be affected if we replace with any equivalent measure. We note that T n weakly converges to T if and only if (Tn1 A T 1 A) ! 0 for each A 2 B and d( ı Tn )/d ! d( ı T)/d in L1 (X; ). Danilenko showed in [34] that (Aut(X; ); dw ) is contractible. It follows easily from the Rokhlin lemma that periodic transformations are dense in Aut(X; ). For each p 1, one can also embed Aut(X; ) into the isometry group of L p (X; ) via a formula similar to (3) but with another power of the Radon–Nikodym derivative in it. The strong operator topology on the isometry group induces the very same weak topology on Aut(X; ) for all p 1 [24]. It is natural to ask which properties of nonsingular transformations are typical in the sense of Baire category. The following technical lemma (see see [24,68]) is an indispensable tool when considering such problems. Lemma 16 The conjugacy class of each aperiodic transformation T is dense in Aut(X; ) endowed with the weak topology. Using this lemma and the Hurewicz ergodic theorem Choksi and Kakutani [24] proved that the ergodic transformations form a dense Gı in Aut(X; ). The same holds for the subgroup Aut0 (X; ) [24,170]. Combined with [107] the above implies that the ergodic transformations of type III is a dense Gı in Aut(X; ). For further refinement of this statement we refer to Sect. “Orbit Theory”. Since the map T 7! T T (p times) from Aut(X; ) to Aut(X p ; ˝p ) is continuous for each p > 0, we deduce that the set E1 of transformations with infinite ergodic index is a Gı in Aut(X; ). It is non-empty by [113]. Since this E1 is invariant under conjugacy, it is dense in Aut(X; ) by Lemma 16. Thus we obtain that E1 is a dense Gı . In a similar way one can show that E1 \ Aut0 (X; ) is a dense Gı in Aut0 (X; ) (see also [24,26,170] for original proofs of these claims). The rigid transformations form a dense Gı in Aut(X; ). It follows that the set of multiply recurrent nonsingular transformations is residual [13]. A finer result was established in [45]: the set of polynomially recurrent transformations is also residual. Given T 2 Aut(X; ), we denote the centralizer fS 2 Aut(X; ) j ST D T Sg of T by C(T). Of course, C(T) is a closed subgroup of Aut(X; ) and C(T) fT n j n 2 Zg. The following problems solved recently (by the efforts of many authors) for probability preserving systems are still open for the nonsingular case. Are the properties: (i) T has square root; (ii) T embeds into a flow;
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(iii) T has non-trivial invariant sub--algebra; (iv) C(T) contains a torus of arbitrary dimension typical (residual) in Aut(X; )? The uniform topology on Aut(X; ), finer than dw , is defined by the metric du (T; S) D (fx : Tx ¤ Sxg) C (fx : T
1
x¤S
1
xg) :
This topology is also complete metric. It depends only on the measure class of . However the uniform topology is not separable and that is why it is of less importance in ergodic theory. We refer to [21,24,27] and [68] for the properties of du . Orbit Theory Orbit theory is, in a sense, the most complete part of nonsingular ergodic theory. We present here the seminal Krieger’s theorem on orbit classification of ergodic nonsingular transformations in terms of ratio sets and associated flows. Examples of transformations of various types III , 0 1 are also given here. Next, we consider the outer conjugacy problem for automorphisms of the orbit equivalence relations. This problem is solved in terms of a simple complete system of invariants. We discuss also a general theory of cocycles (of nonsingular systems) taking values in locally compact Polish groups and present an important orbit classification theorem for cocycles. This theorem is an analogue of the aforementioned result of Krieger. We complete the section by considering ITPFIsystems and their relation to AT-flows. Full Groups. Ratio Set and Types III , 0 1 Let T be a nonsingular transformation of a standard probability space (X; B; ). Denote by Orb T (x) the T-orbit of x, i. e. OrbT (x) D fT n x j n 2 Zg. The full group [T] of T consists of all transformations S 2 Aut(X; ) such that Sx 2 OrbT (x) for a. a. x. If T is ergodic then [T] is topologically simple (or even algebraically simple if T is not of type II1 ) [62]. It is easy to see that [T] endowed with the uniform topology du is a Polish group. If T is ergodic then ([T]; du ) is contractible [34]. The ratio set r(T) of T was defined by Krieger [126] and as we shall see below it is the key concept in the orbit classification (see Definition 1). The ratio set is a subset of [0; C1) defined as follows: t 2 r(T) if and only if for every A 2 B of positive measure and each > 0 there is a subset B A of positive measure and an integer k ¤ 0 such that T k B A and j! k (x) tj < for all x 2 B. It is easy to verify that r(T) depends only on the equivalence class of and not on itself. A basic fact is that 1 2 r(T) if and
only if T is conservative. Assume now T to be conservative and ergodic. Then r(T) \ (0; C1) is a closed subgroup of the multiplicative group (0; C1). Hence r(T) is one of the following sets: (i) f1g; (ii) f0; 1g; in this case we say that T is of type III 0 , (iii) fn j n 2 Zg [ f0g for 0 < < 1; then we say that T is of type III , (iv) [0; C1); then we say that T is of type III 1 . Krieger showed that r(T) D f1g if and only if T is of type II. Hence we obtain a further subdivision of type III into subtypes III0 , III , or III1 . Example 17 (i) Fix 2 (0; 1). Let n (0) :D 1/(1 C ) and n (1) :D /(1 C ) for all n D 1; 2; : : : . Let T be the nonsingular odometer associated with the sequence (2; n )1 nD1 (see Subsect. “Nonsingular Odometers”). We claim that T is of type III . Indeed, the group ˙ of finite permutations of N acts on X by ( x)n D x 1 (n) , for all n 2 N, 2 ˙ and x D (x n )1 nD1 2 X. This action preserves . Moreover, it is ergodic by the Hewitt–Savage 0–1 law. It remains to notice that (d ı T/d)(x) D on the cylinder [0] which is of positive measure. (ii) Fix positive reals 1 and 2 such that log 1 and log 2 are rationally independent. Let n (0) :D 1/(1 C 1 C 2 ), n (1) :D 1 /(1 C 1 C 2 ) and n (2) :D 2 /(1 C 1 C
2 ) for all n D 1; 2; : : : . Then the nonsingular odometer associated with the sequence (3; n )1 nD1 is of type III1 . This can be shown in a similar way as (i). Non-singular odometer of type III0 will be constructed in Example 19 below. Maharam Extension, Associated Flow and Orbit Classification of Type III Systems On X R with the -finite measure , where d(y) D exp (y)dy, consider the transformation d ı T e T(x; y) :D Tx; y log (x) : d We call it the Maharam extension of T (see [136], where these transformations were introduced). It is measure-preserving and it commutes with the flow S t (x; y) :D (x; y C t), t 2 R. It is conservative if and only if T is conservative [136]. However e T is not necessarily ergodic. Let (Z; ) denote the space of e T-ergodic components. Then (S t ) t2R acts nonsingularly on this space. The restriction of (S t ) t2R to (Z; ) is called the associated flow of T. The associated flow is ergodic whenever T is ergodic. It is easy to verify
Ergodic Theory: Non-singular Transformations
that the isomorphism class of the associated flow is an invariant of the orbit equivalence of the underlying system. Proposition 18 ([90]) (i)
T is of type II if and only if its associated flow is the translation on R, i. e. x 7! x C t, x; t 2 R, (ii) T is of type III , 0 < 1 if and only if its associated flow is the periodic flow on the interval [0; log ), i. e. x 7! x C t mod ( log ), (iii) T is of type III1 if and only if its associated flow is the trivial flow on a singleton or, equivalently, e T is ergodic, (iv) T is of type III0 if and only if its associated flow is nontransitive.
We also note that every nontransitive ergodic flow can be realized as the associated flow of a type III0 transformation. However it is somewhat easier to construct a Z2 -action of type III0 whose associated flow is the given one. For this, we take an ergodic nonsingular transformation Q on a probability space (Z; B; ) and a measure-preserving transformation R of an infinite -finite measure space (Y; F ; ) such that there is a continuous homomorphism : R ! C(R) with (d ı (t)/d)(y) D exp (t) for a. a. y (for instance, take a type III1 transformation T and put R :D e T and (t) :D S t ). Let ' : Z ! R be a Borel map with inf Z ' > 0. Define two transformations R0 and Q0 of (Z Y; ) by setting: R0 (x; y) :D (x; Ry) ;
n
Example 19 Let A n D f0; 1; : : : ; 22 g and n (0) D 0:5 n and n (i) D 0:5 22 for all 0 < i 2n . Let T be the n nonsingular odometer associated with (22 C 1; n )1 nD0 . It is straightforward that the associated flow of T is the flow built under the constant function 1 with the probability preserving 2-adic odometer (associated with (2; n )1 nD1 , n (0) D n (1) D 0:5) as the base transformation. In particular, T is of type III0 . A natural problem arises: to compute Krieger’s type (or the ratio set) for the nonsingular odometers – the simplest class of nonsingular systems. Some partial progress was achieved in [56,141,152], etc. However in the general setting this problem remains open. The map : Aut(X; ) 3 T 7! e T 2 Aut(X R; ) is a continuous group homomorphism. Since the set E of ergodic transformations on X R is a Gı in Aut(X R; ) (see Sect. “Topological Group Aut(X, )”), the subset 1 (E ) of type III1 ergodic transformations on X is also Gı . The latter subset is non-empty in view of Example 17(ii). Since it is invariant under conjugacy, we deduce from Lemma 16 that the set of ergodic transformations of type III1 is a dense Gı in (Aut(X; ); dw ) [23,159]. Now we state the main result of this section – Krieger’s theorem on orbit classification for ergodic transformations of type III. It is a far reaching generalization of the basic result by H. Dye: any two ergodic probability preserving transformations are orbit equivalent [60]. Theorem 20 (Orbit equivalence for type III systems [125]–[129]) Two ergodic transformations of type III are orbit equivalent if and only if their associated flows are isomorphic. In particular, for a fixed 0 < 1, any two ergodic transformations of type III are orbit equivalent. The original proof of this theorem is rather complicated. Simpler treatment of it can be found in [90] and [117].
Q0 (x; y) D (Qx; U x y) ;
where U x D ('(x) log(d ı Q/d)(x)). Notice that R0 and Q0 commute. The corresponding Z2 -action generated by these transformations is ergodic. Take any transformation V 2 Aut(Z Y; ) whose orbits coincide with the orbits of the Z2 -action. (According to [29], any ergodic nonsingular action of any countable amenable group is orbit equivalent to a single transformation.) Then V is of type III0 . It is now easy to verify that the associated flow of V is the special flow built under ' ı Q 1 with the base transformation Q 1 . Since Q and ' are arbitrary, we deduce the following from Theorem 14. Theorem 21 Every nontransitive ergodic flow is an associated flow of an ergodic transformation of type III0 . In [129] Krieger introduced a map ˚ as follows. Let T be an ergodic transformation of type III0 . Then the associated flow of T is a flow built under function with a base transformation ˚(T). We note that the orbit equivalence class of ˚(T) is well defined by the orbit equivalent class of T. If ˚ n (T) fails to be of type III0 for some 1 n < 1 then T is said to belong to Krieger’s hierarchy. For instance, the transformation constructed in Example 19 belongs to Krieger’s hierarchy. Connes gave in [28] an example of T such that ˚(T) is orbit equivalent to T (see also [73] and [90]). Hence T is not in Krieger’s hierarchy. Normalizer of the Full Group. Outer Conjugacy Problem Let N[T] D fR 2 Aut(X; ) j R[T]R1 D [T]g ; i. e. N[T] is the normalizer of the full group [T] in Aut(X; ). We note that a transformation R belongs to N[T] if and only if R(Orb T (x)) D OrbT (Rx) for a. a. x.
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To define a topology on N[T] consider the T-orbit equivalence relation RT RX X and a -finite measure R on P RT given by R T D X y2OrbT (x) ı(x;y) d(x). For R 2 N[T], we define a transformation i(R) 2 Aut(RT ; RT ) by setting i(R)(x; y) :D (Rx; Ry). Then the map R 7! i(R) is an embedding of N[T] into Aut(RT ; RT ). Denote by the topology on N[T] induced by the weak topology on Aut(RT ; RT ) via i [34]. Then (N[T]; ) is a Polish group. A sequence Rn converges to R in N[T] if R n ! R weakly (in Aut(X; )) and R n TR1 ! RTR1 uniformly (in n [T]). Given R 2 N[T], denote by e R the Maharam extension of R. Then e R 2 N[e T] and it commutes with (S t ) t2R . Hence it defines a nonsingular transformation mod R on the space (Z; ) of the associated flow W D (Wt ) t2R of T. Moreover, mod R belongs to the centralizer C(W) of W in Aut(Z; ). Note that C(W) is a closed subgroup of (Aut(Z; ); dw ). Let T be of type II1 and let 0 be the invariant -finite measure equivalent to . If R 2 N[T] then it is easy to see that the Radon–Nikodym derivative d0 ı R/d0 is invariant under T. Hence it is constant, say c. Then mod R D log c. Theorem 22 ([86,90]) If T is of type III then the map mod : N[T] ! C(W) is a continuous onto homomorphism. The kernel of this homomorphism is the -closure of [T]. Hence the quotient group N[T]/[T] is (topologically) isomorphic to C(W). In particular, [T] is co-compact in N[T] if and only if W is a finite measure-preserving flow with a pure point spectrum. The following theorem describes the homotopical structure of normalizers. Theorem 23 ([34]) Let T be of type II or III , 0 < 1. The group [T] is contractible. N[T] is homotopically equivalent to C(W). In particular, N[T] is contractible if T is of type II. If T is of type III with 0 < < 1 then 1 (N[T]) D Z. The outer period p(R) of R 2 N[T] is the smallest positive integer n such that R n 2 [T]. We write p(R) D 0 if no such n exists. Two transformations R and R0 in N[T] are called outer conjugate if there are transformations V 2 N[T] and S 2 [T] such that V RV 1 D R0 S. The following theorem provides convenient (for verification) necessary and sufficient conditions for the outer conjugacy. Theorem 24 ([30] for type II and [18] for type III) Transformations R; R0 2 N[T] are outer conjugate if and only if p(R) D p(R0 ) and mod R is conjugate to mod R0 in the centralizer of the associated flow for T.
We note that in the case T is of type II, the second condition in the theorem is just mod R D mod R0 . It is always satisfied when T is of type II1 . Cocycles of Dynamical Systems. Weak Equivalence of Cocycles Let G be a locally compact Polish group and G a left Haar measure on G. A Borel map ' : X ! G is called a cocycle of T. Two cocycles ' and ' 0 are cohomologous if there is a Borel map b : X ! G such that ' 0 (x) D b(Tx)1 '(x)b(x) for a. a. x 2 X. A cocycle cohomologous to the trivial one is called a coboundary. Given a dense subgroup G 0 G, then every cocycle is cohomologous to a cocycle with values in G 0 [81]. Each cocycle ' extends to a (unique) map ˛' : RT ! G such that ˛' (Tx; x) D '(x) for a. a. x and ˛' (x; y)˛' (y; z) D ˛' (x; z) for a. a. (x; y); (y; z) 2 RT . ˛' is called the cocycle of RT generated by '. Moreover, ' and ' 0 are cohomologous via b as above if and only if ˛' and ˛' 0 are cohomologous via b, i. e. ˛' (x; y) D b(x)1 ˛' 0 (x; y)b(y) for RT -a. a. (x; y) 2 RT . The following notion was introduced by Golodets and Sinelshchikov [78,81]: two cocycles ' and ' 0 are weakly equivalent if there is a transformation R 2 N[T] such that the cocycles ˛' and ˛'0 ı (R R) of RT are cohomologous. Let M(X; G) denote the set of Borel maps from X to G. It is a Polish group when endowed with the topology of convergence in measure. Since T is ergodic, it is easy to deduce from Rokhlin’s lemma that the cohomology class of any cocycle is dense in M(X; G). Given ' 2 M(X; G), we define the '-skew product extension T' of T acting on (X G; G ) by setting T' (x; g) :D (Tx; '(x)g). Thus Maharam extension is (isomorphic to) the Radon– Nikodym cocycle-skew product extension. We now specify some basic classes of cocycles [19,35,81,173]: (i) ' is called transient if T' is of type I. (ii) ' is called recurrent if T' is conservative (equivalently, T' is not transient). (iii) ' has dense range in G if T' is ergodic. (iv) ' is called regular if ' cobounds with dense range into a closed subgroup H of G (then H is defined up to conjugacy). These properties are invariant under the cohomology and the weak equivalence. The Radon–Nikodym cocycle ! 1 is a coboundary if and only if T is of type II. It is regular if and only if T is of type II or III , 0 < 1. It has dense range (in the multiplicative group RC ) if and only if T is of type III1 . Notice that ! 1 is never transient (since T is conservative).
Ergodic Theory: Non-singular Transformations
Schmidt introduced in [176] an invariant R(') :D fg 2 G j ' g is recurrentg. He showed in particular that (i) (ii) (iii) (iv)
R(') is a cohomology invariant, R(') is a Borel set in G, R(log !1 ) D f0g for each aperiodic conservative T, there are cocycles ' such that R(') and G n R(') are dense in G, (v) if (X) D 1, Rı T D and ' : X ! R is integrable then R(') D f ' dg. We note that (v) follows from Atkinson theorem [15]. A nonsingular version of this theorem was established in [183]: if T is ergodic and -nonsingular and f 2 L1 () then ˇ ˇ n1 ˇ ˇX ˇ ˇ j f (T x)! j (x)ˇ D 0 for a. a. x lim inf ˇ n!1 ˇ ˇ jD0 R if and only if f d D 0. Since T' commutes with the action of G on X G by inverted right translations along the second coordinate, this action induces an ergodic G-action W' D (W' (g)) g2G on the space (Z; ) of T' -ergodic components. It is called the Mackey range (or Poincaré flow) of ' [66,135,173,188]. We note that ' is regular (and cobounds with dense range into H G) if and only if W' is transitive (and H is the stabilizer of a point z 2 Z, i. e. H D fg 2 G j W' (g)z D zg). Hence every cocycle taking values in a compact group is regular. It is often useful to consider the double cocycle '0 :D ' !1 instead of '. It takes values in the group G RC . Since T'0 is exactly the Maharam extension of T' , it follows from [136] that ' 0 is transient or recurrent if and only if ' is transient or recurrent respectively. Theorem 25 (Orbit classification of cocycles [81]) Let '; ' 0 : X ! G be two recurrent cocycles of an ergodic transformation T. They are weakly equivalent if and only if their Mackey ranges W'0 and W'00 are isomorphic. Another proof of this theorem was presented in [65]. Theorem 26 Let T be an ergodic nonsingular transformation. Then there is a cocycle of T with dense range in G if and only if G is amenable. It follows that if G is amenable then the subset of cocycles of T with dense range in G is a dense Gı in M(X; G) (just adapt the argument following Example 19). The ‘only if ’ part of Theorem 26 was established in [187]. The ‘if ’ part was considered by many authors in particular cases: G is compact [186], G is solvable or amenable almost connected [79], G is amenable unimodular [108], etc. The general case was proved in [78] and [100] (see also a recent treatment in [9]).
Theorem 21 is a particular case of the following result. Theorem 27 ([10,65,80]) Let G be amenable. Let V be an ergodic nonsingular action of G RC . Then there is an ergodic nonsingular transformation T and a recurrent cocycle ' of T with values in G such that V is isomorphic to the Mackey range of the double cocycle ' 0 . Given a cocycle ' 2 M(X; G) of T, we say that a transformation R 2 N[T] is compatible with ' if the cocycles ˛' and ˛' ı (R R) of RT are cohomologous. Denote by D(T; ') the group of all such R. It has a natural Polish topology which is stronger than [41]. Since [T] is a normal subgroup in D(T; '), one can consider the outer conjugacy equivalence relation inside D(T; '). It is called '-outer conjugacy. Suppose that G is Abelian. Then an analogue of Theorem 24 for the '-outer conjugacy is established in [41]. Also, the cocycles ' with D(T; ') D N[T] are described there. ITPFI Transformations and AT-Flows A nonsingular transformation T is called ITPFI 1 if it is orbit equivalent to a nonsingular odometer (associated to a sequence (m n ; n )1 nD1 , see Subsect. “Nonsingular Odometers”). If the sequence mn can be chosen bounded then T is called ITPFI of bounded type. If m n D 2 for all n then T is called ITPFI2 . By [74], every ITPFI-transformation of bounded type is ITPFI2 . A remarkable characterization of ITPFI transformations in terms of their associated flows was obtained by Connes and Woods [31]. We first single out a class of ergodic flows. A nonsingular flow V D (Vt ) t2R on a space (˝; ) is called approximate transitive (AT) if given > 0 and f1 ; : : : ; f n 2 L1C (X; ), there exists f 2 L1C (X; ) and 1 ; : : : ; n 2 L1C (R; dt) such that ˇˇ ˇˇ Z ˇˇ ˇˇ d ı Vt ˇˇ f j ˇˇ < f ı V (t)dt t j ˇˇ ˇˇ d R 1 for all 1 j n. A flow built under a constant ceiling function with a funny rank-one [67] probability preserving base transformation is AT [31]. In particular, each ergodic finite measure-preserving flow with a pure point spectrum is AT. Theorem 28 ([31]) An ergodic nonsingular transformation is ITPFI if and only if its associated flow is AT. The original proof of this theorem was given in the framework of von Neumann algebras theory. A simpler, purely 1 This abbreviates ‘infinite tensor product of factors of type I’ (came from the theory of von Neumann algebras).
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measure theoretical proof was given later in [96] (the ‘only if ’ part) and [88] (the ‘if ’ part). It follows from Theorem 28 that every ergodic flow with pure point spectrum is the associated flow of an ITPFI transformation. If the point spectrum of V is , where is a subgroup of Q and 2 R, then V is the associated flow of an ITPFT2 transformation [91]. Theorem 29 ([54]) Each ergodic nonsingular transformation is orbit equivalent to a Markov odometer (see Subsect. “Markov Odometers”). The existence of non-ITPFI transformations and ITPFI transformations of unbounded type was shown in [127]. In [55], an explicit example of a non-ITPFI Markov odometer was constructed. Smooth Nonsingular Transformations Diffeomorphisms of smooth manifolds equipped with smooth measures are commonly considered as physically natural examples of dynamical systems. Therefore the construction of smooth models for various dynamical properties is a well established problem of the modern (probability preserving) ergodic theory. Unfortunately, the corresponding ‘nonsingular’ counterpart of this problem is almost unexplored. We survey here several interesting facts related to the topic. r For r 2 N [ f1g, denote by DiffC (T ) the group of orientation preserving Cr -diffeomorphisms of the circle T . Endow this set with the natural Polish topology. Fix r T 2 DiffC (T ). Since T D R/Z, there exists a C1 -function f : R ! R such that T(x C Z) D f (x) C Z for all x 2 R. The rotation number (T) of T is the limit lim ( f ı ı f )(x)(mod 1) : „ ƒ‚ …
n!1
n times
The limit exists and does not depend on the choice of x and f . It is obvious that T is nonsingular with respect to r (T ) and Lebesgue measure T . Moreover, if T 2 DiffC
(T) is irrational then the dynamical system (T ; T ; T) is ergodic [33]. It is interesting to ask: which Krieger’s type can such systems have? Katznelson showed in [114] that the subset of type III C 1 -diffeomorphisms and the subset of type II1 C 1 -dif1 (T ). Hawkins and feomorphisms are dense in DiffC Schmidt refined the idea of Katznelson from [114] to construct, for every irrational number ˛ 2 [0; 1) which is not of constant type (i. e. in whose continued fraction expansion the denominators are not bounded) a transformation 2 T 2 DiffC (T ) which is of type III1 and (T) D ˛ [97]. It should be mentioned that class C2 in the construction
is essential, since it follows from a remarkable result of 3 Herman that if T 2 DiffC (T ) then under some condition on ˛ (which determines a set of full Lebesgue measure), T is measure theoretically (and topologically) conjugate to a rotation by (T) [101]. Hence T is of type II1 . In [94], Hawkins shows that every smooth paracompact manifold of dimension 3 admits a type III diffeomorphism for every 2 [0; 1]. This extends a result of Herman [100] on the existence of type III1 diffeomorphisms in the same circumstances. It is also of interest to ask: which free ergodic flows are associated with smooth dynamical systems of type III0 ? Hawkins proved that any free ergodic C 1 -flow on a smooth, connected, paracompact manifold is the associated flow for a C 1 -diffeomorphism on another manifold (of higher dimension) [95]. 2 (T ) A nice result was obtained in [115]: if T 2 DiffC and the rotation number of T has unbounded continued fraction coefficients then (T ; T ; T) is ITPFI. Moreover, a converse also holds: given a nonsingular odometer R, the set of orientation-preserving C 1 -diffeomorphisms of the circle which are orbit equivalent to R is C 1 -dense in the Polish set of all C 1 -orientation-preserving diffeomorphisms with irrational rotation numbers. In contrast to that, Hawkins constructs in [93] a type III0 C 1 -diffeomorphism of the 4-dimensional torus which is not ITPFI. Spectral Theory for Nonsingular Systems While the spectral theory for probability preserving systems is developed in depth, the spectral theory of nonsingular systems is still in its infancy. We discuss below some problems related to L1 -spectrum which may be regarded as an analogue of the discrete spectrum. We also include results on computation of the maximal spectral type of the ‘nonsingular’ Koopman operator for rank-one nonsingular transformations. L1 -Spectrum and Groups of Quasi-Invariance Let T be an ergodic nonsingular transformation of (X; B; ). A number 2 T belongs to the L1 -spectrum e(T) of T if there is a function f 2 L1 (X; ) with f ı T D f . f is called an L1 -eigenfunction of T corresponding to . Denote by E (T) the group of all L1 -eigenfunctions of absolute value 1. It is a Polish group when endowed with the topology of converges in measure. If T is of type II1 then the L1 -eigenfunctions are L2 (0 )-eigenfunctions of T, where 0 is an equivalent invariant probability measure. Hence e(T) is countable. Osikawa constructed in [151] the first examples of ergodic nonsingular transformations with uncountable e(T).
Ergodic Theory: Non-singular Transformations
We state now a nonsingular version of the von Neumann–Halmos discrete spectrum theorem. Let Q T be a countable infinite subgroup. Let K be a compact dual of Qd , where Qd denotes Q with the discrete topology. Let k0 2 K be the element defined by k0 (q) D q for all q 2 Q. Let R : K ! K be defined by Rk D k C k0 . The system (K; R) is called a compact group rotation. The following theorem was proved in [6]. Theorem 30 Assume that the L1 -eigenfunctions of T generate the entire -algebra B. Then T is isomorphic to a compact group rotation equipped with an ergodic quasi-invariant measure. A natural question arises: which subgroups of T can appear as e(T) for an ergodic T? Theorem 31 ([1,143]) e(T) is a Borel subset of T and carries a unique Polish topology which is stronger than the usual topology on T . The Borel structure of e(T) under this topology agrees with the Borel structure inherited from T . There is a Borel map : e(T) 3 7! 2 E (T) such that ıT D for each . Moreover, e(T) is of Lebesgue measure 0 and it can have an arbitrary Hausdorff dimension. A proper Borel subgroup E of T is called (i) weak Dirichlet if lim supn!1 b (n) D 1 for each finite complex measure supported on E; (n)j j(E)j for each fi(ii) saturated if lim supn!1 jb nite complex measure on T , where b (n) denote the nth Fourier coefficient of . Every countable subgroup of T is saturated. Theorem 32 e(T) is -compact in the usual topology on T [104] and saturated [104,139]. It follows that e(T) is weak Dirichlet (this fact was established earlier in [175]). It is not known if every Polish group continuously embedded in T as a -compact saturated group is the eigenvalue group of some ergodic nonsingular transformation. This is the case for the so-called H 2 -groups and the groups of quasi-invariance of measures on T (see below). Given a sequence nj of positive integers and a sequence a j 0, P nj 2 the set of all z 2 T such that 1 jD1 a j j1 z j < 1 is a group. It is called an H 2 -group. Every H 2 -group is Polish in an intrinsic topology stronger than the usual circle topology. Theorem 33 ([104]) (i)
Every H 2 -group is a saturated (and hence weak Dirichlet) -compact subset of T .
P (ii) If 1 jD0 a j D C1 then the corresponding H 2 -group is a proper subgroup of T . P 2 (iii) If 1 jD0 a j (n j /n jC1 ) < 1 then the corresponding H 2 -group is uncountable. (iv) Any H 2 -group is e(T) for an ergodic nonsingular compact group rotation T. It is an open problem whether every eigenvalue group e(T) is an H 2 -group. It is known however that e(T) is close ‘to be an H 2 -group’: if a compact subset L T is disjoint from e(T) then there is an H 2 -group containing e(T) and disjoint from L. Example 34 ([6], see also [151]) Let (X; ; T) be the nonsingular odometer associated to a sequence (2; j )1 jD1 . Let nj be a sequence of positive integers such that n j > P n for all j. For x 2 X, we put h(x) :D n l (x) P i< j i j 0 then either H() is countable or is equivalent to T [137]. Theorem 35 ([6]) Let be an ergodic with respect to the H()-action by translations on T . Then there is a compact group rotation (K; R) and a finite measure on K quasiinvariant and ergodic under R such that e(R) D H(). Moreover, there is a continuous one-to-one homomorphism : e(R) ! E(R) such that ı R D for all 2 e(R). It was shown by Aaronson and Nadkarni [6] that if n1 D 1 and n j D a j a j1 a1 for positive integers a j 2 P 1 with 1 jD1 a j < 1 then the transformation S from Example 34 does not admit a continuous homomorphism : e(S) ! E(S) with ı T D for all 2 e(S). Hence e(S) ¤ H() for any measure satisfying the conditions of Theorem 35.
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Ergodic Theory: Non-singular Transformations
Assume that T is an ergodic nonsingular compact group rotation. Let B0 be the -algebra generated by a subcollection of eigenfunctions. Then B0 is invariant under T and hence a factor (see Sect. “Nonsingular Joinings and Factors”) of T. It is not known if every factor of T is of this form. It is not even known whether every factor of T must have non-trivial eigenvalues. Unitary Operator Associated with a Nonsingular System Let (X; B; ; T) be a nonsingular dynamical system. In this subsection we consider spectral properties of the unitary operator U T defined by (3). First, we note that the spectrum of T is the entire circle T [147]. Next, if U T has an eigenvector then T is of type II1 . Indeed, if there are 2 T and 0 ¤ f 2 L2 (X; ) with U T f D f then the measure , d(x) :D j f (x)j2 d(x), is finite, T-invariant and equivalent to . Hence if T is of type III or II1 then the maximal spectral type T of U T is continuous. Another ‘restriction’ on T was recently found in [166]: no Foïa¸s-Str˘atil˘a measure is absolutely continuous with respect to T if T is of type II1 . We recall that a symmetric measure on T possesses Foïa¸s-Str˘atil˘a property if for each ergodic probability preserving system (Y; ; S) and f 2 L2 (Y; ), if is the spectral measure of f then f is a Gaussian random variable [134]. For instance, measures supported on Kronecker sets possess this property. Mixing is an L2 -spectral property for type II1 transformations: T is mixing Rif and only if T is a Rajchman measure, i. e. b T (n) :D z n dT (z) ! 0 as jnj ! 1. P ki Also, T is mixing if and only if n1 n1 iD0 U T ! 0 in the strong operator topology for each strictly increasing sequence k1 < k2 < [124]. This generalizes a well known theorem of Blum and Hanson for probability preserving maps. For comparison, we note that ergodicity is not an L2 -spectral property of infinite measure preserving systems. Now let T be a rank-one nonsingular transformation associated with a sequence (r n ; w n ; s n )1 nD1 as in Subsect. “Rank-One Transformations. Chacón Maps. Finite Rank”. Theorem 36 ([25,104]) The spectral multiplicity of U T is 1 and the maximal spectral type T of U T (up to a discrete measure in the case T is of type II1 ) is the weak limit of the measures k defined as follows: d k (z) D
k Y
w j (0)jPj (z)j2 dz ;
jD1
where P j (z)
:D
1 C
p w j (1)/w j (0)zR1; j C C
p
R
w j (m j 1)/w j (0)z r j 1; j , z 2 T , R i; j :D ih j1 C s j (0) C C s j (i), 1 i r k 1 and hj is the height of the jth column.
Thus the maximal spectral type of U T is given by a socalled generalized Riesz product. We refer the reader to [25,103,104,148] for a detailed study of Riesz products: their convergence, mutual singularity, singularity to T , etc. It was shown in [6] that H(T ) e(T) for any ergodic nonsingular transformation T. Moreover, T is ergodic under the action of e(T) by translations if T is isomorphic to an ergodic nonsingular compact group rotation. However it is not known: (i) Whether H(T ) D e(T) for all ergodic T. (ii) Whether ergodicity of T under e(T) implies that T is an ergodic compact group rotation. The first claim of Theorem 36 extends to the rank N nonsingular systems as follows: if T is an ergodic nonsingular transformation of rank N then the spectral multiplicity of U T is bounded by N (as in the finite measure-preserving case). It is not known whether this claim is true for a more general class of transformations which are defined as rank N but without the assumption that the Radon– Nikodym cocycle is constant on the tower levels. Entropy and Other Invariants Let T be an ergodic conservative nonsingular transformation of a standard probability space (X; B; ). If P is a finite partition of X, we define the entropy H(P ) of P P as H(P ) D P2P (P) log (P). In the study of measure-preserving systems the classical (Kolmogorov–Sinai) entropy proved to be a very useful invariant for isomorphism [33]. The key fact of the theory is that if ı T D W then the limit limn!1 n1 H( niD1 T i P ) exists for every P . However if T does not preserve , the limit may no longer exist. Some efforts have been made to extend the use of entropy and similar invariants to the nonsingular domain. These include Krengel’s entropy of conservative measure-preserving maps and its extension to nonsingular maps, Parry’s entropy and Parry’s nonsingular version of Shannon–McMillan–Breiman theorem, critical dimension by Mortiss and Dooley, etc. Unfortunately, these invariants are less informative than their classical counterparts and they are more difficult to compute. Krengel’s and Parry’s Entropies Let S be a conservative measure-preserving transformation of a -finite measure space (Y; E ; ). The Krengel en-
Ergodic Theory: Non-singular Transformations
tropy [119] of S is defined by hKr (S) D sup f(E)h(S E ) j 0 < (E) < C1g ; where h(S E ) is the finite measure-preserving entropy of SE . It follows from Abramov’s formula for the entropy of induced transformation that hKr (S) D (E)h(S E ) whenS ever E sweeps out, i. e. i0 S i E D X. A generic transformation from Aut0 (X; ) has entropy 0. Krengel raised a question in [119]: does there exist a zero entropy infinite measure-preserving S and a zero entropy finite measurepreserving R such that hKr (S R) > 0? This problem was recently solved in [44] (a special case was announced by Silva and Thieullen in an October 1995 AMS conference (unpublished)): (i) if hKr (S) D 0 and R is distal then hKr (S R) D 0; (ii) if R is not distal then there is a rank-one transformation S with hKr (S R) D 1. We also note that if a conservative S 2 Aut0 (X; ) commutes with another transformation R such that ı R D c for a constant c ¤ 1 then hKr (S) is either 0 or 1 [180]. Now let T be a type III ergodic transformation of (X; B; ). Silva and Thieullen define an entropy h (T) T), where e T is the Maharam of T by setting h (T) :D hKr (e extension of T (see Subsect. “Maharam Extension, Associated Flow and Orbit Classification of Type III Systems”). Since e T commutes with transformations which ‘multiply’ e T-invariant measure, it follows that h (T) is either 0 or 1. Let T be the standard III -odometer from Example 17(i). Then h (T) D 0. The same is true for a socalled ternary odometer associated with the sequence (3; n )1 nD1 , where n (0) D n (2) D /(1 C 2) and n (1) D /(1 C ) [180]. It is not known however whether every ergodic nonsingular odometer has zero entropy. On the other hand, it was shown in [180] that h (T) D 1 for every K-automorphism. The Parry entropy [158] of S is defined by ˚ hPa (S) :D H(S 1 FjF) j F is a -finite subalgebra of B such that F S 1 F : Parry showed [158] that hPa (S) hKr (S). It is still an open question whether the two entropies coincide. This is the case when S is of rank one (since hKr (S) D 0) and when S is quasi-finite [158]. The transformation S is called quasifinite if there exists a subset of finite measure A Y such that the first return time partition (A n )n>0 of A has finite entropy. We recall that x 2 A n () n is the smallest positive integer such that T n x 2 A. An example of nonquasi-finite ergodic infinite measure preserving transformation was constructed recently in [8].
Parry’s Generalization of Shannon–MacMillan–Breiman Theorem Let T be an ergodic transformation of a standard nonatomic probability space (X; B; ). Suppose that f ı T 2 L1 (X; ) if and only if f 2 L1 (X; ). This means that there is K > 0 such that K 1 < (d ı T)/(d) (x) < K for a. a. x. Let P be a finite partition of X. Denote by C n (x) the W atom of niD0 T i P which contains x. We put !1 D 0. Parry shows in [155] that n X
log C n j (T j x) (! j (x) ! j1 (x))
jD0 n X
! ! j (x)
iD0
! Z ˇ _ ˇ 1 1 d ı T T P log E H P ˇˇ d X iD1
! ˇ _ ˇ 1 i ˇ T P d ˇ iD0
for a. a. x. Parry also shows that under the aforementioned conditions on T, 0 1@ n
n X jD0
0 H@
j _
1 T j PA
iD0
n1 X jD0
0 H@
jC1 _
11 T j P AA
iD1
! ˇ 1 ˇ _ i ˇ T P : !H Pˇ iD1
Critical Dimension The critical dimension introduced by Mortiss [146] measures the order of growth for sums of Radon–Nikodym derivatives. Let (X; B; ; T) be an ergodic nonsingular dynamical system. Given ı > 0, let 8 9 n1 X ˆ > ˆ > ˆ > ! i (x) ˆ > ˆ ˇ > < = ˇ iD0 and (4) X ı :D x 2 X ˇˇ lim inf > 0 ı n!1 ˆ > n ˆ > ˆ > ˆ > ˆ > : ; 8 9 n1 X ˆ > ˆ > ˆ > ! i (x) ˆ > ˆ ˇ > < = ˇ iD0 ı ˇ : (5) X :D x 2 X ˇ lim inf D 0 n!1 ˆ > nı ˆ > ˆ > ˆ > ˆ > : ; Then Xı and X ı are T-invariant subsets. Definition 37 ([57,146]) The lower critical dimension ˛(T) of T is sup fı j (Xı ) D 1g. The upper critical dimension ˇ(T) of T is inf fı j (X ı ) D 1g.
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Ergodic Theory: Non-singular Transformations
It was shown in [57] that the lower and upper critical dimensions are invariants for isomorphism of nonsingular systems. Notice also that ! n X log ! i (x) ˛(T) D lim inf n!1
iD1
log n ! n X log ! i (x)
ˇ(T) D lim sup n!1
iD1
and
Theorem 40 For any > 0, there exists a Hamachi shift S with ˛(S) < and ˇ(S) > 1 . :
log n
Nonsingular Restricted Orbit Equivalence
Moreover, 0 ˛(T) ˇ(T) 1. If T is of type II1 then ˛(T) D ˇ(T) D 1. If T is the standard III -odometer from Example 17 then ˛(T) D ˇ(T) D log(1 C ) /(1 C ) log . Theorem 38 (i) For every 2 [0; 1] and every c 2 [0; 1] there exists a nonsingular odometer of type III with critical dimension equal to c [145]. (ii) For every c 2 [0; 1] there exists a nonsingular odometer of type II1 with critical dimension equal to c [57]. Let T be the nonsingular odometer associated with a sequence (m n ; n )1 nD1 . Let s(n) D m1 m n and let H(P n ) denote the entropy of the partition of the first n coordinates with respect to . We now state a nonsingular version of Shannon–MacMillan–Breiman theorem for T from [57]. Theorem 39 Let mi be bounded from above. Then (i) ˛(T) D lim inf inf
entropy (average coordinate). It also follows from Theorem 39 that if T is an odometer of bounded type then ˛(T 1 ) D ˛(T) and ˇ(T 1 ) D ˇ(T). In [58], Theorem 39 was extended to a subclass of Markov odometers. The critical dimensions for Hamachi shifts (see Subsect. “Nonsingular Bernoulli Transformations – Hamachi’s Example”) were investigated in [59]:
n X
log m i (x i )
iD1
log s(n) H(Pn ) ; D lim inf n!1 log s(n)
In [144] Mortiss initiated study of a nonsingular version of Rudolph’s restricted orbit equivalence [167]. This work is still in its early stages and does not yet deal with any form of entropy. However she introduced nonsingular orderings of orbits, defined sizes and showed that much of the basic machinery still works in the nonsingular setting. Nonsingular Joinings and Factors The theory of joinings is a powerful tool to study probability preserving systems and to construct striking counterexamples. It is interesting to study what part of this machinery can be extended to the nonsingular case. However, the definition of nonsingular joining is far from being obvious. Some progress was achieved in understanding 2-fold joinings and constructing prime systems of any Krieger type. As far as we know the higher-fold nonsingular joinings have not been considered so far. It turned out however that an alternative coding technique, predating joinings in studying the centralizer and factors of the classical measure-preserving Chacón maps, can be used as well to classify factors of Cartesian products of some nonsingular Chacón maps.
n!1
and (ii) ˇ(T) D lim sup inf
n X
log m i (x i )
iD1
n!1
log s(n)
H(Pn ) D lim sup log s(n) n!1 for a. a. x D (x i ) i1 2 X. It follows that in the case when ˛(T) D ˇ(T), the critical dimension coincides with limn!1 H(Pn )/(log s(n)). In [145] this expression (when it exists) was called AC-
Joinings, Nonsingular MSJ and Simplicity In this section all measures are probability measures. A nonsingular joining of two nonsingular systems ˆ on the (X1 ; B1 ; 1 ; T1 ) and (X 2 ; B2 ; 2 ; T2 ) is a measure product B1 B2 that is nonsingular for T1 T2 and satisˆ ˆ 1 B) D 2 (B) for all A 2 fies: (AX 2 ) D 1 (A) and (X B1 and B 2 B2 . Clearly, the product 1 2 is a nonsingular joining. Given a transformation S 2 C(T), the measure S given by S (A B) :D (A \ S 1 B) is a nonsingular joining of (X; ; T) and (X; ıS 1 ; T). It is called a graphjoining since it is supported on the graph of S. Another important kind of joinings that we are going to define now is related to factors of dynamical systems. Recall that given a nonsingular system (X; B; ; T), a sub- -algebra A of B such that T 1 (A) D A mod is called a factor of T.
Ergodic Theory: Non-singular Transformations
There is another, equivalent, definition. A nonsingular dynamical system (Y; C ; ; S) is called a factor of T if there exists a measure-preserving map ' : X ! Y, called a factor map, with 'T D S' a. e. (If ' is only nonsingular, may be replaced with the equivalent measure ı ' 1 , for which ' is measure-preserving.) Indeed, the sub--algebra ' 1 (C ) B is T-invariant and, conversely, any T-invariant sub--algebra of B defines a factor map by immanent properties of standard probability spaces, see e. g. [3]. If ' is a factor map as above,Rthen has a disintegration with respect to ', i. e., D y d(y) for a measurable map y 7! y from Y to the probability measures on X so that y (' 1 (y)) D 1, the measure S'(x) ı T is equivalent to '(x) and dS'(x) ı T d ı T d ı S (x) (x) D ('(x)) d d d'(x)
(6)
ˆ D for a. e. x 2 X. Define now theR relative product ˆ D y y d(y). Then ' on X X by setting ˆ is a nonsingular selfit is easy to deduce from (6) that joining of T. We note however that the above definition of joining is not satisfactory since it does not reduce to the classical definition when we consider probability preserving systems. Indeed, the following result was proved in [168]. Theorem 41 Let (X 1 ; B1 ; 1 ; T1 ) and (X2 ; B2 ; 2 ; T2 ) be two finite measure-preserving systems such that T1 T2 is ergodic. Then for every ; 0 < < 1, there exists a nonˆ of 1 and 2 such that (T1 T2 ; ) ˆ is singular joining ergodic and of type III . ˆ can It is not known however if the nonsingular joining be chosen in every orbit equivalence class. In view of the above, Rudolph and Silva [168] isolate an important subclass of joining. It is used in the definition of a nonsingular version of minimal self-joinings. Definition 42 ˆ of (X 1 ; 1 ; T1 ) and (i) A nonsingular joining (X 2 ; 2 ; T2 ) is rational if there exit measurable functions c 1 : X1 ! RC and c 2 : X2 ! RC such that !ˆ 1 (x1 ; x2 ) D !1 1 (x1 )!1 2 (x2 )c 1 (x1 ) ˆ
ˆ a. e. D !1 1 (x1 )!1 2 (x2 )c 2 (x2 )
(ii) A nonsingular dynamical system (X; B; ; T) has minimal self-joinings (MSJ) over a class M of probability measures equivalent to , if for every 1 ; 2 2 M, ˆ of 1 ; 2 , a. e. ergodic for every rational joining ˆ is either the product of its marginals component of or is the graph-joining supported on T j for some j 2 Z.
Clearly, product measure, graph-joinings and the relative products are all rational joinings. Moreover, a rational joining of finite measure-preserving systems is measure-preserving and a rational joining of type II1 ’s is of type II1 [168]. Thus we obtain the finite measure-preserving theory as a special case. As for the definition of MSJ, it depends on a class M of equivalent measures. In the finite measure-preserving case M D fg. However, in the nonsingular case no particular measure is distinguished. We note also that Definition 42(ii) involves some restrictions on all rational joinings and not only ergodic ones as in the finite measure-preserving case. The reason is that an ergodic component of a nonsingular joining needs not be a joining of measures equivalent to the original ones [2]. For finite measure-preserving transformations, MSJ over fg is the same as the usual 2-fold MSJ [49]. A nonsingular transformation T on (X; B; ) is called prime if its only factors are B and fX; ;g mod . A (nonempty) class M of probability measures equivalent to is said to be centralizer stable if for each S 2 C(T) and 1 2 M, the measure 1 ı S is in M. Theorem 43 ([168]) Let (X; B; ; T) be a ergodic nonatomic dynamical system such that T has MSJ over a class M that is centralizer stable. Then T is prime and the centralizer of T consists of the powers of T. A question that arises is whether if such nonsingular dynamical system (not of type II1 ) exist. Expanding on Ornstein’s original construction from [150], Rudolph and Silva construct in [168], for each 0 1, a nonsingular rank-one transformation T that is of type III and that has MSJ over a class M that is centralizer stable. Type II1 examples with analogues properties were also constructed there. In this connection it is worth to mention the example by Aaronson and Nadkarni [6] of II1 ergodic transformations that have no factor algebras on which the invariant measure is -finite (except for the trivial and the entire ones); however these transformations are not prime. A more general notion than MSJ called graph self-joinings (GSJ), was introduced [181]: just replace the the words “on T j for some j 2 Z” in Definition 3(ii) with “on S for some element S 2 C(T)”. For finite measure-preserving transformations, GSJ over fg is the same as the usual 2fold simplicity [49]. The famous Veech theorem on factors of 2-fold simple maps (see [49]) was extended to nonsingular systems in [181] as follows: if a system (X; B; ; T) has GSJ then for every non-trivial factor A of T there exists a locally compact subgroup H in C(T) (equipped with the weak topology) which acts smoothly (i. e. the partition into H-orbits is measurable) and such that A D fB 2 B j (hB4B) D 0 for all h 2 Hg. It follows that there is a co-
347
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Ergodic Theory: Non-singular Transformations
cycle ' from (X; A; A) to H such that T is isomorphic to the '-skew product extension (T A)' (see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). Of course, the ergodic nonsingular odometers and, more generally, ergodic nonsingular compact group rotation (see Subsect. “L1 -Spectrum and Groups of Quasi-Invariance”) have GSJ. However, except for this trivial case (the Cartesian square is non-ergodic) plus the systems with MSJ from [168], no examples of type III systems with GSJ are known. In particular, no smooth examples have been constructed so far. This is in sharp contrast with the finite measure preserving case where abundance of simple (or close to simple) systems are known (see [39,40,49,182]). Nonsingular Coding and Factors of Cartesian Products of Nonsingular Maps As we have already noticed above, the nonsingular MSJ theory was developed in [168] only for 2-fold self-joinings. The reasons for this were technical problems with extending the notion of rational joinings form 2-fold to n-fold self-joinings. However while the 2-fold nonsingular MSJ or GSJ properties of T are sufficient to control the centralizer and the factors of T, it is not clear whether it implies anything about the factors or centralizer of T T. Indeed, to control them one needs to know the 4-fold joinings of T. However even in the finite measure-preserving case it is a long standing open question whether 2-fold MSJ implies n-fold MSJ. That is why del Junco and Silva [51] apply an alternative – nonsingular coding – techniques to classify the factors of Cartesian products of nonsingular Chacón maps. The techniques were originally used in [48] to show that the classical Chacón map is prime and has trivial centralizer. They were extended to nonsingular systems in [50]. For each 0 < < 1 we denote by T the Chacón map (see Subsect. “Rank-One Transformations. Chacón Maps. Finite Rank”) corresponding the sequence of probability vectors w n D (/(1 C 2); 1/(1 C 2); /(1 C 2)) for all n > 0. One can verify that the maps T are of type III . (The classical Chacón map corresponds to D 1.) All of these transformations are defined on the same standard Borel space (X; B). These transformations were shown to be power weakly mixing in [12]. The centralizer of any finite Cartesian product of nonsingular Chacón maps is computed in the following theorem. Theorem 44 ([51]) Let 0 < 1 < : : : < k 1 and n1 ; : : : ; n k be positive integers. Then the centralizer of the ˝n 1 Cartesian product T˝n : : : T k is generated by maps 1 k of the form U1 : : : U k , where each U i , acting on the
ni -dimensional product space X n i , is a Cartesian product of powers of T i or a co-ordinate permutation on X n i . Let denote the permutation on X X defined by (x; y) D (y; x) and let B2ˇ denote the symmetric factor, i. e. B2ˇ D fA 2 B ˝ B j (A) D Ag. The following theorem classifies the factors of the Cartesian product of any two nonsingular type III , 0 < < 1, or type II1 Chacón maps. Theorem 45 ([51]) Let T1 and T2 be two nonsingular Chacón systems. Let F be a factor algebra of T1 T2 . If 1 ¤ 2 then F is equal mod 0 to one of the four algebras B ˝ B, B ˝ N , N ˝ B, or N ˝ N , where N D f;; Xg. (ii) If 1 D 2 then F is equal mod 0 to one of the following algebras B ˝ C , B ˝ N , N ˝ C , N ˝ N , or (T m Id)B2ˇ for some integer m.
(i)
It is not hard to obtain type III1 examples of Chacón maps for which the previous two theorems hold. However the construction of type II1 and type III0 nonsingular Chacón transformations is more subtle as it needs the choice of ! n to vary with n. In [92], Hamachi and Silva construct type III0 and type II1 examples, however the only property proved for these maps is ergodicity of their Cartesian square. More recently, Danilenko [38] has shown that all of them (in fact, a wider class of nonsingular Chacón maps of all types) are power weakly mixing. In [22], Choksi, Eigen and Prasad asked whether there exists a zero entropy, finite measure-preserving mixing automorphism S, and a nonsingular type III automorphism T, such that T S has no Bernoulli factors. Theorem 45 provides a partial answer (with a mildly mixing only instead of mixing) to this question: if S is the finite measure-preserving Chacón map and T is a nonsingular Chacón map as above, the factors of T S are only the trivial ones, so T S has no Bernoulli factors. Applications. Connections with Other Fields In this – final – section we shed light on numerous mathematical sources of nonsingular systems. They come from the theory of stochastic processes, random walks, locally compact Cantor systems, horocycle flows on hyperbolic surfaces, von Neumann algebras, statistical mechanics, representation theory for groups and anticommutation relations, etc. We also note that such systems sometimes appear in the context of probability preserving dynamics (see also a criterium of distality in Subsect. “Krengel’s and Parry’s Entropies”).
Ergodic Theory: Non-singular Transformations
Mild Mixing An ergodic finite measure-preserving dynamical system (X; B; ; T) is called mildly mixing if for each non-trivial factor algebra A B, the restriction T A is not rigid. For equivalent definitions and extensions to actions of locally compact groups we refer to [3] and [177]. There is an interesting criterium of the mild mixing that involves nonsingular systems: T is mildly mixing if and only if for each ergodic nonsingular transformation S, the product T S is ergodic [71]. Furthermore, T S is then orbit equivalent to S [98]. Moreover, if R is a nonsingular transformation such that R S is ergodic for any ergodic nonsingular S then R is of type II1 (and mildly mixing) [177]. Disjointness and Furstenberg’s Class W ? Two probability preserving systems (X; ; T) and (Y; ; S) are called disjoint if is the only T S-invariant probability measure on X Y whose coordinate projections are and respectively. Furstenberg in [69] initiated studying the class W ? of transformations disjoint from all weakly mixing ones. Let D denote the class of distal transformations and M(W ? ) the class of multipliers of W ? (for the definitions see [75]). Then D M(W ? ) W ? . In [43] and [133] it was shown by constructing explicit examples that these inclusions are strict. We record this fact here because nonsingular ergodic theory was the key ingredient of the arguments in the two papers pertaining to the theory of probability preserving systems. The examples are of the form T';S (x; y) D (Tx; S'(x) y), where T is an ergodic rotation on (X; ), (S g ) g2G a mildly mixing action of a locally compact group G on Y and ' : X ! G a measurable map. Let W' denote the Mackey action of G associated with ' and let (Z; ) be the space of this action. The key observation is that there exists an affine isomorphism of the simplex of T';S -invariant probability measures whose pullback on X is and the simplex of W' S quasi-invariant probability measures whose pullback on Z is and whose Radon–Nikodym cocycle is measurable with respect to Z. This is a far reaching generalization of Furstenberg theorem on relative unique ergodicity of ergodic compact group extensions. Symmetric Stable and Infinitely Divisible Stationary Processes Rosinsky in [163] established a remarkable connection between structural studies of stationary stochastic processes and ergodic theory of nonsingular transformations (and flows). For simplicity we consider only real processes in
discrete time. Let X D (X n )n2Z be a measurable stationary symmetric ˛-stable (S˛S) process, 0 < ˛ < 2. This P means that any linear combination nkD1 a k X j k , j k 2 Z, a k 2 R has an S˛S-distribution. (The case ˛ D 2 corresponds to Gaussian processes.) Then the process admits a spectral representation Z Xn D
f n (y) M(dy) ;
n2Z;
(7)
Y
where f n 2 L˛ (Y; ) for a standard -finite measure space (Y;B;) and M is an independently scattered random measure on B such that E exp (iuM(A)) D exp (juj˛ (A)) for every A 2 B of finite measure. By [163], one can choose the kernel ( f n )n2Z in a special way: there are a -nonsingular transformation T and measurable maps ' : X ! f1; 1g and f 2 L˛ (Y; ) such that f n D U n f , n 2 Z, where U is the isometry of L˛ (X; ) given by Ug D ' (d ı T/d)1/˛ g ı T. If, in addition, the smallest T-invariant -algebra containing f 1 (BR ) coincides with B and Supp f f ı T n : n 2 Zg D Y then the pair (T; ') is called minimal. It turns out that minimal pairs always exist. Moreover, two minimal pairs (T; ') and (T 0 ; ' 0 ) representing the same S˛S process are equivalent in some natural sense [163]. Then one can relate ergodic-theoretical properties of (T; ') to probabilistic properties of (X n )n2Z . For instance, let Y D C t D be the HopfR decomposition of Y (see Theorem 2). We R let X nD : D D f n (y) M(dy) and X nC :D C f n (y) M(dy). Then we obtain a unique (in distribution) decomposition of X into the sum X D C X C of two independent stationary S˛S-processes. Another kind of decomposition was considered in [171]. Let P be the largest invariant subset of Y such that T P has a finite invariant measure. Partitioning Y into P and N :D Y n N and restricting the integration in (7) to P and N we obtain a unique (in distribution) decomposition of X into the sum X P C X N of two independent stationary S˛S-processes. Notice that the process X is ergodic if and only if (P) D 0. Recently, Roy considered a more general class of infinitely divisible (ID) stationary processes [165]. Using Maruyama’s representation of the characteristic function of an ID process X without Gaussian part he singled out the Lévy measure Q of X. Then Q is a shift invariant -finite measure on RZ . Decomposing the dynamical system (RZ ; ; Q) in various natural ways (Hopf decomposition, 0-type and positive type, so-called ‘rigidity free’ part and its complement) he obtains corresponding decompositions for the process X. Here stands for the shift on RZ .
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Poisson Suspensions Poisson suspensions are widely used in statistical mechanics to model ideal gas, Lorentz gas, etc (see [33]). Let (X; B; ) be a standard -finite non-atomic measure space and (X) D 1. Denote by e X the space of unordered countable subsets of X. It is called the space of configurations. Fix t > 0. Let A 2 B have positive finite measure and let j 2 ZC . Denote by [A; j] the subset of all configurations e x2e X such that #(e x \ A) D j. Let e B be the -algebra generated by all [A; j]. We define a probability measure et on e B by two conditions:
Boundaries of Random Walks
j
exp (t(A)); (i) et ([A; j]) D (t(A)) j! (ii) if A1 ; : : : ; A p are pairwise disjoint Tp Qp et ([A k ; j k ]). et ( kD1 [A k ; j k ]) D kD1
singular transformation of a standard probability space (X; B; ) and let f : X ! Rn a measurable function. DeP n fine for m 1, Ym : X ! Rn by Ym :D m1 nD0 f ı T . In other words, (Ym )m1 is the random walk associated with the (non-stationary) process ( f ı T n )n0 . Let us call this random walk recurrent if the cocycle f of T is recurrent (see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). It was shown in [176] that in the case ı T D , i. e. the process is stationary, this definition is equivalent to the standard one.
then
If T is a -preserving transformation of X and e x D (x1 ; x2 ; : : : ) is a configuration then we set e T! :D T is a e -preserving (Tx1 ; Tx2 ; : : : ). It is easy to verify that e transformation of e X. The dynamical system (e X; e B; e ; e T) is called the Poisson suspension above (X; B; ; T). It is ergodic if and only if T has no invariant sets of finite positive measure. There is a canonical representation of L2 (e X; e ) as the Fock space over L2 (X; ) such that the unitary operator Ue of U T . Thus, the T is the ‘exponent’ P 1 n , where is (n!) maximal spectral type of Ue n0 T is a measure of the maximal spectral type of U T . It is easy to see that a -finite factor of T corresponds to a factor (called Poissonian) of e T. Moreover, a -finite measurepreserving joining (with -finite projections) of two infinite measure-preserving transformations T 1 and T 2 gene2 [52,164]. e1 and T erates a joining (called Poissonian) of T Thus we see a similarity with the well studied theory of Gaussian dynamical systems [134]. However, the Poissonian case is less understood. There was a recent progress in this field. Parreau and Roy constructed Poisson suspensions whose ergodic self-joinings are all Poissonian [154]. In [111] partial solutions of the following (still open) problems are found: (i) whether the Pinsker factor of e T is Poissonian, (ii) what is the relationship between Krengel’s entropy of T, Parry’s entropy of T and Kolmogorov–Sinai entropy of e T.
Recurrence of Random Walks with Non-stationary Increments Using nonsingular ergodic theory one can introduce the notion of recurrence for random walks obtained from certain non-stationary processes. Let T be an ergodic non-
Boundaries of random walks on groups retain valuable information on the underlying groups (amenability, entropy, etc.) and enable one to obtain integral representation for harmonic functions of the random walk [112,186,187]. Let G be a locally compact group and a probability measure on G. Let T denote the (one-sided) shift on the probability space (X; B X ; ) :D (G; BG ; )ZC and ' : X ! G a measurable map defined by (y0 ; y1 ; : : : ) 7! y0 . Let T' be the '-skew product extension of T acting on the space (X G; G ) (for noninvertible transformations the skew product extension is defined in the very same way as for invertible ones, see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). Then T' is isomorphic to the homogeneous random walk on G with jump probability . Let I (T' ) denote the sub- -algebra of T' -invariant sets and T let F (T' ) :D n>0 T'n (B X ˝ BG ). The former is called the Poisson boundary of T' and the latter one is called the tail boundary of T' . Notice that a nonsingular action of G by inverted right translations along the second coordinate is well defined on each of the two boundaries. The two boundaries (or, more precisely, the G-actions on them) are ergodic. The Poisson boundary is the Mackey range of ' (as a cocycle of T). Hence the Poisson boundary is amenable [187]. If the support of generates a dense subgroup of G then the corresponding Poisson boundary is weakly mixing [4]. As for the tail boundary, we first note that it can be defined for a wider family of non-homogeneous random walks. This means that the jump probability is no longer fixed and a sequence (n )n>0 of probability measures on G is considered instead. Now let Q (X; B X ; ) :D n>0 (G; BG ; ). The one-sided shift on X may not be nonsingular now. Instead of it, we consider the tail equivalence relation R on X and a cocycle ˛ : R ! G given by ˛(x; y) D x1 x n y 1 n y 1 , where x D (x i ) i>0 and y D (y i ) i>0 are R-equivalent and n in the smallest integer such that x i D y i for all i > n. The tail boundary of the random walk on G with time dependent jump
Ergodic Theory: Non-singular Transformations
probabilities (n )n>0 is the Mackey G-action associated with ˛. In the case of homogeneous random walks this definition is equivalent to the initial one. Connes and Woods showed [32] that the tail boundary is always amenable and AT. It is unknown whether the converse holds for general G. However it is true for G D R and G D Z: the class of AT-flows coincides with the class of tail boundaries of the random walks on R and a similar statement holds for Z [32]. Jaworsky showed [109] that if G is countable and a random walk is homogeneous then the tail boundary of the random walk possesses a so-called SAT-property (which is stronger than AT). Classifying -Finite Ergodic Invariant Measures The description of ergodic finite invariant measures for topological (or, more generally, standard Borel) systems is a well established problem in the classical ergodic theory [33]. On the other hand, it seems impossible to obtain any useful information about the system by analyzing the set of all ergodic quasi-invariant (or just -finite invariant) measures because this set is wildly huge (see Subsect. “The Glimm–Effros Theorem”). The situation changes if we impose some restrictions on the measures. For instance, if the system under question is a homeomorphism (or a topological flow) defined on a locally compact Polish space then it is natural to consider the class of ( -finite) invariant Radon measures, i. e. measures taking finite values on the compact subsets. We give two examples. First, the seminal results of Giordano, Putnam and Skau on the topological orbit equivalence of compact Cantor minimal systems were extended to locally compact Cantor minimal (l.c.c.m.) systems in [37] and [138]. Given a l.c.c.m. system X, we denote by M(X) and M1 (X) the set of invariant Radon measures and the set of invariant probability measures on X. Notice that M1 (X) may be empty [37]. It was shown in [138] that two systems X and X 0 are topologically orbit equivalent if and only if there is a homeomorphism of X onto X 0 which maps bijectively M(X) onto M(X 0 ) and M1 (X) onto M1 (X 0 ). Thus M(X) retains an important information on the system – it is ‘responsible’ for the topological orbit equivalence of the underlying systems. Uniquely ergodic l.c.c.m. systems (with unique up to scaling infinite invariant Radon measure) were constructed in [37]. The second example is related to study of the smooth horocycle flows on tangent bundles of hyperbolic surfaces. Let D be the open disk equipped with the hyperbolic metric jdzj/(1 jzj2 ) and let Möb(D) denote the group of Möbius transformations of D. A hyperbolic surface can be written in the form M :D nMöb(D), where is a tor-
sion free discrete subgroup of Möb(D). Suppose that is a nontrivial normal subgroup of a lattice 0 in Möb(D). Then M is a regular cover of the finite volume surface M0 : D 0 nMöb(D). The group of deck transformations G D 0 / is finitely generated. The horocycle flow (h t ) t2R and the geodesic flow (g t ) t2R defined on the unit tangent bundle T 1 (D) descend naturally to flows, say h and g, on T 1 (M). We consider the problem of classification of the h-invariant Radon measures on M. According to Ratner, h has no finite invariant measures on M if G is infinite (except for measures supported on closed orbits). However there are infinite invariant Radon measures, for instance the volume measure. In the case when G is free Abelian and 0 is co-compact, every homomorphism ' : G ! R determines a unique up to scaling ergodic invariant Radon measure (e.i.r.m.) m on T 1 (M) such that m ı dD D exp ('(D))m for all D 2 G [16] and every e.i.r.m. arises this way [172]. Moreover all these measures are quasi-invariant under g. In the general case, an interesting bijection is established in [131] between the e.i.r.m. which are quasi-invariant under g and the ‘nontrivial minimal’ positive eigenfunctions of the hyperbolic Laplacian on M. Von Neumann Algebras There is a fascinating and productive interplay between nonsingular ergodic theory and von Neumann algebras. The two theories alternately influenced development of each other. Let (X; B; ; T) be a nonsingular dynamical system. Given ' 2 L1 (X; ) and j 2 Z, we define operators A' and U j on the Hilbert space L2 (Z Z; ) by setting (A' f )(x; i) :D '(T i x) f (x; i) ; (U j f )(x; i) :D f (x; i j) : Then U j A' U j D A'ıT j . Denote by M the von Neumann algebra (i. e. the weak closure of the -algebra) generated by A' , ' 2 L1 (X; ) and U j , j 2 Z. If T is ergodic and aperiodic then M is a factor, i. e. M \ M0 D C1, where M0 denotes the algebra of bounded operators commuting with M. It is called a Krieger’s factor. Murray–von Neumann–Connes’ type of M is exactly the Krieger’s type of T. The flow of weights of M is isomorphic to the associated flow of T. Two Krieger’s factors are isomorphic if and only if the underlying dynamical systems are orbit equivalent [129]. Moreover, a number of important problems in the theory of von Neumann algebras such as classification of subfactors, computation of the flow of weights and Connes’ invariants, outer conjugacy for
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automorphisms, etc. are intimately related to the corresponding problems in nonsingular orbit theory. We refer to [42,66,73,74,89,142] for details. Representations of CAR Representations of canonical anticommutation relations (CAR) is one of the most elegant and useful chapters of mathematical physics, providing a natural language for many body quantum physics and quantum field theory. By a representation of CAR we mean a sequence of bounded linear operators a1 ; a2 ; : : : in a separable Hilbert space K such that a j a k C a k a j D 0 and a j ak C a k a j D ı j;k . Consider f0; 1g as a group with addition mod 2. Then X D f0; 1gN is a compact Abelian group. Let :D fx D (x1 ; x2 ; : : : ) : limn!1 x n D 0g. Then is a dense countable subgroup of X. It is generated by elements k whose k-coordinate is 1 and the other ones are 0. acts on X by translations. Let be an ergodic -quasi-invariant measure on X. Let (C k ) k1 be Borel maps from X to the group of unitary operators in a Hilbert space H satisfying C k (x) D C k (x C ı k ), C k (x)C l (x C ı l ) D C l (x)C k (x C ı k ), k ¤ l for a. a. x. In other words, (C k ) k1 f :D defines a cocycle of the -action. We now put H 2 f L (X; ) ˝ H and define operators ak in H by setting (a k f )(x) D (1)x 1 CCx k1 (1 x k ) s d ı ı k (x) f (x C ı k ) ; C k (x) d f and x D (x1 ; x2 ; where f : X ! H is an element of H : : : ) 2 X. It is easy to verify that a1 ; a2 ; : : : is a representation of CAR. The converse was established in [72] and [77]: every factor-representation (this means that the von Neumann algebra generated by all ak is a factor) of CAR can be represented as above for some ergodic measure , Hilbert space H and a -cocycle (C k ) k1 . Moreover, using nonsingular ergodic theory Golodets [77] constructed for each k D 2; 3; : : : ; 1, an irreducible representation of CAR such that dim H D k. This answered a question of Gårding and Wightman [72] who considered only the case k D 1. Unitary Representations of Locally Compact Groups Nonsingular actions appear in a systematic way in the theory of unitary representations of groups. Let G be a locally compact second countable group and H a closed normal subgroup of G. Suppose that H is commutative (or, more generally, of type I, see [53]). Then the natural action of G by conjugation on H induces a Borel G-action, say ˛, on the dual space b H – the set of unitarily equivalent
classes of irreducible unitary representations of H. If now U D (U g ) g2G is a unitary representation of G in a separable Hilbert space then by applying Stone decomposition theorem to U H one can deduce that ˛ is nonsingular with respect to a measure of the ‘maximal spectral b Moreover, if U is irreducible then ˛ type’ for U H on H. is ergodic. Whenever is fixed, we obtain a one-to-one correspondence between the set of cohomology classes of irreducible cocycles for ˛ with values in the unitary group on a Hilbert space H and the subset of b G consisting of classes of those unitary representations V for which the measure associated to V H is equivalent to . This correspondence is used in both directions. From information about cocycles we can deduce facts about representations and vise versa [53,118]. Concluding Remarks While some of the results that we have cited for nonsingular Z-actions extend to actions of locally compact Polish groups (or subclasses of Abelian or amenable ones), many natural questions remain open in the general setting. For instance: what is Rokhlin lemma, or the pointwise ergodic theorem, or the definition of entropy for nonsingular actions of general countable amenable groups? The theory of abstract nonsingular equivalence relations [66] or, more generally, nonsingular groupoids [160] and polymorphisms [184] is also a beautiful part of nonsingular ergodic theory that has nice applications: description of semifinite traces of AF-algebras, classification of factor representations of the infinite symmetric group [185], path groups [14], etc. Nonsingular ergodic theory is getting even more sophisticated when we pass from Z-actions to noninvertible endomorphisms or, more generally, semigroup actions (see [3] and references therein). However, due to restrictions of space we do not consider these issues in our survey. Bibliography 1. Aaronson J (1983) The eigenvalues of nonsingular transformations. Isr J Math 45:297–312 2. Aaronson J (1987) The intrinsic normalizing constants of transformations preserving infinite measures. J Analyse Math 49:239–270 3. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. Amer Math Soc, Providence ´ 4. Aaronson J, Lemanczyk M (2005) Exactness of Rokhlin endomorphisms and weak mixing of Poisson boundaries, Algebraic and Topological Dynamics. Contemporary Mathematics, vol 385. Amer Math Soc, Providence 77–88 5. Aaronson J, Lin M, Weiss B (1979) Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products. Isr J Math 33:198–224
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Ergodic Theory: Basic Examples and Constructions
Ergodic Theory: Basic Examples and Constructions MATTHEW N ICOL1 , KARL PETERSEN2 Department of Mathematics, University of Houston, Houston, USA 2 Department of Mathematics, University of North Carolina, Chapel Hill, USA
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Article Outline Glossary Definition of the Subject Introduction Examples Constructions Future Directions Bibliography Glossary A transformation T of a measure space (X; B; ) is measure-preserving if (T 1 A) D (A) for all measurable A 2 B. A measure-preserving transformation (X; B; ; T) is ergodic if T 1 (A) D A (mod ) implies (A) D 0 or (Ac ) D 0 for each measurable set A 2 B. A measure-preserving transformation (X; B; ; T) of a probability space is weak-mixing if lim n!1 n1 P n1 i iD0 j(T A \ B) (A)(B)j D 0 for all measurable sets A; B 2 B. A measure-preserving transformation (X; B; ; T) of a probability space is strong-mixing if limn!1 (T n A\B) D (A)(B) for all measurable sets A; B 2 B. A continuous transformation T of a compact metric space X is uniquely ergodic if there is only one T-invariant Borel probability measure on X. A continous transformation of a topological space X is topologically mixing for any two open sets U; V X there exists N > 0 such that T n (U) \ V ¤ ;, for each n N. Suppose (X; B; ) is a probability space. A finite partition P of X is a finite collection of disjoint (mod , i. e., up to sets of measure 0) measurable sets fP1 ; : : : ; Pn g such that X D [Pi (mod ). The entropy of P with reP spect to is H(P ) D i (Pi ) ln (Pi ) (other bases are sometimes used for the logarithm). The metric (or measure-theoretic) entropy of T with respect to P is h (T; P ) D lim n!1 n1 H(P _ _ T nC1 (P )), where P _ _ T nC1 (P ) is the partition of X into sets of points with the same coding with
respect to P under T i , i D 0; : : : ; n 1. That is x, y are in the same set of the partition P _ _ T nC1 (P ) if and only if T i (x) and T i (y) lie in the same set of the partition P for i D 0; : : : ; n 1. The metric entropy h (T) of (X; B; ; T) is the supremum of h (T; P ) over all finite measurable partitions P . If T is a continuous transformation of a compact metric space X, then the topological entropy of T is the supremum of the metric entropies h (T), where the supremum is taken over all T-invariant Borel probability measures. A system (X; B; ; T) is loosely Bernoulli if it is isomorphic to the first-return system to a subset of positive measure of an irrational rotation or a (positive or infinite entropy) Bernoulli system. Two systems are spectrally isomorphic if the unitary operators that they induce on their L2 spaces are unitarily equivalent. A smooth dynamical system consists of a differentiable manifold M and a differentiable map f : M ! M. The degree of differentiability may be specified. Two submanifolds S1 , S2 of a manifold M intersect transversely at p 2 M if Tp (S1 ) C Tp (S2 ) D Tp (M). An ( -) small Cr perturbation of a Cr map f of a manifold M is a map g such that dC r ( f ; g) < i. e. the distance between f and g is less than in the Cr topology. A map T of an interval I D [a; b] is piecewise smooth (Ck for k 1) if there is a finite set of points a D x1 < x2 < < x n D b such that Tj(x i ; x iC1 ) is Ck for each i. The degree of differentiability may be specified. A measure on a measure space (X; B) is absolutely continuous with respect to a measure on (X; B) if (A) D 0 implies (A) D 0 for all measurable A 2 B. A Borel measure on a Riemannian manifold M is absolutely continuous if it is absolutely continuous with respect to the Riemannian volume on M. A measure on a measure space (X; B) is equivalent to a measure on (X; B) if is absolutely continuous with respect to and is absolutely continuous with respect to .
Definition of the Subject Measure-preserving systems are a common model of processes which evolve in time and for which the rules governing the time evolution don’t change. For example, in Newtonian mechanics the planets in a solar system undergo motion according to Newton’s laws of motion: the planets
Ergodic Theory: Basic Examples and Constructions
move but the underlying rule governing the planets’ motion remains constant. The model adopted here is to consider the time-evolution as a transformation (either a map in discrete time or a flow in continuous time) on a probability space or more generally a measure space. This is the setting of the subject called ergodic theory. Applications of this point of view include the areas of statistical physics, classical mechanics, number theory, population dynamics, statistics, information theory and economics. The purpose of this chapter is to present a flavor of the diverse range of examples of measure-preserving transformations which have played a role in the development and application of ergodic theory and smooth dynamical systems theory. We also present common constructions involving measurepreserving systems. Such constructions may be considered a way of putting ‘building-block’ dynamical systems together to construct examples or decomposing a complicated system into simple ‘building-blocks’ to understand it better. Introduction In this chapter we collect a brief list of some important examples of measure-preserving dynamical systems, which we denote typically by (X; B; ; T) or (T; X; B; ) or slight variations. These examples have played a formative role in the development of dynamical systems theory, either because they occur often in applications in one guise or another or because they have been useful simple models to understand certain features of dynamical systems. There is a fundamental difference in the dynamical properties of those systems which display hyperbolicity: roughly speaking there is some exponential divergence of nearby orbits under iteration of the transformation. In differentiable systems this is associated with the derivative of the transformation possessing eigenvalues of modulus greater than one on a ‘dynamically significant’ subset of phase space. Hyperbolicity leads to complex dynamical behavior such as positive topological entropy, exponential divergence of nearby orbits (“sensitivity to initial conditions”) often coexisting with a dense set of periodic orbits. If ; are sufficiently regular functions on the phase space X of a hyperbolic measure-preserving transformation (T; X; ), then typically we have fast decay of correlations in the sense that ˇZ ˇ Z Z ˇ ˇ ˇ (T n x) (x) d d ˇ Ca(n) d ˇ ˇ X
where a(n) ! 0. If a(n) ! 0 at an exponential rate we say that the system has exponential decay of correlations. A theme in dynamical systems is that the time se-
ries formed by sufficiently regular observations on systems with some degree of hyperbolicity often behave statistically like independent identically distributed random variables. At this point it is appropriate to point out two pervasive differences between the usual probabilistic setting of a stationary stochastic process fX n g and the (smooth) dynamical systems setting of a time series of observations on a measure-preserving system f ı T n g. The most crucial is that for deterministic dynamical systems the time series is usually not an independent process, which is a common assumption in the strictly probabilistic setting. Even if some weak-mixing is assumed in the probabilistic setting it is usually a mixing condition on the -algebras Fn D (X1 ; : : : ; X n ) generated by successive random variables, a condition which is not natural (and usually very difficult to check) for dynamical systems. Mixing conditions on dynamical systems are given more naturally in terms of the mixing of the sets of the -algebra B of the probability space (X; B; ) under the action of T and not by mixing properties of the -algebras generated by the random variables f ı T n g. The other difference is that in the probabilistic setting, although fX n g satisfy moment conditions, usually no regularity properties, such as the Hölder property or smoothness, are assumed. In contrast in dynamical systems theory the transformation T is often a smooth or piecewise smooth transformation of a Riemannian manifold X and the observation : X ! R is often assumed continuous or Hölder. The regularity of the observation turns out to play a crucial role in proving properties such as rates of decay of correlation, central limit theorems and so on. An example of a hyperbolic transformation is an expanding map of the unit interval T(x) D (2x) (where (x) is x modulo the integers). Here the derivative has modulus 2 at all points in phase space. This map preserves Lebesgue measure, has positive topological entropy, Lebesgue almost every point x has a dense orbit and periodic points for the map are dense in [0; 1). Non-hyperbolic systems are of course also an important class of examples, and in contrast to hyperbolic systems they tend to model systems of ‘low complexity’, for example systems displaying quasiperiodic behavior. The simplest non-trivial example is perhaps an irrational rotation of the unit interval [0; 1) given by a map T(x) D (x C ˛), ˛ 2 R n Q. T preserves Lebesgue measure, every point has a dense orbit (there are no periodic orbits), yet the topological entropy is zero and nearby points stay the same distance from each other under iteration under T. There is a natural notion of equivalence for measurepreserving systems. We say that measure-preserving systems (T; X; B; ) and (S; Y; C ; ) are isomorphic if (possi-
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bly after deleting sets of measure 0 from X and Y) there is a one-to-one onto measurable map : X ! Y with measurable inverse 1 such that ı T D S ı a.e. and ( 1 (A)) D (A) for all A 2 C . If X, Y are compact topological spaces we say that T is topologically conjugate to S if there exists a homeomorphism : X ! Y such that ı T D S ı . In this case we call a conjugacy. If is Cr for some r 1 we will call a Cr -conjugacy and similarly for other degrees of regularity. We will consider X D [0; 1)(mod 1) as a representation of the unit circle S 1 D fz 2 C : jzj D 1g (under the map x ! e2 i x ) and similarly represent the k-dimensional torus T k D S 1 S 1 (k-times). If the -algebra is clear from the context we will write (T; X; ) instead of (T; X; B; ) when denoting a measure-preserving system. Examples Rigid Rotation of a Compact Group If G is a compact group equipped with Haar measure and a 2 G, then the transformation T(x) D ax preserves Haar measure and is called a rigid rotation of G. If G is abelian and the transformation is ergodic (in this setting transitivity implies ergodicity), then the transformation is uniquely ergodic. Such systems always have zero topological entropy. The simplest example of such a system is a circle rotation. Take X D [0; 1)(mod 1), with T(x) D (x C ˛) where ˛ 2 R : Then T preserves Lebesgue (Haar) measure and is ergodic (in fact uniquely ergodic) if and only if ˛ is irrational. Similarly, the map T(x1 ; : : : ; x k ) D (x1 C ˛1 ; : : : ; x k C ˛ k );
defined by (k1 1; k2 1; : : : ) D (0; 0; : : : ) if each entry in the Z k i component is k i 1 , while (k1 1; k2 1; : : : ; k n 1; x1 ; x2 ; : : : ) n times
‚ …„ ƒ D (0; 0; : : : ; 0; x1 C 1; x2 ; x3 ; : : : ) when x1 6D k nC1 1. The map may be thought of as “add one and carry” and also as mapping each point to its successor in a certain order. See Sect. “Adic Transformations” for generalizations. If each k i D 2 then the system is called the dyadic (or von Neumann-Kakutani) adding machine or 2-odometer. Adding machines give examples of naturally occurring minimal systems of low orbit complexity in the sense that the topological entropy of an adding machine is zero. In fact if f is a continuous map of an interval with zero topological entropy and S is a closed, topologically transitive invariant set without periodic orbits, then the restriction of f to S is topologically conjugate to the dyadic adding machine (Theorem 11.3.13 in [46]). We say a non-empty set is an attractor for a map T if there is an open set U containing such that D \n0 T n (U) (other definitions are found in the literature). The dyadic adding machine is topologically conjugate to the Feigenbaum attractor at the limit point of period doubling bifurcations (see Sect. “Unimodal Maps”). Furthermore, attractors for continuous unimodal maps of the interval are either periodic orbits, transitive cycles of intervals, or Cantor sets on which the dynamics is topologically conjugate to an adding machine [32].
where ˛1 ; : : : ; ˛ k 2 R ; preserves k-dimensional Lebesgue (Haar) measure and is ergodic (uniquely ergodic) if and only if there are no integers m1 ; : : : ; m k , not all 0, which satisfy m1 ˛1 C C m k ˛ k 2 Z. Adding Machines Let fk i g i2N be a sequence of integers with k i 2. Equip each cyclic group Z k i with the discrete topology and form Q the product space ˙ D 1 iD1 Z k i equipped with the product topology. An adding machine corresponding to the seQ quence fk i g i2N is the topological space ˙ D 1 iD1 Z k i together with the map : ˙ !˙
Interval Exchange Maps A map T : [0; 1] ! [0; 1] is an interval exchange transformation if it is defined in the following way. Suppose that is a permutation of f1; : : : ; ng and l i > 0, i D 1; : : : ; n, is a sequence of subintervals of I (open or closed) with P i l i D 1. Define ti by l i D t i t i1 with t0 D 0. Suppose also that is an n-vector with entries ˙1. T is defined by sending the interval t i1 x < t i1 of length li to the interval X X l( j) x < l( j) ( j)(i)
with orientation preserved if the ith entry of is C1 and orientation reversed if the ith entry of is -1. Thus on each
Ergodic Theory: Basic Examples and Constructions
interval li , T has the form T(x) D i x Ca i , where i is ˙1. If i D 1 for each i, the transformation is called orientation preserving. The transformation T has finitely many discontinuities (at the endpoints of each li ), and modulo this set of discontinuities is smooth. T is also invertible (neglecting the finite set of discontinuities) and preserves Lebesgue measure. These maps have zero topological entropy and arise naturally in studies of polygonal billiards and more generally area-preserving flows. There are examples of minimal but non-ergodic interval exchange maps [56,69]. Full Shifts and Shifts of Finite Type Given a finite set (or alphabet) A D f0; : : : ; d 1g, take X D ˝ C (A) D AN (or X D AZ ) the sets of one-sided (two-sided) sequences, respectively, with entries from A. For example sequences in AN have the form x D x0 x1 : : : x n : : : . A cylinder set C(y n 1 ; : : : ; y n k ), y n i 2 A, of length k is a subset of X defined by fixing k entries; for example, C(y n 1 ; : : : ; y n k ) D fx : x n 1 D y n 1 ; : : : ; x n k D y n k g: We define the set Ak to consist of all cylinders C(y1 ; : : : ; y k ) determined by fixing the first k entries, i. e. an element of Ak is specified by fixing the first k entries of a sequence x0 x k by requiring x i D y i , i D 0; : : : ; k. Let p D (p0 ; : : : ; p d1 ) be a probability vector: all P p i 0 and d1 iD0 p i D 1. For any cylinder B D C(b1 ; : : : ; b k ) 2 Ak , define g k (B) D p b 1 : : : p b k :
(1)
It can be shown that these functions on Ak extend to a shift-invariant measure p on AN (or AZ ) called product measure. (See the article on Measure Preserving Systems.) The space AN or AZ may be given a metric by defining ( 1 if x0 6D y0 ; d(x; y) D 1 if x n 6D y n and x i D y i for jij < n : 2jnj The shift (:x0 x1 x n ) D :x1 x2 x n is ergodic with respect to p . The measure-preserving system (˝; B; ; ) (with B the -algebra of Borel subsets of ˝(A), or its completion), is denoted by B(p) and is called the Bernoulli shift determined by p. This system models an infinite number of independent repetitions of an experiment with finitely many outcomes, the ith of which has probability pi on each trial. These systems are mixing of all orders (i. e. n is mixing for all n 1) and have countable Lebesgue spectrum (hence are all spectrally isomorphic). Kolmogorov
and Sinai showed that two of them cannot be isomorphic unless they have the same entropy; Ornstein [81] showed the converse. B(1/2; 1/2) is isomorphic to the Lebesguemeasure-preserving transformation x ! 2x mod 1 on [0; 1]; similarly, B(1/3; 1/3; 1/3) is isomorphic to x ! 3x mod 1. Furstenberg asked whether the only nonatomic measure invariant for both x ! 2x mod 1 and x ! 3x mod 1 on [0; 1] is Lebesgue measure. Lyons [68] showed that if one of the actions is K; then the measure must be Lebesgue, and Rudolph [100] showed the same thing under the weaker hypothesis that one of the actions has positive entropy. For further work on this question, see [50,87]. This construction can be generalized to model onestep finite-state Markov stochastic processes as dynamical systems. Again let A D f0; : : : ; d1g, and let p D (p0 ; : : : ; p d1 ) be a probability vector. Let P be a d d stochastic matrix with rows and columns indexed by A. This means that all entries of P are nonnegative, and the sum of the entries in each row is 1. We regard P as giving the transition probabilities between pairs of elements of A. Now we define for any cylinder B D C(b1 ; : : : ; b k ) 2 Ak p;P (B) D p b 1 Pb 1 b 2 Pb 2 b 3 : : : Pb k1 b k :
(2)
It can be shown that p;P extends to a measure on the Borel -algebra of ˝ C (A), and its completion. (See the article on Measure Preserving Systems.) The resulting stochastic process is a (one-step, finite-state) Markov process. If p and P also satisfy pP D p ;
(3)
then the Markov process is stationary. In this case we call the (one or two-sided) measure-preserving system the Markov shift determined by p and P. Aperiodic and irreducible Markov chains (those for which a power of the transition matrix P has all entries positive) are strongly mixing, in fact are isomorphic to Bernoulli shifts (usually by means of a complicated measure-preserving recoding). More generally we say a set AZ is a subshift if it is compact and invariant under . A subshift is said to be of finite type (SFT) if there exists an d d matrix M D (a i j ) such that all entries are 0 or 1 and x 2 if and only if a x i x iC1 D 1 for all i 2 Z. Shifts of finite type are also called topological Markov chains. There are many invariant measures for a non-trivial shift of finite type. For example the orbit of each periodic point is the support of an invariant measure. An important role in the theory, derived from motivations of statistical mechanics, is played by equilibrium measures (or equilibrium states) for continuous functions : ! R, i. e. those measures
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R which maximize fh () C dg over all shift-invariant probability measures, where h () is the measure-theoretic entropy of with respect to . The study of full shifts or shifts of finite type has played a prominent role in the development of the hyperbolic theory of dynamical systems as physical systems with ‘chaotic’ dynamics ‘typically’ possess an invariant set with induced dynamics topologically conjugate to a shift of finite type (see the discussion by Smale in p. 147 in [108]). Dynamical systems in which there are transverse homoclinic connections are a common example (Theorem 5.3.5 in [45]). Furthermore in certain settings positive metric entropy implies the existence of shifts of finite type. One result along these lines is a theorem of Katok [53]. Let hto p ( f ) denote the topological entropy of a map f and h ( f ) denote metric entropy with respect to an invariant measure . Theorem 1 (Katok) Suppose T : M ! M is a C 1C diffeomorphism of a closed manifold and is an invariant measure with positive metric entropy (i. e. h (T) > 0). Then for any 0 < < h (T) there exists an invariant set topologically conjugate to a transitive shift of finite type with hto p (Tj) > h (T) . More Examples of Subshifts We consider some further examples of systems that are given by the shift transformation on a subset of the set of (usually doubly-infinite) sequences on a finite alphabet, usually f0; 1g: Associated with each subshift is its language, the set of all finite blocks seen in all sequences in the subshift. These languages are extractive (or factorial ) (every subword of a word in the language is also in the language) and insertive (or extendable) (every word in the language extends on both sides to longer words in the language). In fact these two properties characterize the languages (subsets of the set of finite-length words on an alphabet) associated with subshifts. Prouhet–Thue–Morse An interesting (and often rediscovered) element of f0; 1gZC is produced as follows. Start with 0 and at each stage write down the opposite (00 D 1; 10 D 0) or mirror image of what is available so far. Or, repeatedly apply the substitution 0 ! 01; 1 ! 10: 0 01 0 1 10 0 1 10 0110 :: :
The nth entry is the sum, mod 2, of the digits in the dyadic expansion of n: Using Keane’s block multiplication [55] according to which if B is a block, B 0 D B; B 1 D B0 ; and B (!1 ! n ) D (B !1 ) (B !n ); we may also obtain this sequence as 0 01 01 01 : The orbit closure of this sequence is uniquely ergodic (there is a unique shift-invariant Borel probability measure, which is then necessarily ergodic). It is isomorphic to a skew product (see Sect. “Skew Products”) over the von Neumann-Kakutani adding machine, or odometer (see Sect. “Adding Machines”). Generalized Morse systems, that is, orbit closures of sequences like 0001001001 ; are also isomorphic to skew products over compact group rotations. Chacon System This is the orbit closure of the sequence generated by the substitution 0 ! 0010; 1 ! 1: It is uniquely ergodic and is one of the first systems shown to be weakly mixing but not strongly mixing. It is prime (has no nontrivial factors) [30], and in fact has minimal self joinings [31]. It also has a nice description by means of cutting up the unit interval and stacking the pieces, using spacers (see Sect. “Cutting and Stacking”). This system has singular spectrum. It is not known whether or not its Cartesian square is loosely Bernoulli. Sturmian Systems Take the orbit closure of the sequence ! n D [1˛;1) (n˛); where ˛ is irrational. This is a uniquely ergodic system that is isomorphic to rotation by ˛ on the unit interval. These systems have minimal complexity in the sense that the number of n-blocks grows as slowly as possible (n C 1) [29]. Toeplitz Systems A bi-infinite sequence (x i ) is a Toeplitz sequence if the set of integers can be decomposed into arithmetic progressions such that each xi is constant on each arithmetic progression. A shift space X is a Toeplitz shift if it is the closure of the orbit of a Toeplitz sequence. It is possible to construct Toeplitz shifts which are uniquely ergodic and isomorphic to a rotation on a compact abelian group [34]. Sofic Systems These are images of SFT’s under continuous factor maps (finite codes, or block maps). They correspond to regular languages—languages whose words are recognizable by finite automata. These are the same as the languages defined by regular expressions—finite expressions built up from ; (empty set), (empty word), + (union of two languages), (all concatenations of words
Ergodic Theory: Basic Examples and Constructions
from two languages), and (all finite concatenations of elements). They also have the characteristic property that the family of all follower sets of all blocks seen in the system is a finite family; similarly for predecessor sets. These are also generated by phase-structure grammars which are linear, in the sense that every production is either of the form A ! Bw or A ! w; where A and B are variables and w is a string of terminals (symbols in the alphabet of the language). (A phase-structure grammar consists of alphabets V of variables and A of terminals, a set of productions, which is finite set of pairs of words (˛; w); usually written ˛ ! w; of words on V [ A; and a start symbol S. The associated language consists of all words on the alphabet A of terminals which can be made by starting with S and applying a finite sequence of productions.) Sofic systems typically support many invariant measures (for example they have many periodic points) but topologically transitive ones (those with a dense orbit) have a unique measures of maximal entropy (see [65]). Context-free Systems These are generated by phasestructure grammars in which all productions are of the form A ! w; where A is a variable and w is a string of variables and terminals. Coded Systems These are systems all of whose blocks are concatenations of some (finite or infinite) list of blocks. These are the same as the closures of increasing sequences of SFT’s [60]. Alternatively, they are the closures of the images under finite edge-labelings of irreducible countable-state topological Markov chains. They need not be context-free. Squarefree languages are not coded, in fact do not contain any coded systems of positive entropy. See [13,14,15]. Smooth Expanding Interval Maps Take X D [0; 1)(mod 1), m 2 N, m > 1 and define
If m D 2 the system is isomorphic to a model of tossing a fair coin, which is a common example of randomness. To see this let P D fP0 D [0; 1/2); P1 D [1/2; 1]g be a partition of [0; 1] into two subintervals. We code the orbit under T of any point x 2 [0; 1) by 0’s and 1’s by letting x k D i if T k x 2 Pi ; k D 0; 1; 2; : : : . The map : X ! f0; 1gN which associates a point x to its itinerary in this way is a measure-preserving map from (X; ) to f0; 1gN equipped with the Bernoulli measure from p0 D p1 D 12 . The map satisfies ı T D ı , a.e. and is invertible a.e., hence is an isomorphism. Furthermore, reading the binary expansion of x is equivalent to following the orbit of x under T and noting which element of the partition P is entered at each time. Borel’s theorem on normal numbers (base m) may be seen as a special case of the Birkhoff Ergodic Theorem in this setting. Piecewise C2 Expanding Maps The main statistical features of the examples in Sect. “Smooth Expanding Interval Maps” generalize to a broader class of expanding maps of the interval. For example: Let X D [0; 1] and let P D fI1 ; : : : ; I n g (n 2) be a partition of X into intervals (closed, half-open or open) such that I i \ I j D ; if i 6D j. Let I oi denote the interior of I i . Suppose T : X ! X satisfies: (a) For each i D 1; : : : ; n, TjI i has a C2 extension to the 0 closure I¯i of I i and jT (x)j ˛ > 1 for all x 2 I oi . (b) T(I j ) D [ i2P j I i Lebesgue a.e. for some non-empty subset Pj f1; : : : ; ng. (c) For each I j there exists nj such that T n j (I j ) D [0; 1] Lebesgue a.e. Then T has an invariant measure which is absolutely continuous with respect to Lebesgue measure m, and there d C. Furthermore T is exists C > 0 such that C1 dm ergodic with respect to and displays the same statistical properties listed above for the C2 expanding maps [20]. (See the “Folklore Theorem” in the article on Measure Preserving Systems.)
T(x) D (mx): Then T preserves Lebesgue measure (recall that T preserves if (T 1 A) D (A) for all A 2 B). Furthermore it can be shown that T is ergodic. This simple map exemplifies many of the characteristics of systems with some degree of hyperbolicity. It is isomorphic to a Bernoulli shift. The map has positive topological entropy and exponential divergence of nearby orbits, and Hölder functions have exponential decay of correlations and satisfy the central limit theorem and other strong statistical properties [20].
More Interval Maps Continued Fraction Map This is the map T : [0; 1] ! [0; 1] given by Tx D 1/x mod 1, and it corresponds to the shift [0; a1 ; a2 ; : : : ] ! [0; a2 ; a3 ; : : : ] on the continued fraction expansions of points in the unit interval (a map on N N ): It preserves a unique finite measure equivalent to Lebesgue measure, the Gauss measure dx/(log 2)(1 C x): It is Bernoulli with entropy 2 /6 log 2 (in fact the natural partition into intervals is a weak Bernoulli generator, for the definition and details see [91]). By using the Ergodic
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Theorem, Khintchine and Lévy showed that
(a1 a n )1/n !
1
Y kD1
1 1C 2 k C 2k
there are good summaries by Bertrand-Mathis [9] and Blanchard [12]. p
k log log 2
a.e. as n ! 1 ;
pn 1 2 if [0; a1 ; : : : ; a n ] D ; then log q n ! a.e. ; qn n 12 log 2 ˇ ˇ ˇ 1 p n (x) ˇˇ 2 log ˇˇx a.e. ; ! ˇ n q n (x) 6 log 2 and if m is Lebesgue measure (or any equivalent measure) and is Gauss measure, then for each interval I; m(T n I) ! (I); in fact exponentially fast, with a best constant 0:30366 : : : see [10,75]. The Farey Map This is the map U : [0; 1] ! [0; 1] given by Ux D x/(1 x) if 0 x 1/2; Ux D (1 x)/x if 1/2 x 1: It is ergodic for the -finite infinite measure dx/x (Rényi and Parry). It is also ergodic for the Minkowski measure d, which is a measure of maximal entropy. This map corresponds to the shift on the Farey tree of rational numbers which provide the intermediate convergence (best one-sided) as well as the continued fraction (best two-sided) rational approximations to irrational numbers. See [61,62]. f -Expansions Generalizing the continued fraction map, let f : [0; 1] ! [0; 1] and let fI n g be a finite or infinite partition of [0; 1] into subintervals. We study the map f by coding itineraries with respect to the partition fI n g: For many examples, absolutely continuous (with respect to Lebesgue measure) invariant measures can be found and their dynamical properties determined. See [104]. ˇ-Shifts This is the special case of f -expansions when f (x) D ˇx mod 1 for some fixed ˇ > 1: This map of the interval is called the ˇ-transformation. With a proper choice of partition, it is represented by the shift on a certain subshift of the set of all sequences on the alphabet D D f0; 1; : : : ; bˇcg, called the ˇ-shift. A point x is expanded as an infinite series in negative powers of ˇ with coefficients from this set; dˇ (x)n D bˇ f n (x)c: (By convention terminating expansions are replaced by eventually periodic ones.) A one-sided sequence on the alphabet D is in the ˇ-shift if and only if all of its shifts are lexicographically less than or equal to the expansion dˇ (1) of 1 base ˇ. A one-sided sequence on the alphabet D is the valid expansion of 1 for some ˇ if and only if it lexicographically dominates all its shifts. These were first studied by Bissinger [11], Everett [35], Rényi [93] and Parry [84,85];
For ˇ D 1C2 5 ; dˇ (1) D 10101010 : : : . For ˇ D 32 ; dˇ (1) D 101000001 : : : (not eventually periodic). Every ˇ-shift is coded. The topological entropy of a ˇ-shift is log ˇ: There is a unique measure of maximal entropy log ˇ. A ˇ-shift is a shift of finite type if and only if the ˇ-expansion of 1 is finite. It is sofic if and only if the expansion of 1 is eventually periodic. If ˇ is a Pisot–Vijayaragavhan number (algebraic integer all of whose conjugates have modulus less than 1), then the ˇ-shift is sofic. If the ˇ-shift is sofic, then ˇ is a Perron number (algebraic integer of maximum modulus among its conjugates). Theorem 2 (Parry [86]) Every strongly transitive (for every nonempty open set U, [n>0 T n U D X) piecewise monotonic map on [0; 1] is topologically conjugate to a ˇ-transformation. Gaussian Systems Consider a real-valued stationary process f f k : 1 < k < 1g on a probability space (˝; F ; P): The process (and the associated measure-preserving system consisting of the shift and a shift-invariant measure on RZ ) is called Gaussian if for each d 1, any d of the f k form an Rd valued Gaussian random variable on ˝ : this means that with E( f k ) D m for all k and Z Ai j D
˝
( f k i m)( f k j m)dP D C(k i k j ) for i; j D 1; : : : ; d ;
where C() is a function, for each Borel set B R, Pf! : ( f k 1 (!); : : : ; f k d (!)) 2 Bg
Z 1 1 D p exp (x (m; : : : ; m)) tr d/2 2 2 det A B A1 (x (m; : : : ; m)) dx1 dx d : where A is a matrix with entries (A i j ). The function C(k) is positive semidefinite and hence has an associated measure on [0; 2) such that Z 2 e i k t d (t) : C(k) D 0
Theorem 3 (de la Rue [33]) The Gaussian system is ergodic if and only if the “spectral measure” is continuous (i. e., nonatomic), in which case it is also weakly mixing. It
Ergodic Theory: Basic Examples and Constructions
is mixing if and only if C(k) ! 0 as jkj ! 1: If is singular with respect to Lebesgue measure, then the entropy is 0; otherwise the entropy is infinite. For more details see [28]. Hamiltonian Systems (This paragraph is from the article on Measure Preserving Systems.) Many systems that model physical situations can be studied by means of Hamilton’s equations. The state of the entire system at any time is specified by a vector (q; p) 2 R2n , the phase space, with q listing the coordinates of the positions of all of the particles, and p listing the coordinates of their momenta. We assume there is a time-independent Hamiltonian function H(q, p) such that the time development of the system satisfies Hamilton’s equations: dq i @H ; D dt @p i
dp i @H ; D dt @q i
i D 1; : : : ; n :
(4)
Often in applications the Hamiltonian function is the sum of kinetic and potential energy: H(q; p) D K(p) C U(q) :
(5)
Solving these equations with initial state (q, p) for the system produces a flow (q; p) ! Tt (q; p) in phase space which moves (q, p) to its position Tt (q; p)t units of time later. According to Liouville’s formula Theorem 3.2 in [69], this flow preserves Lebesgue measure on R2n . Calculating dH/dt by means of the Chain Rule X @H dp i dH @H dq i D C dt @p i dt @q i dt i
and using Hamilton’s equations shows that H is constant on orbits of the flow, and thus each set of constant energy X(H0 ) D f(q; p) : H(q; p) D H0 g is an invariant set. There is a natural invariant measure on a constant energy set X(H0 ) for the restricted flow, namely the measure given by rescaling the volume element dS on X(H0 ) by the factor 1/jjOHjj. Billiard Systems These form an important class of examples in ergodic theory and dynamical systems, motivated by natural questions in physics, particularly the behavior of gas models. Consider the motion of a particle inside a bounded region D in Rd with piecewise smooth (C1 at least) boundaries. In the case of planar billiards we have d D 2. The particle moves in a straight line with
constant speed until it hits the boundary, at which point it undergoes a perfectly elastic collision with the angle of incidence equal to the angle of reflection and continues in a straight line until it next hits the boundary. It is usual to normalize and consider unit speed, as we do in this discussion for convenience. We take coordinates (x, v) given by the Euclidean coordinates in x 2 D together with a direction vector v 2 S d1 . A flow t is defined with respect to Lebesgue almost every (x, v) by translating x a distance t defined by the direction vector v, taking account of reflections at boundaries. t preserves a measure absolutely continuous with respect to Riemannian volume on (x, v) coordinates. The flow we have described is called a billiard flow. The corresponding billiard map is formed by taking the Poincaré map corresponding to the cross-section given by the boundary @D. We will describe the planar billiard map; the higher dimensional generalization is clear. The billiard map is a map T : @D ! @D, where @D is coordinatized by (s; ), s 2 [0; L), where L is the length of @D and 2 (0; ) measures the angle that inward pointing vectors make with the tangent line to @D at s. Given a point (s; ), the angle defines an oriented line l(s; ) which intersects @D in two points s and s0 . Reflecting l in the tangent line to @D at the point s0 gives another oriented line passing through s0 with angle 0 (measured with respect to the angular coordinate system based at s0 ). The billiard map is the map T(s; ) D (s 0 ; 0 ). T preserves a measure D sin ds d . The billiard flow may be modeled as a suspension flow over the billiard map (see Sect. “Suspension Flows”). If the region D is a polygon in the plane (or polyhedron in Rd ), then @D consists of the faces of the polyhedron. The dynamical behavior of the billiard map or flow in regions with only flat (non-curved) boundaries is quite different to that of billiard flows or maps in regions D with strictly convex or strictly concave boundaries. The topological entropy of a flat polygonal billiard is zero. Research interest focuses on the existence and density of periodic or transitive orbits. It is known that if all the angles between sides are rational multiples of then there are periodic orbits [17,74,112] and they are dense in the phase space [16]. It is also known that a residual set of polygonal billiards are topologically transitive and ergodic [58,117]. On the other hand, billiard maps in which @D has strictly convex components are physical examples of nonuniformly hyperbolic systems (with singularities). The meaning of concave or convex varies in the literature. We will consider a billiard flow inside a circle to be a system with a strictly concave boundary, while a billiard flow on the torus from which a circle has been excised to be a billiard flow with strictly convex boundary.
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Ergodic Theory: Basic Examples and Constructions
The class of billiards with some strictly convex boundary components, sometimes called dispersing billiards or Sinai billiards, was introduced by Sinai [106] who proved many of their fundamental properties. Lazutkin [63] proved that planar billiards with generic strictly concave boundary are not ergodic. Nevertheless Bunimovich [22,23] produced a large of billiard systems, Bunimovich billiards, with strictly concave boundary segments (perhaps with some flat boundaries as well) which were ergodic and non-uniformly hyperbolic. For more details see [26,54,66,109]. We will discuss possibly the simplest example of a dispersing billiard, namely a toral billiard with a single convex obstacle. Take the torus T 2 and consider a single strictly convex subdomain S with C 1 boundary. The domain of the billiard map is [0; L)(0; ), where L is the length of @S. The measure sin( )ds d is preserved. If the curvature of @S is everywhere non-zero, then the billiard map T has positive topological entropy, periodic points are dense, and in fact the system is isomorphic to a Bernoulli shift [41]. KAM-Systems and Stably Non-Ergodic Behavior A celebrated theorem of Kolmogorov, Arnold and Moser (the KAM theorem) implies that the set of ergodic areapreserving diffeomorphisms of a compact surface without boundary is not dense in the Cr topology for r 4. This has important implications, in that there are natural systems in which ergodicity is not generic. The constraint of perturbing in the class of area-preserving diffeomorphisms is an appropriate imposition in many physical models. We will take the version of the KAM theorem as given in Theorem 5.1 in [69] (original references include [3,59,79]). An elliptic fixed point for an area-preserving diffeomorphism T of a surface M is called a nondegenerate elliptic fixed point if there is a local Cr , r 4, change of coordinates h so that in polar coordinates hT h1 (r; ) D (r; C ˛0 C ˛1 r) C F(r; ) ; where all derivatives of F up to order 3 vanish, ˛1 6D 0 and ˙2 ˛0 6D 0; ˙ 2 ; ; 3 . A map of the form (r; ) D (r; C ˛0 C ˛1 r) ; where ˛1 6D 0, is called a twist map. Note that a twist map leaves invariant the circle r D k, for any constant k, and rotates each invariant curve by a rigid rotation ˛1 r, the magnitude of the rotation depending upon r. With respect to two-dimensional Lebesgue measure a twist map is certainly not ergodic. Theorem 4 Suppose T is a volume-preserving diffeomorphism of class Cr , r 4, of a surface M. If x is a non-degenerate elliptic fixed point, then for every > 0 there exists
a neighborhood U of x and a set U0; U with the properties: (a) U0; is a union of T-invariant simple closed curves of class Cr1 containing x in their interior. (b) The restriction of T to each such invariant curve is topologically conjugate to an irrational rotation. (c) m(U U0; ) m(U ), where m is Lebesgue measure on M. It is possible to prove the existence of a Cr volume preserving diffeomorphism (r 4) with a non-degenerate elliptic fixed point and also show that if T possesses a non-degenerate elliptic fixed point then there is a neighborhood V of T in the Cr topology on volume-preserving diffeomorphisms such that each T 0 2 V possesses a non-degenerate elliptic fixed point Chapter II, Sect. 6 in [69]. As a corollary we have Corollary 1 Let M be a compact surface without boundary and Diffr (M) the space of Cr area-preserving diffeomorphisms with the Cr topology. Then the set of T 2 Diffr (M) which are ergodic with respect to the probability measure determined by normalized area is not dense in Diffr (M) for r 4. Smooth Uniformly Hyperbolic Diffeomorphisms and Flows Time series of measurements on deterministic dynamical systems sometimes display limit laws exhibited by independent identically distributed random variables, such as the central limit theorem, and also various mixing properties. The models of hyperbolicity we discuss in this section have played a key role in showing how this phenomenon of ‘chaotic behavior’ arises in deterministic dynamical systems. Hyperbolic sets and their associated dynamics have also been pivotal in studies of structural stability. A smooth system is Cr structurally stable if a small perturbation in the Cr topology gives rise to a system which is topologically conjugate to the original. When modeling a physical system it is desirable that slight changes in the modeling parameters do not greatly affect the qualitative or quantitative behavior of the ensemble of orbits considered as a whole. The orbit of a point may change drastically under perturbation (especially if the system has sensitive dependence on initial conditions) but the collection of all orbits should ideally be ‘similar’ to the original unperturbed system. In the latter case one would hope that statistical properties also vary only slightly under perturbation. Structural stability is one, quite strong, notion of stability. The conclusion of a body of work on structural
Ergodic Theory: Basic Examples and Constructions
stability is that a system is C1 structurally stable if and only if it is uniformly hyperbolic and satisfies a technical assumption called strong transversality (see below for details). Suppose M is a C1 compact Riemannian manifold equipped with metric d and tangent space TM with norm kk. Suppose also that U M is a non-empty open subset and T : U ! T(U) is a C1 diffeomorphism. A compact T invariant set U is called a hyperbolic set if there is a splitting of the tangent space Tp M at each point p 2 into two invariant subspaces, Tp M D E u (p) ˚ E s (p), and a number 0 < < 1 such that for n 0 kD p T n vk Cn kvk for v 2 E s (p) ; kD p T n vk < Cn kvk for v 2 E u (p) : The subspace Eu is called the unstable or expanding subspace and the subspace Es the stable or contracting subspace. The stable and unstable subspaces may be integrated to produce stable and unstable manifolds W s (p) D fy : d(T n p; T n y) ! 0g u
W (p) D fy : d(T
n
p; T
n
as n ! 1 ;
y) ! 0g
as n ! 1 :
The stable and unstable manifolds are immersions of Euclidean spaces of the same dimension as E s (p) and E u (p), respectively, and are of the same differentiability as T. Moreover, Tp (W s (p)) D E s (p) and Tp (W u (p)) D E u (p). It is also useful to define local stable manifolds and local unstable manifolds by W s (p) D fy 2 W s (p) : d(T n p; T n y) < g for all n 0 ; W u (p) D fy 2 W u (p) : d(T n p; T n y) < g for all n 0 : Finally we discuss the notion of strong transversality. We say a point x is non-wandering if for each open neighborhood U of x there exists an n > 0 such that T n (U) \ U 6D ;. The NW set of non-wandering points is called the non-wandering set. We say a dynamical system has the strong transversal property if W s (x) intersects W u (y) transversely for each pair of points x; y 2 NW. In the Cr , r 1 topology Robbin [94], de Melo [78] and Robinson [95,96] proved that dynamical systems with the strong transversal property are structurally stable, and Robinson [97] in addition showed that strong transversality was also necessary. Mañé [70] showed that a C1 structurally stable diffeomorphism must be uniformly hyperbolic and Hayashi [47] extended this to flows. Thus a C1 diffeomorphism or flow on a compact manifold is structurally stable if and only it is uniformly hyperbolic and satisfies the strong transversality condition.
Geodesic Flow on Manifold of Negative Curvature The study of the geodesic flow on manifolds of negative sectional curvature by Hedlund and Hopf was pivotal to the development of the ergodic theory of hyperbolic systems. Suppose that M is a geodesically complete Riemannian manifold. Let p;v (t) be the geodesic with p;v (0) D p and ˙ p;v (0) D v, where ˙ p;v denotes the derivative with respect to time t. The geodesic flow is a flow t on the tangent bundle TM of M, t : R TM ! TM, defined by t (p; v) D ( p;v (t); ˙p;v (t)) : where (p; v) 2 TM. Since geodesics have constant speed, if kvk D 1 then k p;v (t)k D 1 for all t, and thus the unit tangent bundle T 1 M D f(p; v) 2 TM : kvk D 1g is preserved under the geodesic flow. The geodesic flow and its restriction to the unit tangent bundle both preserve a volume form, Liouville measure. In 1934 Hedlund [48] proved that the geodesic flow on the unit tangent bundle of a surface of strictly negative constant sectional curvature is ergodic, and in 1939 Hopf [49] extended this result to manifolds of arbitrary dimension and strictly negative (not necessarily constant) curvature. Hopf’s technique of proof of ergodicity (Hopf argument) was extremely influential and used the foliation of the tangent space into stable and unstable manifolds. For a clear exposition of this technique, and the property of absolute continuity of the foliations into stable and unstable manifolds, see [66]. The geodesic flow on manifolds of constant negative sectional curvature is an Anosov flow (see Sect. “Anosov Systems”). We remark that for surfaces sectional curvature is the same as Gaussian curvature. Recently the time-one map of the geodesic flow on the unit tangent bundle of a surface with constant negative curvature, which is a partially hyperbolic system (see Sect. “Partially Hyperbolic Dynamical Systems”), was shown to be stably ergodic [44], so the geodesic flow is still playing a major role in the development of ergodic theory. Horocycle Flow All surfaces endowed with a Riemannian metric of constant negative curvature are quotients of the upper half-plane H C :D fx C iy 2 C : y > 0g with dx 2 Cdy 2
, whose sectional curvature is the metric ds 2 D y2 1. The orientation-preserving isometries of this metric are exactly the linear fractional (also known as Möbius) transformations: az C b a b 2 H C] : 2 SL(2; R); [z 2 H C 7! c d cz C d Since each matrix ˙I corresponds to the identity transformation, we consider matrices in PSL(2; R) :D SL(2; R)/f˙Ig.
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The unit tangent bundle, S H C , of the upper halfplane can be identified with PSL(2; R). Then the geodesic flow corresponds to the transformations t e 0 t 2 R 7! 0 et seen as acting on PSL(2; R). The unstable foliation of an element A 2 PSL(2; R) Š S H C is given by 1 t A; t 2 R ; 0 1 and the flow along this foliation, given by 1 t ; t 2 R 7! 0 1 is called the horocycle flow. Similarly for the flow induced on the unit tangent bundle of each quotient of the upper half-plane by a discrete group of linear fractional transformations. The geodesic and horocycle flows acting on a (finitevolume) surface of constant negative curvature form the fundamental example of a transverse pair of actions. The geodesic flow often has many periodic orbits and many invariant measures, has positive entropy, and is in fact Bernoulli with respect to the natural measure [82], while the horocycle flow is often uniquely ergodic [39,72] and of entropy zero, although mixing of all orders [73]. See [46] for more details. Markov Partitions and Coding If (X; T; B; ) is a dynamical system then a finite partition of X induces a coding of the orbits and a semi-conjugacy with a subshift on a symbol space (it may not of course be a full conjugacy). For hyperbolic systems a special class of partitions, Markov partitions, induce a conjugacy for the invariant dynamics to a subshift of finite type. A Markov partition P for an invariant subset of a diffeomorphism T of a compact manifold M is a finite collection of sets Ri , 1 i n called rectangles. The rectangles have the property, for some > 0, if x; y 2 R i then W s (x) \ W u (y) 2 R i . This is sometimes described as being closed under local product structure. We let W u (x; R i ) denote W u (x) \ R i and W s (x; R i ) denote W s (x) \ R i . Furthermore we require for all i, j: (1) (2) (3) (4)
Each Ri is the closure of its interior. [i R i R i \ R j D @R i \ @R j if i 6D j if x 2 R oi and T(x) 2 R oj then W u (T(x); R j )
T(W u (x; R i )) and W s (x; R i ) T 1 (W u (T(x); R j ))
Anosov Systems An Anosov diffeomorphism [2] is a uniformly hyperbolic system in which the entire manifold is a hyperbolic set. Thus an Anosov diffeomorphism is a C1 diffeomorphism T of M with a DT-invariant splitting (which is a continuous splitting) of the tangent space TM(x) at each point p into a disjoint sum Tp M D E u (p) ˚ E s (p) and there exist constants 0 < < 1, constant C > 0 such that kDT n vk < Cn kvk for all v 2 E s (p) and kDT n wk Cn kwk for all w 2 E u (p). A similar definition holds for Anosov flows : R M ! M. A flow is Anosov if there is a splitting of the tangent bundle into flow-invariant subspaces E u ; E s ; E c so D p t E sp D Es t (p) , D p t E up D Eu t (p) and D p t E cp D Ec t (p) , and at each point p 2 M Tp M D E sp ˚ E up ˚ E cp k(D p t )vk < C t kvk for v 2 E s (p) k(D p t )vk < C t kvk for v 2 E u (p) for some 0 < < 1. The tangent to the flow direction E c (p) is a neutral direction: k(D p t )vk D kvk for v 2 E c (p) : Anosov proved that Anosov flows and diffeomorphisms which preserve a volume form are ergodic [2] and are also structurally stable. Sinai [105] constructed Markov partitions for Anosov diffeomorphisms and hence coded trajectories via a subshift of finite type. Using ideas from statistical physics in [107] Sinai constructed Gibbs measures for Anosov systems. An SRB measure (see Sect. “Physically Relevant Measures and Strange Attractors”) is a type of Gibbs measure corresponding to the potential log j det(DTj E u )j and is characterized by the property of absolutely continuous conditional measures on unstable manifolds. The simplest examples of Anosov diffeomorphisms are perhaps the two-dimensional hyperbolic toral automorphisms (the n > 2 generalization is clear). Suppose A is a 2 2 matrix with integer entries a b c d such that det(A) D 1 and A has no eigenvalues of modulus 1. Then A defines a transformation of the two-dimensional torus T 2 D S 1 S 1 such that if v 2 T 2 , v vD 1 ; v2
Ergodic Theory: Basic Examples and Constructions
then (av1 C bv2 ) : Av D (cv1 C dv2 ) A preserves Lebesgue (or Haar) measure and is ergodic. A prominent example of such a matrix is 2 1
1 1
;
which is sometimes called the Arnold Cat Map. Each point with rational coordinates (p1 /q1 ; p2 /q2 ) is periodic. p There are two eigenvalues 1/ < 1 < D (3 C 5)/2 with orthogonal eigenvectors, and the projections of the eigenspaces from R2 to T 2 are the stable and unstable subspaces. Axiom A Systems In the case of Anosov diffeomorphisms the splitting into contracting and expanding bundles holds on the entire phase space M. A system T : M ! M is an Axiom A system if the non-wandering set NW is a hyperbolic set and periodic points are dense in the nonwandering set. NW M may have Lebesgue measure zero. A set M is locally maximal if there exists an open set U such that D \n2Z T n (U). The solenoid and horseshoe discussed below are examples of locally maximal sets. Bowen [18] constructed Markov partitions for Axiom A diffeomorphisms . Ruelle imported ideas from statistical physics, in particular the idea of an equilibrium state and the variational principle, to the study of Axiom A systems (see [101,102]) This work extended the notion of Gibbs measure and other ideas from statistical mechanics, introduced by Sinai for Anosov systems [107], into Axiom A systems. One achievement of the Axiom A program was the Smale Decomposition Theorem, which breaks the dynamics of Axiom A systems into locally maximal sets and describes the dynamics on each [18,19,108]. Theorem 5 (Spectral Decomposition Theorem) If T is Axiom A then there is a unique decomposition of the nonwandering set NW of T NW D 1 [ [ k as a disjoint union of closed, invariant, locally maximal hyperbolic sets i such that T is transitive on each i . Furthermore each i may be further decomposed into a disj joint union of closed sets i , j D 1; : : : ; n i such that T n i is j
topologically mixing on each i and T cyclically permutes j
the i .
Horseshoe Maps This type of map was introduced by Steven Smale in the 1960’s and has played a pivotal role in the development of dynamical systems theory. It is perhaps the canonical example of an Axiom A system [108] and is conjugate to a full shift on 2 symbols. Let S be a unit square in R2 and let T be a diffeomorphism of S onto its image such that S \ T(S) consists of two disjoint horizontal strips S0 and S1 . Think of stretching S uniformly in the horizontal direction and contracting uniformly in the vertical direction to form a long thin rectangle, and then bending the rectangle into the shape of a horseshoe and laying the straight legs of the horseshoe back on the unit square S. This transformation may be realized by a diffeomorphism and we may also require that T restricted to T 1 S i , i D 0; 1, acts as a linear map. The restriction of T i to the maximal invariant set H D \1 iD1 T S is a Smale horseshoe map. H is a Cantor set, the product of a Cantor set in the horizontal direction and a Cantor set in the vertical direction. The conjugacy with the shift on two symbols is realized by mapping x 2 H to its itinerary with respect to the sets S0 and S1 under powers of T (positive and negative powers). Solenoids The solenoid is defined on the solid torus X in R3 which we coordinatize as a circle of two-dimensional solid disks, so that X D f( ; z) : 2 [0; 1) and jzj 1; z 2 Cg The transformation T : X ! X is given by 1 1 T( ; z) D 2 (mod 1) ; z C e2 i 4 2 Geometrically the transformation stretches the torus to twice its length, shrinks its diameter by a factor of 4, then twists it and doubles it over, placing the resultant object without self-intersection back inside the original solid torus. T(X) intersects each disk D c D f( ; z) : D cg in two smaller disks of 14 the diameter. The transformation T contracts volume by a factor of 2 upon each application, yet there is expansion in the direction ( ! 2 ). The solenoid A D \n0 T n (X) has zero Lebesgue measure, is T-invariant and is (locally) topologically a line segment cross a two-dimensional Cantor set (A intersects each disk Dc in a Cantor set). The set A is an attractor, in that all points inside X limit under iteration by T upon A. T : A ! A is an Axiom A system. Partially Hyperbolic Dynamical Systems Partially hyperbolic dynamical systems are a generalization of uniformly hyperbolic systems in that an invariant central direction is allowed but the contraction in the
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central direction is strictly weaker than the contraction in the contracting direction and the expansion in the central direction is weaker than the expansion in the expanding direction. More precisely, suppose M is a C1 compact (adapted) Riemannian manifold equipped with metric d and tangent space TM with norm kk. A C1 diffeomorphism T of M is a partially hyperbolic diffeomorphism if there is a nontrivial continuous DT invariant splitting of the tangent space Tp M at each point p into a disjoint sum Tp M D E u (p) ˚ E c (p) ˚ E s (p) and continuous positive functions m; M; ˜ ; such that
Es
is contracted: kD p T n v s k m(p) < 1; if v s 2 E s (x) n f0g then kv sk Eu is expanded: kD Tv u k if v u 2 E u (x) n f0g then kvp u k M(p) > 1; Ec is uniformly dominated by Eu and Es : ˜
(p) if v c 2 E c (x) n f0g then there are numbers (p); kD p Tv c k ˜ such that m(p) < (p) kv c k (p) < M(p) .
The notion of partial hyperbolicity was introduced by Brin and Pesin [21] who proved existence and properties, including absolute continuity, of invariant foliations in this setting. There has been intense recent interest in partially hyperbolic systems primarily because significant progress has been made in establishing that certain volume-preserving partially hyperbolic systems are ‘stably ergodic’—that is, they are ergodic and under small (Cr topology) volume-preserving perturbations remain ergodic. This phenomenon had hitherto been restricted to uniformly hyperbolic systems. For recent developments, and precise statements, on stable ergodicity of partially hyperbolic systems see [24,92]. Compact Group Extensions of Uniformly Hyperbolic Systems A natural example of a partially hyperbolic system is given by a compact group extension of an Anosov diffeomorphism. If the following terms are not familiar see Sect. “Constructions” on standard constructions. Suppose that (T; M; ) is an Anosov diffeomorphism, G is a compact connected Lie group and h : M ! G is a differentiable map. The skew product F : M G ! M G given by F(x; g) D (Tx; h(x)g) has a central direction in its tangent space corresponding to the Lie algebra LG of G (as a group element h acts isometrically on G so there is no expansion or contraction)
and uniformly expanding and contracting bundles corresponding to those of the tangent space of T : M ! M. Thus T(M G) D E u ˚ LG ˚ E s . Time-One Maps of Anosov Flows Another natural context in which partial hyperbolicity arises is in time-one maps of uniformly hyperbolic flows. Suppose t : R M ! M is an Anosov flow. The diffeomorphism 1 : M ! M is a partially hyperbolic diffeomorphism with central direction given by the flow direction. There is no expansion or contraction in the central direction. Non-Uniformly Hyperbolic Systems The assumption of uniform hyperbolicity is quite restrictive and few ‘chaotic systems’ found in applications are likely to exhibit uniform hyperbolicity. A natural weakening of this assumption, and one that is non-trivial and greatly extends the applicability of the theory, is to require the hyperbolic splitting (no longer uniform) to hold only at almost every point of phase space. A systematic theory was built by Pesin [89,90] on the assumption that the system has non-zero Lyapunov exponents almost everywhere where is a Lebesgue equivalent invariant probability measure . Recall that a number is a Lyapunov exponent for p 2 M if kD p T n vk en for some unit vector v 2 Tp M. Oseledet’s theorem [83] (see also p. 232 in [113]), which is also called the Multiplicative Ergodic Theorem, implies that if T is a C1 diffeomorphism of M then for any T-invariant ergodic measure almost every point has well-defined Lyapunov exponents. One of the highlights of Pesin theory is the following structure theorem: If T : M ! M is a C 1C diffeomorphism with a T-invariant Lebesgue equivalent Borel measure such that T has non-zero Lyapunov exponents with respect to then T has at most a countable number of ergodic components fC i g on each of which the restriction of T is either Bernoulli or Bernoulli times a rotation (by which we mean the support of i D jC i consists of a finite number ni of sets fS1i ; : : : S ni i g cyclically permuted and T n i is Bernoulli when restricted to each S ij ) [90,114]. This structure theorem has been generalized to SRB measures with non-zero Lyapunov exponents [64,90]. Physically Relevant Measures and Strange Attractors (This paragraph is from the article on Measure Preserving Systems.) For Hamiltonian systems and other volumepreserving systems it is natural to consider ergodicity (and other statistical properties) of the system with respect to Lebesgue measure. In dissipative systems a measure equiv-
Ergodic Theory: Basic Examples and Constructions
alent to Lebesgue may not be invariant (for example the solenoid). Nevertheless Lebesgue measure has a distinguished role since sampling by experimenters is done with respect to Lebesgue measure. The idea of a physically relevant measure is that it determines the statistical behavior of a positive Lebesgue measure set of orbits, even though the support of may have zero Lebesgue measure. An example of such a situation in the uniformly hyperbolic setting is the solenoid , where the attracting set has Lebesgue measure zero and is (locally) topologically the product of a two-dimensional Cantor set and a line segment. Nevertheless determines the behavior of all points in a solid torus in R3 . More generally, suppose that T : M ! M is a diffeomorphism on a compact Riemannian manifold and that m is a version of Lebesgue measure on M, given by a smooth volume form. Although Lebesgue measure m is a distinguished physically relevant measure, m may not be invariant under T, and the system may even be volume contracting in the sense that m(T n A) ! 0 for all measurable sets A. Nevertheless an experimenter might observe long-term “chaotic” behavior whenever the state of the system gets close to some compact invariant set X which attracts a positive m-measure of orbits in the sense that these orbits limit on X. Possibly m(X) D 0, so that X is effectively invisible to the observer except through its effects on orbits not contained in X. The dynamics of T restricted to X can in fact be quite complicated—maybe a full shift, or a shift of finite type, or some other complicated topological dynamical system. Suppose there is a T-invariant measure supported on X such that for all continuous functions : M ! R Z n1 1X k ı T (x) ! d ; n X
(6)
kD0
for a positive m-measure of points x 2 M. Then the longterm equilibrium dynamics of an observable set of points x 2 M (i. e. a set of points of positive m measure) is described by (X; T; ). In this situation is described as a physical measure. There has been a great deal of research on the properties of systems with attractors supporting physical measures. In the dissipative non-uniformly hyperbolic setting the theory of ‘physically relevant’ measures is best developed in the theory of SRB (for Sinai, Ruelle and Bowen) measures. These dynamically invariant measures may be supported on a set of Lebesgue measure zero yet determine the asymptotic behavior of points in a set of positive Lebesgue measure. If T is a diffeomorphism of M and is a T-invariant Borel probability measure with positive Lyapunov expo-
nents which may be integrated to unstable manifolds, then we call an SRB measure if the conditional measure induces on the unstable manifolds is absolutely continuous with respect to the Riemannian volume element on these manifolds. The reason for this definition is technical but is motivated by the following observations. Suppose that the diffeomorphism has no zero Lyapunov exponents with respect to . Since T is a diffeomorphism, this implies T has negative Lyapunov exponents as well as positive Lyapunov exponents and corresponding local stable manifolds as well as local unstable manifolds. Suppose that a T-invariant set A consists of a union of unstable manifolds and is the support of an ergodic SRB measure and that : M ! R is a continuous function. Since has absolutely continuous conditional measures on unstable manifolds with respect to conditional Lebesgue measure on the unstable manifolds, almost every point x in the union of unstable manifolds U satisfies Z n1 1X lim ı T j (x) D d : (7) n!1 n jD0
If y 2 W s (x) for such an x 2 U then d(T n x; T n y) ! 1 and hence (7) implies Z n1 1X ı T j (y) D d : lim n!1 n jD0
Furthermore, if the holonomy between unstable manifolds defined by sliding along stable manifolds is absolutely continuous (takes sets of zero Lebesgue measure on W u to sets of zero Lebesgue measure on W u ), there is a positive Lebesgue measure of points (namely an unstable manifold and the union of stable manifolds through it) satisfying (7). Thus an SRB measure with absolutely continuous holonomy maps along stable manifolds is a physically relevant measure. If the stable foliation possesses this property it is called absolutely continuous. An Axiom A attractor for a C2 diffeomorphism is an example of an SRB attractor [19,101,102,107]. The examples we have given of SRB measures and attractors and measures have been uniformly hyperbolic. Recently much progress has been made in understanding the statistical properties of non-uniformly hyperbolic systems by using a tower (see Sect. “Induced Transformations”) to construct SRB measures. We refer to Young’s original papers [115,116], the book by Baladi [5] and to [114] for a recent survey on SRB measures in the nonuniformly hyperbolic setting. Unimodal Maps Unimodal Maps of an interval are simple examples of non-uniformly hyperbolic dynamical sys-
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Ergodic Theory: Basic Examples and Constructions
tems that have played an important role in the development of dynamical systems theory. Suppose I R is an interval; for simplicity we take I D [0; 1]. A unimodal map is a map T : [0; 1] ! [0; 1] such that there exists a point 0 < c < 1 and T is C2 ; T 0 (x) > 0 for x < c, T 0 (x) < 0 for x > c; T 0 (c) D 0. Such a map is clearly not uniformly expanding, as jT 0 (x)j < 1 for points in a neighborhood of c. The family of maps T (x) D x(1 x), 0 < 4, is a family of unimodal maps with c D 1/2 and T2 (1/2) D 1/2, T4 (1/2) D 1. We could have taken the interval I to be [1; 1] or indeed any interval with an obvious modification of the definition above. A well-studied family of unimodal maps in this setting is the logistic family f a : [1; 1] ! [1; 1], f a (x) D 1 ax 2 , a 2 (0; 2]. The families f a and T are equivalent under a smooth coordinate change, so statements about one family may be translated into statements about the other. Unimodal maps are studied because of the insights they offer into transitions from regular or periodic to chaotic behavior as a parameter (e. g. or a) is varied, the existence of absolutely continuous measures, and rates of decay of correlations of regular observations for non-uniformly hyperbolic systems. Results of Jakobson [52] and Benedicks and Carleson [6] implies that in the case of the logistic family there is a positive Lebesgue measure set of a such that f a has an absolutely continuous ergodic invariant measure a . It has been shown by Keller and Nowicki [57] (see alsoYoung [116]) that if f a is mixing with respect to a then the decay of correlations for Lipshitz observations on I is exponential. It is also known that the set of a such that f a is mixing with respect to a has positive Lebesgue measure. There is a well-developed theory concerning bifurcations the maps T undergo as varies [27]. We briefly describe the period-doubling route to chaos in the family T (x) D x(1 x). For a nice account see [46].p We let c denote the fixed point 1 . For 3 < 1 C 6, all points in [0; 1] except for 0; c and their preimages are attracted to a unique periodic orbit O(p ) of period 2. There is a monotone sequence of parameter values n ( 1 D 3) such that for n < nC1 , T has a unique attracting periodic orbit O(n ) of period 2n and for each k D 1; 2; : : : ; n 1 a unique repelling orbit of period 2k . All points in the interval [0; 1] except for the repelling periodic orbits and their preimages are attracted to the attracting periodic orbit of period 2n . At
D n the periodic orbit O(n ) undergoes a period-doubling bifurcation. Feigenbaum [36] found that the limit n1 ı D n 4:699 : : : exists and that in a wide class n nC1 of unimodal maps this period-doubling cascade occurs and the differences between successive bifurcation parameters give the same limiting ratio, an example of universality. At the end of the period-doubling cascade at a parameter 1 3:569 : : :, T1 has an invariant Cantor set C (the Feigenbaum attractor) which is topologically conjugate to the dyadic adding machine coexisting with isolated repelling orbits of period 2n , n D 0; 1; 2; : : : . There is a unique repelling orbit of period 2n for n 1 along with two fixed points. The Cantor set C is the !-limit set for all points that are not periodic or preimages of periodic orbits. C is the set of accumulation points of periodic orbits. Despite this picture of incredible complexity the topological entropy is zero for 1 . For > 1 the map T has positive topological entropy and infinitely many periodic orbits whose periods are not powers of 2. For each 1 , T possesses an invariant Cantor set which is repelling for > 1 . We say that T is hyperbolic if there is only one attracting periodic orbit and the only recurrent sets are the attracting periodic orbit, repelling periodic orbits and possibly a repelling invariant Cantor set. It is known that the set of 2 [0; 4] for which T is hyperbolic is open and dense [43]. Remarkably, by Jakobson’s result [52] there is also a positive Lebesgue measure set of parameters for which T has an absolutely continuous invariant measure with a positive Lyapunov exponent. Intermittent Maps Maps of the unit interval T : [0; 1] ! [0; 1] which are expanding except at the point x D 0, where they are locally of form T(x) x C x 1C˛ , ˛ > 0, have been extensively studied both for the insights they give into rates of decay of correlations for nonuniformly hyperbolic systems (hyperbolicity is lost at the point x D 0, where the derivative is 1) and for their use as models of intermittent behavior in turbulence [71]. A fixed point where the derivative is 1 is sometimes called an indifferent fixed point. It is a model of intermittency in the sense that orbits close to 1 will stay close for many iterates (since the expansion is very weak there) and hence a time series of observations will be quite uniform for long periods of time before displaying chaotic type behavior after moving away from the indifferent fixed into that part of the domain where the map is uniformly expanding. A particularly simple model [67] is provided by ( x(1 C 2˛ x ˛ ) if x 2 [0; 1/2) ; T(x) D 2x 1 if x 2 [1/2; 1] :
Ergodic Theory: Basic Examples and Constructions
For ˛ D 0 the map is uniformly expanding and Lebesgue measure is invariant. In this case the rate of decay of correlations for Hölder observations is exponential. For 0 < ˛ < 1 the map has an SRB measure ˛ with support the unit interval. For ˛ 1 there are no absolutely continuous invariant probability measures though there are -finite absolutely continuous measures. Upper and lower polynomial bounds on the rate of decay of observations on such maps have been given as a function of 0 < ˛ < 1 and the regularity of the observable. For details see [51,67,103]. Hénon Diffeomorphisms The Henón family of diffeomorphisms was introduced and studied as Poincaré maps for the Lorenz system of equations. It is a two-parameter two-dimensional family which shares many characteristics with the logistic family and for small b > 0 may be considered a two-dimensional ‘perturbation’ of the logistic family. The parametrized mapping is defined as Ta;b (x; y) D (1 ax 2 C y; bx) ; so Ta;b : R2 ! R2 with 0 < a < 2 and b > 0. Benedicks and Carleson [7] showed that for a positivemeasure set of parameters (a, b), T a;b has a topologically transitive attractor a;b . Benedicks and Young [8] later proved that for a positive-measure set of parameters (a, b), T a;b has a topologically transitive SRB attractor a;b with SRB measure a;b and that (Ta;b ; a;b ; a;b ) is isomorphic to a Bernoulli shift. Complex Dynamics Complex dynamics is concerned with the behavior of rational maps ˛1 z d C ˛2 z d1 C ˛dC1 ˇ1 z d C ˇ2 z d1 C ˇdC1 ¯ to itself, in which the of the extended complex plane C domain is C completed with the point at infinity (called the Riemann sphere). Recall that a family F of meromorphic functions is called normal on a domain D if every sequence possesses a subsequence that converges uniformly ¯ S 2 ) on compact subsets of (in the spherical metric C ¯ if it is normal on D. A family is normal at a point z 2 C ¯ of a rational a neighborhood of z. The Fatou set F(R) C ¯ ¯ ¯ map R : C ! C is the set of points z 2 C such that the family of forward iterates fR n gn0 is normal at z. The Julia set J(R) is the complement of the Fatou set F(R). The Fatou set is open and hence the Julia set is a closed set. Another characterization in the case d > 1 is that J(R) is the clo¯ ! C. ¯ sure of the set of all repelling periodic orbits of R : C
Both F(R) and J(R) are invariant under R. The dynamics of greatest interest is the restriction R : J(R) ! J(R). The Julia set often has a complicated fractal structure. In the case that R a (z) D z2 a, a 2 C, the Mandelbrot set is defined as the set of a for which the orbit of the origin 0 is bounded. The topology of the Mandelbrot set has been the subject of intense research. The study of complex dynamics is important because of the fascinating and complicated dynamics displayed and also because techniques and results in complex dynamics have direct implications for the behavior of one-dimensional maps. For more details see [25]. Infinite Ergodic Theory We may also consider a measure-preserving transformation (T; X; ) of a measure space such that (X) D 1. For example X could be the real line equipped with Lebesgue measure. This setting also arises with compact X in applications. For example, suppose T : [0; 1] ! [0; 1] is the simple model of intermittency given in Sect. “Intermittent Maps” and 2 (1; 2). Then T possesses an absolutely continuous invariant measure with support [0; 1], but ([0; 1]) D 1. The Radon–Nikodym derivative of with respect to Lebesgue measure m exists but is not in L1 (m). In this setting we say a measurable set A is a wandering set for T if fT n Ag1 nD0 are disjoint. Let D(T) be the measurable union of the collection of wandering sets for T. The transformation T is conservative with respect to if (X n D(T)) D X (mod ) (see the article on Measure Preserving Systems). It is usually necessary to assume T conservative with respect to to say anything interesting about its behavior. For example if T(x) D x C ˛, ˛ > 0, is a translation of the real line then D(T) D X. The definition of ergodicity in this setting remains the same: T is ergodic if A 2 B and T 1 A D A mod implies that (A) D 0 or (Ac ) D 0. However the equivalence of ergodicity of T with respect to and the equality of time and space averages for L1 () functions no longer holds. Thus in general ergodic does not imply that lim
n!1
Z n1 1X ı T j (x) D d a.e. x 2 X n X iD0
for all 2 L1 (). In the example of the intermittent map with 2 (1; 2) the orbit of Lebesgue almost every x 2 X is dense in X, yet the fraction of time spent near the indifferent fixed point x D 0 tends to one for Lebesgue almost every x 2 X. In fact it may be shown Sect. 2.4 in [1] that when (x) D 1 there are no constants a n > 0 such that Z n1 1 X lim ı T j (x) D d a.e. x 2 X n!1 a n X iD0
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Nevertheless it is sometimes possible to obtain distributional limits, rather than almost sure limits, of Birkhoff sums under suitable normalization. We refer the reader to Aaronson’s book [1] for more details. Constructions We give examples of some of the standard constructions in dynamical systems. Often these constructions appear in modeling situations (for example skew products are often used to model systems which react to inputs from other systems, continuous time systems are often modeled as suspension flows over discrete-time dynamics) or to reduce systems to simpler components (often a factor system or induced system is simpler to study). Unless stated otherwise, in the sequel we will be discussing measure-preserving transformations on Lebesgue spaces (see the article on Measure Preserving Systems). Products Given measure-preserving systems (X; B; ; T) and (Y; C ; ; S), their product consists of their completed product measure space with the transformation T S : X Y ! X Y defined by (T S)(x; y) D (Tx; Sy) for all (x; y) 2 X Y. Neither ergodicity nor transitivity is in general preserved by taking products; for example the product of an irrational rotation on the unit circle with itself is not ergodic. For a list of which mixing properties are preserved in various settings by forming a product see [113]. Given any countable family of measure-preserving transformations on probability spaces, their direct product is defined similarly. Factors We say that a measure-preserving system (Y; C ; ; S) is a factor of a measure-preserving system (X; B; ; T) if (possibly after deleting a set of measure 0 from X) there is a measurable onto map : X ! Y such that 1 C B; T D S; T
1
and
(8)
D :
For Lebesgue spaces, there is a correspondence of factors of (X; B; ; T) and T-invariant complete sub- -algebras of B. According to Rokhlin’s theory of Lebesgue spaces [98] factors also correspond to certain partitions of X (see the article on Measure Preserving Systems). A factor map : X ! Y between Lebesgue spaces is an isomorphism if and only if it has a measurable inverse, or equivalently 1 C D B up to sets of measure 0.
Skew Products If (X; B; ; T) is a measure-preserving system, (Y; C ; ) is a measure-space, and fS x : x 2 Xg is a family of measurepreserving maps Y ! Y such that the map that takes (x, y) to S x y is jointly measurable in the two variables x and y, then we may define a skew product system consisting of the product measure space of X and Y equipped with product measure together with the measure-preserving map T Ë S : X Y ! X Y defined by (T Ë S)(x; y) D (Tx; S x y) :
(9)
The space Y is called the fiber of the skew product and the space X the base. Sometimes in the literature the word skew product has a more general meaning and refers to the structure (T Ë S)(x; y) D (Tx; S x y) (without any assumption of measure-preservation), where the action of the map on the fiber Y is determined or ‘driven’ by the map T : X ! X. Some common examples of skew products include: Random Dynamical Systems Suppose that X indexes a collection of mappings S x : Y ! Y. We may have a transformation T : X ! X which is a full shift. Then the sequence of mappings fS T n x g may be considered a (random) choice of a mapping Y ! Y from the set fS x : x 2 Xg. The projection onto Y of the orbits of (Tx; S x y) give the orbits of a point y 2 Y under a random composition of maps S T n x ı ıS T x ıS x . More generally we could consider the choice of maps Sx that are composed to come from any ergodic dynamical system, (T; X; ) to model the effect of perturbations by a stationary ergodic ‘noise’ process. Group Extensions of Dynamical Systems Suppose Y is a group, is a measure on Y invariant under a left group action, and S x y :D g(x)y is given by a groupvalued function g : X ! Y. In this setting g is often called a cocycle, since upon defining g (n) (x) by (T Ë S)(n) (x; y) D (T n x; g (n) (x)y) we have a cocycle relation, namely g (mCn) (x) D g (m) (T n x)g (n) (x). Group extensions arise often in modeling systems with symmetry [37]. Common examples are provided by a random composition of matrices from a group of matrices (or more generally from a set of matrices which may form a group or not).
Induced Transformations Since by the Poincaré Recurrence Theorem (see [113]) a measure-preserving transformation (T; X; ; B) on a probability space is recurrent, given any set B of positive
Ergodic Theory: Basic Examples and Constructions
measure, the return-time function n B (x) D inffn 1 : T n x 2 Bg
(10)
is finite a.e. We may define the first-return map by TB x D T n B (x) x :
(11)
Then (after perhaps discarding as usual a set of measure 0) TB : B ! B is a measurable transformation which preserves the probability measure B D /(B). The system (B; B \ B; B ; TB ) is called an induced, first-return or derived transformation. If (T; X; ; B) is ergodic then (B; B \ B; B ; TB ) is ergodic, but the converse is not in general true. The construction of the transformation T B allows us to represent the forward orbit of points in B via a tower or skyscraper over B. For each n D 1; 2; : : : , let B n D fx 2 B : n B (x) D ng :
(12)
Then fB1 ; B2 ; : : : g form a partition of B, which we think of as the bottom floor or base of the tower. The next floor is made up of TB2 ; TB3 ; : : : , which form a partition of TB n B, and so on. All these sets are disjoint. A column is a part of the tower of the form B n [ TB n [ [ T n1 B n for some n D 1; 2; : : : . The action of T on the entire tower is pictured as mapping each x not at the top of its column straight up to the point Tx above it on the next level, and mapping each point on the top level to T n B x 2 B. An equivalent way to describe the transformation on the tower is to write for each n and j < n, T j B n as f(x; j) : x 2 B n g, and then the transformation F on the tower is ( (x; l C 1) if l < n B (x) 1 ; F(x; l) D n (x) B x; 0) if l D n B (x) 1 : (T If T preserves a measure , then F preserves dl, where l is counting measure. Sometimes the process of inducing yields an induced map which is easier to analyze (perhaps it has stronger hyperbolicity properties) than the original system. Sometimes it is possible to ‘lift’ ergodic or statistical properties from an induced system to the original system, so the process of inducing plays an important role in the study of statistical properties of dynamical systems [77]. It is possible to generalize the tower construction and relax the condition that nB (x) is the first-return time function. We may take a measurable set B X of positive measure and define for almost every point x 2 B a height or ceiling function R : B ! N and take a countable partition fX n g of B into the sets on which R is constant. We
define the tower as the set :D f(x; l) : x 2 B; 0 l < R(x)g and the tower map F : ! by ( (x; l C 1) if l < R(x) 1 ; F(x; l) D R(x) x; 0) if l D R(x) 1 : (T R In this setting, if B R(x) d < 1, we may define an F-in dl, where dl variant probability measure on as C(R;B) is counting measure and C(R, B) is the normalizing conR stant C(R; B) D (B) B R(x)d. This viewpoint is connected with the construction of systems by cutting and stacking—see Sect. “Cutting and Stacking”. Suspension Flows The tower construction has an analogue in which the height function R takes values in R rather than N. Such towers are commonly used to model dynamical systems with continuous time parameter. Let (T; X; ) be a measure-preserving system and R : X ! (0; 1) a measurable “ceiling” function on X. The set X R D f(x; t) : 0 R(x) < tg ;
(13)
with measure given locally by the product of on X with Lebesgue measure m on R, is a measure space in a natural way. If is a finite measure and R is integrable with respect to then is a finite measure. We define an action of R on X R by letting each point x flow at unit speed up the vertical lines f(x; t) : 0 t < R(x)g under the graph of R until it hits the ceiling, then jump to Tx, and so on. More precisely, defining R n (x) D R(x) C C R(T n x), 8 (x; s C t) ˆ ˆ ˆ ˆ ˆ if 0 s C t < R(x); ˆ ˆ ˆ ˆ ˆ ˆ 0, there is a measurable set B X such that the sets B; TB; : : : ; T n1 B are pairwise k disjoint and ([n1 kD0 T B) > 1 . Inverse Limits Suppose that for each i D 1; 2; : : : we have a Lebesgue probability space (X i ; B i ; i ) and a measure-preserving transformation Ti : X i ! X i . Suppose also that for each i j there is a factor map ji : (T j ; X j ; B j ; j ) ! (Ti ; X i ; B i ; i ; ), such that each j j is the identity on X j
Ergodic Theory: Basic Examples and Constructions
and ji k j D ki whenever k j i. Let 1 X i : ji x j D x i for all j ig : X D fx 2 ˘ iD1
(15)
For each j, let j : X ! X j be the projection defined by j x D x j. Let B be the smallest -algebra of subsets of X which 1 contains all the 1 j B j . Define on each j B j by ( 1 j B) D j (B)
for all B 2 B j :
(16)
Because ji j D i for all j i, the 1 j B j are increasing, and so their union is an algebra. The set function can, with some difficulty, be shown to be countably additive on this algebra: since we are dealing with Lebesgue spaces, by means of measure-theoretic isomorphisms it is possible to replace the entire situation by compact metric spaces and continuous maps, then use regularity of the measures involved—see p. 137 ff. in [88]. Thus by Carathéodory’s Theorem (see the article on Measure Preserving Systems) extends to all of B. Define T : X ! X by T(x j ) D (T j x j ). Then (T; X; B; ) is a measure-preserving system such that any system which has all the (T j ; X j ; B j ; j ) as factors, also has (T; X; B; ) a factor. Natural Extension The natural extension is a way to produce an invertible system from a non-invertible system. The original system is a factor of its natural extension and its orbit structure and ergodic properties are captured by the natural extension, as will be seen from its construction. Let (T; X; B; ) be a measure-preserving transformation of a Lebesgue probability space. Define ˝ : D f(x0 ; x1 ; x2 ; : : : ) : x n D T(x nC1 ); x n 2 X; n D 0; 1; 2; : : : g with : ˝ ! ˝ defined by ((x0 ; x1 ; x2 ; : : : )) D (T(x0 ); x0 ; x1 ; : : : ). The map is invertible on ˝. Given the invariant measure we define the invariant measure ˜ on ˝ by defining it first on for the natural extension cylinder sets C(A0 ; A1 ; : : : ; A k ) by ˜ (C(A 0 ; A1 ; : : : ; A k )) D (T k (A0 )\T kC1 (A1 ) \T kCi (A i )\ \A k ) and then extending it to ˝ using Kolmogorov’s extension theorem. We think of (x0 ; x1 ; x2 ; : : : ) as being an inverse branch of x0 2 X under the mapping T : X ! X. The ˜ if maps ; 1 : ˝ ! ˝ are ergodic with respect to (T; X; B; ) is ergodic [113]. If : ˝ ! X is projection onto the first component i. e. (x0 ; : : : ; x n ; : : : ) D x0 then
ı n (x0 ; : : : ; x n ; : : : ) D T n (x0 ) for all x0 and thus the natural extension yields information about the orbits of X under T. The natural extension is an inverse limit. Let (X; B; ) be a Lebesgue probability space and T : X ! X a map such that T 1 B B and T 1 D . For each i D 1; 2; : : : let (Ti ; X i ; B i ; i ) D (T; X; B; ), and ji D ˆ X; ˆ Bˆ ; ) ˆ T ji for each j > i. Then the inverse limit (T; of this system is an invertible measure-preserving system which is the natural extension of (T; X; B; ). We have Tˆ 1 (x1 ; x2 ; : : : ) D (x2 ; x3 ; : : : ) :
(17)
ˆ X; ˆ Bˆ ; ) ˆ The original system (T; X; B; ) is a factor of (T; (using any i as the factor map), and any factor mapping from an invertible system onto (T; X; B; ) consists ˆ X; ˆ Bˆ ; ) ˆ followed by projecof a factor mapping onto (T; tion onto the first coordinate. Joinings Given measure-preserving systems (T; X; B; ) and (S; Y; C ; ), a joining of the two systems is a T S-invariant measure P on their product measurable space that projects to and , respectively, under the projections of X Y to X and Y, respectively. That is, if 1 : X Y ! X is the projection onto the first component i. e. 1 (x; y) D x then P(11 (A)) D (A) for all A 2 B and similarly for 2 : X Y ! Y. This concept is the ergodic-theoretic version of the notion in probability theory of a coupling. The product measure is always a joining of the two systems. If product measure is the only joining of the two systems, then we say that they are disjoint and write X ? Y [40]. If D is any family of systems, we write D? for the family of all measure-preserving systems which are disjoint from every system in D. Extensive recent accounts of the use of joinings in ergodic theory are in [42,99,110]. Future Directions The basic examples and constructions presented here are idealized, and many of the underlying assumptions (such as uniform hyperbolicity) are seldom satisfied in applications, yet they have given important insights into the behavior of real-world physical systems. Recent developments have improved our understanding of the ergodic properties of non-uniformly and partially hyperbolic systems. The ergodic properties of deterministic systems will continue to be an active research area for the foreseeable future. The directions will include, among others: establishing statistical and ergodic properties under weakened dependence assumptions; the study of systems which dis-
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play ‘anomalous statistics’; the study of the stability and typicality of ergodic behavior and mixing in dynamical systems; the ergodic theory of infinite-dimensional systems; advances in number theory (see the sections on Szemerédi and Ramsey theory); research into models with non-singular rather than invariant measures; and infinitemeasure systems. Other chapters in this Encyclopedia discuss in more detail these and other topics. Bibliography Primary Literature 1. Aaronson J (1997) An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol 50. American Mathematical Society, Providence. MR 1450400 (99d:28025) 2. Anosov DV (1967) Geodesic flows on closed riemannian manifolds of negative curvature. Trudy Mat Inst Steklov 90:209. MR 0224110 (36 #7157) 3. Arnol0 d VI (1963) Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat Nauk 18(6 (114)):91–192. MR 0170705 (30 #943) 4. Bailey S, Keane M, Petersen K, Salama IA (2006) Ergodicity of the adic transformation on the euler graph. Math Proc Cambridge Philos Soc 141(2):231–238. MR 2265871 (2007m:37010) 5. Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol 16. World Scientific Publishing Co Inc, River Edge. MR 1793194 (2001k:37035) 6. Benedicks M, Carleson L (1985) On iterations of 1 ax(X; B; )2 on (1; 1). Ann Math (2) 122(1):1–25. MR 799250 (87c:58058) 7. Benedicks M, Carleson L (1991) The dynamics of the hénon map. Ann Math (2) 133(1):73–169. MR 1087346 (92d:58116) 8. Benedicks M, Young LS (1993) Sina˘ı-bowen-ruelle measures for certain hénon maps. Invent Math 112(3):541–576. MR 1218323 (94e:58074) 9. Bertrand-Mathis A (1986) Développement en base ; répartition modulo un de la suite (x (X; B; )n)n0 ; langages codés et -shift. Bull Soc Math France 114(3):271–323. MR 878240 (88e:11067) 10. Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, N.Y., reprint of the 1965 original. MR 524567 (80b:28017) 11. Bissinger BH (1944) A generalization of continued fractions. Bull Amer Math Soc 50:868–876. MR 0011338 (6,150h) 12. Blanchard F (1989) ˇ -expansions and symbolic dynamics. Theoret Comput Sci 65(2):131–141. MR 1020481 (90j:54039) 13. Blanchard F, Hansel G (1986) Systèmes codés. Theoret Comput Sci 44(1):17–49. MR 858689 (88m:68029) 14. Blanchard F, Hansel G (1986) Systèmes codés et limites de systèmes sofiques. C R Acad Sci Paris Sér I Math 303(10):475–477. MR 865864 (87m:94009) 15. Blanchard F, Hansel G (1991) Sofic constant-to-one extensions of subshifts of finite type. Proc Amer Math Soc 112(1):259–265. MR 1050016 (91m:54050) 16. Boshernitzan M, Galperin G, Krüger T, Troubetzkoy S (1998) Periodic billiard orbits are dense in rational polygons. Trans Amer Math Soc 350(9):3523–3535. MR 1458298 (98k:58179)
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Ergodic Theory: Interactions with Combinatorics and Number Theory TOM W ARD School of Mathematics, University of East Anglia, Norwich, UK Article Outline Glossary Definition of the Subject Introduction Ergodic Theory Frequency of Returns Ergodic Ramsey Theory and Recurrence Orbit-Counting as an Analogous Development Diophantine Analysis as a Toolbox Future Directions Bibliography Glossary Almost everywhere (abbreviated a. e.) A property that makes sense for each point x in a measure space (X; B; ) is said to hold almost everywhere (or a. e.) if the set N X on which it does not hold satisfies N 2 B and (N) D 0. ˇ Cech–Stone compactification of N, ˇN A compact Hausdorff space that contains N as a dense subset with the property that any map from N to a compact Hausdorff space K extends uniquely to a continuous map ˇN ! K. This property and the fact that ˇN is a compact Hausdorff space containing N characterizes ˇN up to homeomorphism. Curvature An intrinsic measure of the curvature of a Riemannian manifold depending only on the Riemannian metric; in the case of a surface it determines whether the surface is locally convex (positive curvature), locally saddle-shaped (negative) or locally flat (zero). Diophantine approximation Theory of the approximation of real numbers by rational numbers: how small can the distance from a given irrational real number to a rational number be made in terms of the denominator of the rational? Equidistributed A sequence is equidistributed if the asymptotic proportion of time it spends in an interval is proportional to the length of the interval. Ergodic A measure-preserving transformation is ergodic if the only invariant functions are equal to a constant a. e.; equivalently if the transformation exhibits the convergence in the quasi-ergodic hypothesis.
Ergodic theory The study of statistical properties of orbits in abstract models of dynamical systems; more generally properties of measure-preserving (semi-) group actions on measure spaces. Geodesic (flow) The shortest path between two points on a Riemannian manifold; such a geodesic path is uniquely determined by a starting point and the initial tangent vector to the path (that is, a point in the unit tangent bundle). The transformation on the unit tangent bundle defined by flowing along the geodesic defines the geodesic flow. Haar measure (on a compact group) If G is a compact topological group, the unique measure defined on the Borel sets of G with the property that (A C g) D (A) for all g 2 G and (G) D 1. Measure-theoretic entropy A numerical invariant of measure-preserving systems that reflects the asymptotic growth in complexity of measurable partitions refined under iteration of the map. Mixing A measure-preserving system is mixing if measurable sets (events) become asymptotically independent as they are moved apart in time (under iteration). (Quasi) Ergodic hypothesis The assumption that, in a dynamical system evolving in time and preserving a natural measure, there are some reasonable conditions under which the ‘time average’ along orbits of an observable (that is, the average value of a function defined on the phase space) will converge to the ‘space average’ (that is, the integral of the function with respect to the preserved measure). Recurrence Return of an orbit in a dynamical system close to its starting point infinitely often. S-Unit theorems A circle of results stating that linear equations in fields of zero characteristic have only finitely many solutions taken from finitely-generated multiplicative subgroups of the multiplicative group of the field (apart from infinite families of solutions arising from vanishing sub-sums). Topological entropy A numerical invariant of topological dynamical systems that measures the asymptotic growth in the complexity of orbits under iteration. The variational principle states that the topological entropy of a topological dynamical system is the supremum over all invariant measures of the measure-theoretic entropies of the dynamical systems viewed as measurable dynamical systems. Definition of the Subject Number theory is a branch of pure mathematics concerned with the properties of numbers in general, and integers in particular. The areas of most relevance to this ar-
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ticle are Diophantine analysis (the study of how real numbers may be approximated by rational numbers, and the consequences for solutions of equations in integers); analytic number theory, and in particular asymptotic estimates for the number of primes smaller than X as a function of X; equidistribution, and questions about how the digits of real numbers are distributed. Combinatorics is concerned with identifying structures in discrete objects; of most interest here is that part of combinatorics connected with Ramsey theory, asserting that large subsets of highly structured objects must automatically contain large replicas of that structure. Ergodic theory is the study of asymptotic behavior of group actions preserving a probability measure; it has proved to be a powerful part of dynamical systems with wide applications. Introduction Ergodic theory, part of the mathematical study of dynamical systems, has pervasive connections with number theory and combinatorics. This article briefly surveys how these arise through a small sample of results. Unsurprisingly, many details are suppressed, and of course the selection of topics reflects the author’s interests far more than it does the full extent of the flow of ideas between ergodic theory and number theory. In addition the selection of topics has been chosen in part to be complementary to those in related articles in the Encyclopedia. A particularly enormous lacuna is the theory of arithmetic dynamical systems itself – the recent monograph by Silverman [94] gives a comprehensive overview. More sophisticated aspects of this connection – in particular the connections between ergodic theory on homogeneous spaces and Diophantine analysis – are covered in the articles Ergodic Theory on Homogeneous Spaces and Metric Number Theory and Ergodic Theory: Rigidity; more sophisticated overviews of the connections with combinatorics may be found in the article Ergodic Theory: Recurrence. Ergodic Theory While the early origins of ergodic theory lie in the quasiergodic hypothesis of classical Hamiltonian dynamics, the mathematical study of ergodic theory concerns various properties of group actions on measure spaces, including but not limited to several special branches: 1. The classical study of single measure-preserving transformations. 2. Measure-preserving actions of Zd ; more generally of countable amenable groups.
3. Measure-preserving actions of Rd and more general amenable groups, called flows. 4. Measure-preserving actions of lattices in Lie groups. 5. Measure-preserving actions of Lie groups. The ideas and conditions surrounding the quasi-ergodic hypothesis were eventually placed on a firm mathematical footing by developments starting in 1890. For a single measure-preserving transformation T : X ! X of a probability space (X; B; ), Poincaré [74] showed a recurrence theorem: if E 2 B is any measurable set, then for a. e. x 2 E there is an infinite set of return times, 0 < n1 < n2 < with T n j (x) 2 E (of course Poincaré noted this in a specific setting, concerned with a natural invariant measure for the “three-body” problem in planetary motion). Poincaré’s qualitative result was made quantitative in the 1930s, when von Neumann [105] used the approach of Koopman [52] to show the mean ergodic theorem: if f 2 L2 () then there is some f 2 L2 () for which N1 1 X f ı T n f ! 0 as N nD0
N ! 1;
2
clearly f then R R has the property that k f f ı Tk2 D 0 and f d D f d. Around the same time, Birkhoff [13] showed the more delicate pointwise ergodic theorem: for any g 2 L1 () there is some g 2 L1 () for which N1 1 X g(T n x) ! g(x) a. e. ; N nD0
R again it is then clear that g(Tx) D g(x) a. e. and g d D R g d. The map T is called ergodic if the invariance condition forces the function (f or g) to be equal to a constant a. e. Thus an ergodic map has the property that the time or erP N1 godic average (1/N) nD0 f ı T j converges to the space R average d. An overview of ergodic theorems and their many extensions may be found in the article Ergodic Theorems. Thus ergodic theory at its most basic level makes strong statements about the asymptotic behavior of orbits of a dynamical system as seen by observables (measurable functions on the space X). Applying the ergodic theorem to the indicator function of a measurable set A shows that ergodicity guarantees that a. e. orbit spends an asymptotic proportion of time in A equal to the volume (A) of that set (as measured by the invariant measure). This points to the start of the pervasive connections between ergodic theory and number theory – but as this and other articles relate, the connections extend far beyond this.
Ergodic Theory: Interactions with Combinatorics and Number Theory
Frequency of Returns
First Digits
In this section we illustrate the way in which a dynamical point of view may unify, explain and extend quite disparate results from number theory.
The astronomer Newcomb [67] noted that the first digits of large collections of numerical data that are not dimensionless have a specific and non-uniform distribution:
Normal Numbers Borel [15] showed (as a consequence of what became the Borel–Cantelli Lemma in probability) that a. e. real number (with respect to Lebesgue measure) is normal to every base: that is, has the property that any block of k digits in the base-r expansion appears with asymptotic frequency rk . Continued Fraction Digits Analogs of normality results for the continued fraction expansion of real numbers were found by Khinchin, Kuz’min, Lévy and others. Any irrational x 2 [0; 1] has a unique expansion as a continued fraction 1
a1 (x) C a2 (x) C
1 a3 (x) C
and, just as in the case of the familiar base-r expansion, it turns out that the digits (a n (x)) obey precise statistical rules for a. e. x. Gauss conjectured that the appearance of individual digits would obey the law 1 jfk : 1 k N; a k (x) D jgj N 2 log(1 C j) log j log(2 C j) ! : log 2
(1)
lim (a1 (x)a2 (x) : : : a n (x))1/n D
log n/ log 2 1 Y (n C 1)2 nD1
n(n C 2)
D 2:68545 : : :
n!1
1 jfk : 1 k N; fa k g 2 [a; b]gj ! (b a) N as N ! 1 ;
Z 1 N 1 X f (a k ) ! f (t)dt N 0
as
N !1
kD1
for all continuous functions f . Weyl showed that the sequence fn˛g is equidistributed if and only if ˛ is irrational. This result was refined and extended in many directions; for example, Hlawka [44] and others found rates for the convergence in terms of the discrepancy of the sequence, Weyl [110] proved equidistribution for fn2 ˛g, and Vinogradov for fp n ˛g where pn is the nth prime.
for a. e. x :
Lévy [60] showed that the denominator q n (x) of the nth convergent (p n (x))/(q n (x)) (the rational obtained by truncating the continued fraction expansion of x at the nth term) grows at a specific exponential rate, lim
Weyl [109] (and, separately, Bohl [14] and Sierpi´nski [91]) found an important instance of equidistribution. Writing fg for the fractional part, a sequence (a n ) of real numbers is said to be equidistributed modulo 1 if, for any interval [a; b] [0; 1),
equivalently if
This was eventually proved by Kuz’min [54] and Lévy [59], and the probability distribution of the digits is the Gauss– Kuz’min law. Khinchin [50] developed this further, showing for example that n!1
This is now known as Benford’s Law, following his popularization and possible rediscovery of the phenomenon [6]. In both cases, this was an empirical observation eventually made rigorous by Hill [42]. Arnold (App. 12 in [3]), pointed out the dynamics behind this phenomena in certain cases, best illustrated by the statistical behavior of the sequence 1; 2; 4; 8; 1; 3; 6; 1; : : : of first digits of powers of 2. Empirically, the digit 1 appears about 30% of the time, while the digit 9 appears about 5% of the time. Equidistribution
1
xD
“The law of probability of the occurrence of numbers is such that all mantissæ of their logarithms are equally probable.”
1 2 log q n (x) D n 12 log 2
for a. e. x :
The Ergodic Context All the results of this section are manifestations of various kinds of convergence of ergodic averages. Borel’s theorem on normal numbers is an immediate consequence of the fact that Lebesgue measure on [0; 1) is invariant and ergodic for the map x 7! bx modulo 1 with b > 2. The asymptotic properties of continued fraction digits are all
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a consequence of the fact that the Gauss measure defined by Z dx 1 (A) D for A [0; 1] log 2 A 1 C x ˚ is invariant and ergodic for the Gauss map x 7! x1 , and the orbit of an irrational number under the Gauss map determine the digits appearing in the continued fraction expansion much as the orbit under the map x 7! bx (mod 1) determines the digits in the base b expansion. The results on equidistribution and the frequency of first digits are related to ergodic averaging of a different sort. For example, writing R˛ (t) D t C ˛ modulo 1 for the circle rotation by ˛, the first digit of 2n is the digit j if and only if log10 j Rlog10 (2) (0) < log10 ( j C 1) : Thus the asymptotic frequency of appearance concerns the orbit of a specific point. In order to see what this means, consider a continuous map T : X ! X of a compact metric space (X; d). The space M(T) of Borel probability measures on the Borel -algebra of (X; d) is a non-empty compact convex set in the weak*-topology, each extreme point is an ergodic measure for T, and these ergodic measures are mutually singular. If M(T) is not a singleton and 1 ; 2 2 M(T) are distinct ergodic measures, then for R R a continuous function f with X f d1 ¤ X f d2 it is P N1 f (T n x) must clear thatRthe ergodic averages N1 nD0 R converge to X f d1 a. e. with respect to 1 and to X f d2 a. e. with respect to 2 . Thus the presence of many invariant measures for a continuous map means that ergodic averages along the orbits of specific points need not converge to the space average with respect to a chosen invariant measure. In the extreme situation of unique ergodicity (a single invariant measure, which is necessarily an extreme point of M(T) and hence ergodic) the convergence of ergodic averages is much more uniform. Indeed, if T is uniquely ergodic with M(T) D fg then, for any continuous function f : X ! R, Z N1 1 X n f (T x) ! f d uniformly in x N nD0 X (see Oxtoby [69]). The circle rotation R˛ is uniquely ergodic for irrational ˛, leading to the equidistribution results. The ergodic viewpoint on equidistribution also places equidistribution results in a wider context. Weyl’s result that fn2 ˛g (indeed, the fractional part of any polynomial
with at least one irrational coefficient) is equidistributed for irrational ˛ was given another proof by Furstenberg [29] using the notion of unique ergodicity. These methods were then used in the study of nilsystems (translations on quotients of nilpotent Lie groups) by Auslander, Green and Hahn [4] and Parry [70], and these nilsystems play an essential role for polynomial (and other non-conventional) ergodic averaging (see Host and Kra [45] and Leibman [58] in the polynomial case; Host and Kra [46] in the multiple linear case). Remarkably, nilsystems are starting to play a role within combinatorics – an example is the work on the asymptotic number of 4-step arithmetic progressions in the primes by Green and Tao [40]. Pointwise ergodic theorems have also been found along sequences other than N; notably for integer-valued polynomials and along the primes for L2 functions by Bourgain [16,17]. For more details, see the survey paper of del Junco on ergodic theorems. Ergodic Ramsey Theory and Recurrence In 1927 van der Waerden proved a conjecture attributed to Baudet: if the natural numbers are written as a disjoint union of finitely many sets, N D C1 t C2 t t C r ;
(2)
then there must be one set Cj that contains arbitrarily long arithmetic progressions. That is, there is some j 2 f1; : : : ; rg such that for any k > 1 there are a > 1 and n > 1 with a; a C n; a C 2n; : : : ; a C (k 1)n 2 C j : The original proof appears in van der Waerden’s paper [103], and there is a discussion of how he found the proof in [104]. Work of Furstenberg and Weiss [34] and others placed the theorem of van der Waerden in the context of topological dynamics, giving alternative proofs. Specifically, van der Waerden’s theorem is a consequence of topological multiple recurrence: the return of points under iteration in a topological dynamical system close to their starting point along finite sequences of times. The same approach readily gives dynamical proofs of Rado’s extension [76] of van der Waerden’s theorem, and of Hindman’s theorem [43]. The theorems of Rado and Hindman introduce a new theme: given a set A D fn1 ; n2 ; : : : g of natural numbers, write FS(A) for the set of numbers obtained as finite sums n i 1 C C n i j with i1 < i2 < < i j . Rado showed that for any large n there is some Cs containing some FS(A) for a set A of cardinality n. Hindman showed that there is some Cs containing some FS(A) for an infinite set A.
Ergodic Theory: Interactions with Combinatorics and Number Theory
In the theorem of van der Waerden, it is clear that for any reasonable notion of “proportion” or “density” one of the sets Cj must occupy a positive proportion of N. A set A N is said to have positive upper density if there are sequences (M i ) and (N i ) with N i M i ! 1 as i ! 1 such that lim
i!1
1 jfa 2 A: M i < a < N i gj > 0 : Ni Mi
Erd˝os and Turán [26] conjectured the stronger statement that any subset of N with positive upper density must contain arbitrary long arithmetic progressions. This statement was shown for arithmetic progressions of length 3 by Roth [79] in 1952, then for length 4 by Szemerédi [96] in 1969. The general result was eventually proved by Szemerédi [97] in 1975 in a lengthy and extremely difficult argument. Furstenberg saw that Szemerédi’s Theorem would follow from a deep extension of the Poincaré recurrence phenomena described in Sect. “Ergodic Theory” and proved that extension [30] (see also the survey article by Furstenberg, Katznelson and Ornstein [33]). The multiple recurrence result of Furstenberg says that for any measure-preserving system (X; B; ; T) and set A 2 B with (A) > 0, and for any k 2 N, lim inf
NM!1
N X
1 NMC1
A \ T n A \ T 2n A \ \ T kn A > 0 :
nDM
An immediate consequence is that in the same setting there must be some n > 1 for which (3) A \ T n A \ T 2n A \ \ T kn A > 0 : A general correspondence principle, due to Furstenberg, shows that statements in combinatorics like Szemerédi’s Theorem are equivalent to statements in ergodic theory like (3). This opened up a significant new field of ergodic Ramsey theory, in which methods from dynamical systems and ergodic theory are used to produce new results in infinite combinatorics. For an overview, see the articles Ergodic Theory on Homogeneous Spaces and Metric Number Theory, Ergodic Theory: Rigidity, Ergodic Theory: Recurrence and the survey articles of Bergelson [7,8,10]. The field is too large to give an overview here, but a few examples will give a flavor of some of the themes. Call a set R Z a set of recurrence if, for any finite measure-preserving invertible transformation T of a finite
measure space (X; B; ) and any set A 2 B with (A) > 0, there are infinitely many n 2 R for which (A \ T n A) > 0. Thus Poincaré recurrence is the statement that N is a set of recurrence. Furstenberg and Katznelson [31] showed that if T1 ; : : : ; Tk form a family of commuting measurepreserving transformations and A is a set of positive measure, then lim inf N!1
N1 1 X (T1n A \ \ Tkn A) > 0 : N nD0
This remarkable multiple recurrence implies a multi-dimensional form of Szemerédi’s theorem. Recently, Gowers has found a non-ergodic proof of this [38]. Furstenberg also gave an ergodic proof of Sárközy’s theorem [80]: if p 2 Q[t] is a polynomial with p(Z) Z and p(0) D 0, then fp(n)gn>0 is a set of recurrence. This was extended to multiple polynomial recurrence by Bergelson and Leibman [11]. Topology and Coloring Theorems ˇ The existence of idempotent ultrafilters in the Cech–Stone compactification ˇN gives rise to an algebraic approach to many questions in topological dynamics (this notion has its origins in the work of Ellis [24]). Using these methods, results like Hindman’s finite sums theorem find elegant proofs, and many new results in combinatorics have been found. For example, in the partition (2) there must be one set Cj containing a triple x; y; z solving x y D z2 . A deeper application is to improve a strengthening of Kronecker’s theorem. To explain this, recall that a set S is called IP if there is a sequence (n i ) of natural numbers (which do not need to be distinct) with the property that S contains all the terms of the sequence and all finite sums of terms of the sequence with distinct indices. A set S is called IP if it has non-empty intersection with every IP set, and if there is some t 2 Z for which S t is a set S is called IPC ) is an extreme form of ‘fatness’ IP . Thus being IP (or IPC for a set. Now let 1; ˛1 ; : : : ; ˛ k be numbers that are linearly independent over the rationals, and for any d 2 N and kd non-empty intervals I i j [0; 1] (1 i d; 1 j k), let D D fn 2 N : fn i ˛ j g 2 I i j for all i; jg : Kronecker showed that if d D 1 then D is non-empty; Hardy and Littlewood showed that D is infinite, and Weyl showed that D has positive density. Bergelson [9] uses these algebraic methods to improve the result by showing set. that D is an IPC
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Polynomialization and IP-sets
Sets of Primes
As mentioned above, Bergelson and Leibman [11] extended multiple recurrence to a polynomial setting. For example, let
In a remarkable development, Szemerédi’s theorem and some of the ideas behind ergodic Ramsey theory joined results of Goldston and Yıldırım [37] in playing a part in Green and Tao’s proof [39] that the set of primes contains arbitrarily long arithmetic progressions. This profound result is surveyed from an ergodic point of view in the article of Kra [53]. As with Szemerédi’s theorem itself, this result has been extended to a polynomial setting by Tao and Ziegler [101]. Given integer-valued polynomials f1 ; : : : ; f k 2 Z[t] with
fp i; j : 1 i k; 1 j tg be a collection of polynomials with rational coefficients and p i; j (Z) Z, p i; j (0) D 0. Then if (A) > 0, we have 11 1 0 0 N k t X \ Y 1 p i; j (n) A AA > 0 : @ @ Tj lim inf N!1 N nD1 iD1
jD1
Using the Furstenberg correspondence principle, this gives a multi-dimensional polynomial Szemerédi theorem: If P : Zr ! Z` is a polynomial mapping with the property that P(0) D 0, and F Zr is a finite configuration, then any set S Z` of positive upper Banach density contains a set of the form u C P(nF) for some u 2 Z` and n 2 N. In a different direction, motivated in part by Hindman’s theorem, the multiple recurrence results generalize to IP-sets. Furstenberg and Katznelson [32] proved a linear IP-multiple recurrence theorem in which the recurrence is guaranteed to occur along an IP-set. A combinatorial proof of this result has been found by Nagle, Rödl and Schacht [66]. Bergelson and McCutcheon [12] extended these results by proving a polynomial IP-multiple recurrence theorem. To formulate this, make the following definitions. Write F for the family of non-empty finite subsets of N, so that a sequence indexed by F is an IP-set. More generally, an F -sequence (n˛ )˛2F taking values in an abelian group is called an IP-sequence if n˛[ˇ D n˛ C nˇ whenever ˛ \ ˇ D ;. An IP-ring is a set of the form S F (1) D f i2ˇ ˛ i : ˇ 2 F g where ˛1 < ˛2 < is a sequence in F , and ˛ < ˇ means a < b for all a 2 ˛, b 2 ˇ; write F<m for the set of m-tuples (˛1 ; : : : ; ˛m ) from F m with ˛ i < ˛ j for i < j. Write PE(m; d) for the collection of all expressions of the form T(˛1 ; : : : ; ˛m ) D (b) p i ((n ˛ j )1bk; 1 jm ) Qr m, T ; (˛1 ; : : : ; ˛m ) 2 (F [ ;)< iD1 i where each pi is a polynomial in a k m matrix of variables with integer coefficients and zero constant term with degree d. Then for every m; t 2 N, there is an IP-ring F (1) , and an a D a(A; m; t; d) > 0, such that, for every set of polynomial expressions fS0 ; : : : ; S t g PE(m; d), ! t \ 1 S i (˛1 ; : : : ; ˛m ) A > 0 : IP lim m (˛1 ;:::;˛ m )2(F (1) )
0 with MT (N) D k T log N C C T C O(N ı ) :
jni 1j
iD1
and
jjN
it is shown in [21] that for an ergodic S-integer map T with K D Q and S finite, there are constants k T 2 Q and CT such that
(10)
htop (TA ) D
r X
log maxf1; j i jg
(11)
iD1
where 1 ; : : : ; r are the eigenvalues of A. It follows that the convergence in (10) is clear under the assumption that T A is hyperbolic (that is, no eigenvalue has modulus one). Without this assumption the convergence is less clear: for r > 4 the automorphism T A may be ergodic without being hyperbolic. That is, while no eigenvalues are unit
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roots some may have unit modulus. As pointed out by Lind [61] in his study of these quasihyperbolic automorphisms, the convergence (10) does still hold for these systems, but this requires a significant Diophantine result (the theorem of Gel’fond [35] suffices; one may also use Baker’s theorem [5]). Even further from hyperbolicity lie the family of S-integer systems [21,107]; their orbit-growth properties are intimately tied up with Artin’s conjecture on primitive roots and prime divisors of linear recurrence sequences. Mixing and Additive Relations in Fields The problem of higher-order mixing for commuting group automorphisms provides a striking example of the dialogue between ergodic theory and number theory, in which deep results from number theory have been used to solve problems in ergodic theory, and questions arising in ergodic theory have motivated further developments in number theory. An action T of a countable group on a probability space (X; B; ) is called k-fold mixing or mixing on (k C 1) sets if A0 \ T g 1 A1 \ \ T g k A k ! (A0 ) (A k ) (12) as g i g 1 j ! 1 for i ¤ j with the convention that g0 D 1 , for any sets A0 ; : : : ; A k 2 B; g n ! 1 in means that for any finite set F there is an N with n > N H) g n … F. For k D 1 the property is called simply mixing. This notion for single transformations goes back to the foundational work of Rohlin [78], where he showed that ergodic group endomorphisms are mixing of all orders (and so the notion is not useful for distinguishing between group endomorphisms as measurable dynamical systems). He raised the (still open) question of whether any measure-preserving transformation can be mixing without being mixing of all orders. A class of group actions that are particularly easy to understand are the algebraic dynamical systems studied systematically by Schmidt [83]: here X is a compact abelian group, each T g is a continuous automorphism of X, and is the Haar measure on X. Schmidt [82] related mixing properties of algebraic dynamical systems with D Zd to statements in arithmetic, and showed that a mixing action on a connected group could only fail to mix in a certain way. Later Schmidt and the author [85] showed that for X
connected, mixing implies mixing of all orders. The proof proceeds by showing that the result is exactly equivalent to the following statement: if K is a field of characteristic zero, and G is a finitely generated subgroup of the multiplicative group K , then the equation a1 x1 C C a n x n D 1
(13)
for fixed a1 ; : : : ; a n 2 K has a finite number of soluP tions x1 ; : : : ; x n 2 G for which no subsum i2I a i x i with I ¨ f1; : : : ; ng vanishes. The bound on the number of solutions to (13) follows from the profound extensions to W. Schmidt’s subspace theorem in Diophantine geometry [86] by Evertse and Schlickewei (see [28,81,102] for the details). The argument in [85] may be cast as follows: failure of k-fold mixing in a connected algebraic dynamical system implies (via duality) an infinite set of solutions to an equation of the shape (13) in some field of characteristic zero. The S-unit theorem means that this can only happen if there is some proper subsum that vanishes infinitely often. This infinite family of solutions to a homogeneous form of (13) with fewer terms can then be translated back via duality to show that the system fails to mix for some strictly lower order, proving that mixing implies mixing of all orders by induction. Mixing properties for algebraic dynamical systems without the assumption of connectedness are quite different, and in particular it is possible to have mixing actions that are not mixing of all orders. This is a simple consequence of the fact that the constituents of a disconnected algebraic dynamical system are associated with fields of positive characteristic, where the presence of the Frobenius automorphism can prevent higher-order mixing. Ledrappier [57] pointed this out via examples of the following shape. Let n 2 X D x 2 F2Z : x(aC1;b) C x(a;b) C x(a;bC1) D0
o (mod 2)
and define the Z2 -action T to be the natural shift action, (T (n;m) x)(a;b) D x(aCn;bCm) : It is readily seen that this action is mixing with respect to the Haar measure. The condition x(aC1;b) C x(a;b) C x(a;bC1) D 0 (mod 2) implies that, for any k > 1, x(0;2k )
! 2k X 2k D x( j;0) D x(0;0) Cx(2k ;0) j jD0
(mod 2) (14)
Ergodic Theory: Interactions with Combinatorics and Number Theory
since every entry in the 2 k th row of Pascal’s triangle is even apart from the first and the last. Now let A D fx 2 X : x(0;0) D 0g and let x 2 X be any element with x(0;0) D 1. Then X is the disjoint union of A and A C x , so (A) D (A C x ) D
1 2
stants b1 ; : : : ; b m 2 K and some (g1 ; : : : ; g m ) 2 G m with the following properties: g j ¤ 1 for 1 j m; g i g 1 j ¤ 1 for 1 i < j m; there are infinitely many k for which
:
However, (14) shows that x 2 A \ T(2k ;0) A H) x 2 T(0;2k ) A ; so A \ T(2k ;0) A \ T(0;2k ) (A C x ) D ; for all k > 1, which shows that T cannot be mixing on three sets. The full picture of higher-order mixing properties on disconnected groups is rather involved; see Schmidt’s monograph [83]. A simple illustration is the construction by Einsiedler and the author [23] of systems with any prescribed order of mixing. When such systems fail to be mixing of all orders, they fail in a very specific way – along dilates of a specific shape (a finite subset of Zd ). In the example above, the shape that fails to mix is f(0; 0); (1; 0); (0; 1)g. This gives an order of mixing as detected by shapes; computing this is in principle an algebraic problem. On the other hand, there is a more natural definition of the order of mixing, namely the largest k for which (12) holds; computing this is in principle a Diophantine problem. A conjecture emerged (formulated explicitly by Schmidt [84]) that for any algebraic dynamical system, if every set of cardinality r > 2 is a mixing shape, then the system is mixing on r sets. This question motivated Masser [64] to prove an appropriate analogue of the S-unit theorem on the number of solutions to (13) in positive characteristic as follows. Let H be a multiplicative group and fix n 2 N. An infinite subset A H n is called broad if it has both of the following properties: if h 2 H and 1 j n, then there are at most finitely many (a1 ; : : : ; a n ) in A with a j D h; if n > 2, h 2 H and 1 i < j n then there are at most finitely many (a1 ; : : : ; a n ) 2 H with a i a1 j D h. Then Masser’s theorem says the following. Let K be a field of characteristic p > 0, let G be a finitely-generated subgroup of K and suppose that the equation a1 x1 C C a n x n D 1 has a broad set of solutions (x1 ; : : : ; x n ) 2 G n for some constants a1 ; : : : ; a n 2 K . Then there is an m n, con-
k b1 g1k C b2 g2k C C b m g m D1:
The proof that shapes detect the order of mixing in algebraic dynamics then proceeds much as in the connected case. Future Directions The interaction between ergodic theory, number theory and combinatorics continues to expand rapidly, and many future directions of research are discussed in the articles Ergodic Theory on Homogeneous Spaces and Metric Number Theory, Ergodic Theory: Rigidity and Ergodic Theory: Recurrence. Some of the directions most relevant to the examples discussed in this article include the following. The recent developments mentioned in Subsect. “Sets of Primes” clearly open many exciting prospects involving finding new structures in arithmetically significant sets (like the primes). The original conjecture of Erd˝os and P 1 Turán [26] asked if a2A N a D 1 is sufficient to force the set A to contain arbitrary long arithmetic progressions, and remains open. This would of course imply both Szemerédi’s theorem [97] and the result of Green and Tao [39] on arithmetic progressions in the primes. More generally, it is clear that there is still much to come from the dialogue subsuming the four parallel proofs of Szemerédi’s: one by purely combinatorial methods, one by ergodic theory, one by hypergraph theory, and one by Fourier analysis and additive combinatorics. For an overview, see the survey papers of Tao [98,99,100]. In the context of the orbit-counting results in Sect. “Orbit-Counting as an Analogous Development”, a natural problem is to on the one hand obtain finer asymptotics with better control of the error terms, and on the other to extend the situations that can be handled. In particular, relaxing the hypotheses related to hyperbolicity (or negative curvature) is a constant challenge. Bibliography Primary Literature 1. Adler RL, Weiss B (1970) Similarity of automorphisms of the torus. Memoirs of the American Mathematical Society, No. 98. American Mathematical Society, Providence
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75. Pollicott M, Sharp R (1998) Exponential error terms for growth functions on negatively curved surfaces. Amer J Math 120(5):1019–1042 76. Rado R (1933) Studien zur Kombinatorik. Math Z 36(1):424– 470 77. Ratner M (1973) Markov partitions for Anosov flows on n-dimensional manifolds. Israel J Math 15:92–114 78. Rohlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 79. Roth K (1952) Sur quelques ensembles d’entiers. C R Acad Sci Paris 234:388–390 80. Sárközy A (1978) On difference sets of sequences of integers. III. Acta Math Acad Sci Hungar 31(3-4):355–386 81. Schlickewei HP (1990) S-unit equations over number fields. Invent Math 102(1):95–107 82. Schmidt K (1989) Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pacific J Math 137(2):371–385 83. Schmidt K (1995) Dynamical systems of algebraic origin, vol 128. Progress in Mathematics. Birkhäuser, Basel 84. Schmidt K (2001) The dynamics of algebraic Zd -actions. In: European Congress of Mathematics, vol I (Barcelona, 2000), vol 201. Progress in Math. Birkhäuser, Basel, pp 543–553 85. Schmidt K, Ward T (1993) Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent Math 111(1):69– 76 86. Schmidt WM (1972) Norm form equations. Ann Math 96(2):526–551 87. Selberg A (1949) An elementary proof of the prime-number theorem. Ann Math 50(2):305–313 88. Selberg A (1956) Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J Indian Math Soc 20:47–87 89. Sharp R (1991) An analogue of Mertens’ theorem for closed orbits of Axiom A flows. Bol Soc Brasil Mat 21(2):205–229 90. Sharp R (1993) Closed orbits in homology classes for Anosov flows. Ergodic Theory Dyn Syst 13(2):387–408 ´ W (1910) Sur la valeur asymptotique d’une certaine 91. Sierpinski somme. Bull Intl Acad Polonmaise des Sci et des Lettres (Cracovie), pp 9–11 92. Sina˘ı JG (1966) Asymptotic behavior of closed geodesics on compact manifolds with negative curvature. Izv Akad Nauk SSSR Ser Mat 30:1275–1296 93. Sina˘ı JG (1968) Construction of Markov partitionings. Funkcional Anal i Priložen 2(3):70–80 94. Silverman JH (2007) The Arithmetic of Dynamical Systems, vol 241. Graduate Texts in Mathematics. Springer, New York 95. Smale S (1967) Differentiable dynamical systems. Bull Amer Math Soc 73:747–817 96. Szemerédi E (1969) On sets of integers containing no four elements in arithmetic progression. Acta Math Acad Sci Hungar 20:89–104 97. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:199–245 98. Tao T (2005) The dichotomy between structure and randomness, arithmetic progressions, and the primes. arXiv: math/0512114v2 99. Tao T (2006) Arithmetic progressions and the primes. Collect Math, vol extra, Barcelona, pp 37–88 100. Tao T (2007) What is good mathematics? arXiv:math/ 0702396v1
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Ergodic Theory: Recurrence
Ergodic Theory: Recurrence N IKOS FRANTZIKINAKIS, RANDALL MCCUTCHEON Department of Mathematics, University of Memphis, Memphis, USA Article Outline Glossary Definition of the Subject Introduction Quantitative Poincaré Recurrence Subsequence Recurrence Multiple Recurrence Connections with Combinatorics and Number Theory Future Directions Bibliography Glossary Almost every, essentially Given a Lebesgue measure space (X; B; ), a property P(x) predicated of elements of X is said to hold for almost every x 2 X, if the set X n fx : P(x) holdsg has zero measure. Two sets A; B 2 B are essentially disjoint if (A \ B) D 0. Conservative system Is an infinite measure preserving system such that for no set A 2 B with positive measure are A; T 1 A; T 2 A; : : : pairwise essentially disjoint. (cn )-Conservative system If (c n )n2N is a decreasing sequence of positive real numbers, a conservative ergodic measure preserving transformation T is (c n )conservative if for some non-negative function f 2 P n L1 (), 1 nD1 c n f (T x) D 1 a. e. Doubling map If T is the interval [0; 1] with its endpoints identified and addition performed modulo 1, the (non-invertible) transformation T : T ! T , defined by Tx D 2x mod 1, preserves Lebesgue measure, hence induces a measure preserving system on T . Ergodic system Is a measure preserving system (X; B; ; T) (finite or infinite) such that every A 2 B that is T-invariant (i. e. T 1 A D A) satisfies either (A) D 0 or (X n A) D 0. (One can check that the rotation R˛ is ergodic if and only if ˛ is irrational, and that the doubling map is ergodic). Ergodic decomposition Every measure preserving system (X; X ; ; T) can be expressed as an integral of ergodic systems; for example, one can write D R t d(t), where is a probability measure on [0; 1] and t are T-invariant probability measures on (X; X ) such that the systems (X; X ; t ; T) are ergodic for t 2 [0; 1].
Ergodic theorem States that if (X; B; ; T) is a measure 2 preserving system and 1 PN f 2 L (), then lim N!1 nf P T D 0, where P f denotes the f L 2 () nD1 N orthogonal projection of the function f onto the subspace f f 2 L2 () : T f D f g. Hausdorff a-measure Let (X; B; ; T) be a measure preserving system endowed with a -compatible metric d. The Hausdorff a-measure H a (X) of X is an outer measure defined for all subsets of X as follows: First, for P a A X and " > 0 let H a;"(A) D inff 1 iD1 r i g, where the infimum is taken over all countable coverings of A by sets U i X with diameter r i < ". Then define H a (A) D lim sup"!0 H a;"(A). Infinite measure preserving system Same as measure preserving system, but (X) D 1. Invertible system Is a measure preserving system (X; B; ; T) (finite or infinite), with the property that there exists X0 2 X, with (X n X 0 ) D 0, and such that the transformation T : X0 ! X0 is bijective, with T 1 measurable. Measure preserving system Is a quadruple (X; B; ; T), where X is a set, B is a -algebra of subsets of X (i. e. B is closed under countable unions and complementation), is a probability measure (i. e. a countably additive function from B to [0; 1] with (X) D 1), and T : X ! X is measurable (i. e. T 1 A D fx 2 X : Tx 2 Ag 2 B for A 2 B), and -preserving (i. e. (T 1 A) D (A)). Moreover, throughout the discussion we assume that the measure space (X; B; ) is Lebesgue (see Sect. 1.0 in [2]). -Compatible metric Is a separable metric on X, where (X; B; ) is a probability space, having the property that open sets are measurable. Positive definite sequence Is a complex-valued sequence (a n )n2Z such that for any n1 ; : : : ; n k 2 Z and z1 ; : : : ; P z k 2 C, ki; jD1 a n i n j z i z j 0. Rotations on T If T is the interval [0; 1] with its endpoints identified and addition performed modulo 1, then for every ˛ 2 R the transformation R˛ : T ! T , defined by R˛ x D x C ˛mod1, preserves Lebesgue measure on T and hence induces a measure preserving system on T . Syndetic set Is a subset E Z having bounded gaps. If G is a general discrete group, a set E G is syndetic if G D FE for some finite set F G. Upper density Is the number d() D lim sup N!1 (j \ fN; : : : ; Ngj)/(2N C 1), where Z (assuming the limit to exist). Alternatively for measurable E
Rm , D(E) D lim sup l (S)!1 (m(S \ E))/(m(S)), where S ranges over all cubes in Rm , and l(S) denotes the length of the shortest edge of S.
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Notation The following notation will be used throughout the article: T f D f ı T, fxg D x [x], D- limn!1 (a n ) D a $ d fn : ja n aj > "g D 0 for every " > 0.
a finite (or conservative) measure preserving system, every set of positive measure (or almost every point) comes back to itself infinitely many times under iteration. Despite the profound importance of these results, their proofs are extremely simple.
Definition of the Subject
Theorem 1 (Poincaré Recurrence for Sets) Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then (A \ T n A) > 0 for infinitely many n 2 N.
The basic principle that lies behind several recurrence phenomena is that the typical trajectory of a system with finite volume comes back infinitely often to any neighborhood of its initial point. This principle was first exploited by Poincaré in his 1890 King Oscar prize-winning memoir that studied planetary motion. Using the prototype of an ergodic-theoretic argument, he showed that in any system of point masses having fixed total energy that restricts its dynamics to bounded subsets of its phase space, the typical state of motion (characterized by configurations and velocities) must recur to an arbitrary degree of approximation. Among the recurrence principle’s more spectacularly counterintuitive ramifications is that isolated ideal gas systems that do not lose energy will return arbitrarily closely to their initial states, even when such a return entails a decrease in entropy from equilibrium, in apparent contradiction to the second law of thermodynamics. Such concerns, previously canvassed by Poincaré himself, were more infamously expounded by Zermelo [74] in 1896. Subsequent clarifications by Boltzmann, Maxwell and others led to an improved understanding of the second law’s primarily statistical nature. (For an interesting historical/philosophical discussion, see [68]; also [10]. For a probabilistic analysis of the likelihood of observing second law violations in small systems over short time intervals, see [28]). These discoveries had a profound impact in dynamics, and the theory of measure preserving transformations (ergodic theory) evolved from these developments. Since then, the Poincaré recurrence principle has been applied to a variety of different fields in mathematics, physics, and information theory. In this article we survey the impact it has had in ergodic theory, especially as pertains to the field of ergodic Ramsey theory. (The heavy emphasis herein on the latter reflects authorial interest, and is not intended to transmit a proportionate image of the broader landscape of research relating to recurrence in ergodic theory.) Background information we assume in this article can be found in the books [35,63,71] ( Measure Preserving Systems). Introduction In this section we shall give several formulations of the Poincaré recurrence principle using the language of ergodic theory. Roughly speaking, the principle states that in
Proof Since T is measure preserving, the sets A; T 1 A; T 2 A; : : : have the same measure. These sets cannot be pairwise essentially disjoint, since then the union of finitely many of them would have measure greater than (X) D 1. Therefore, there exist m; n 2 N, with n > m, such that (T m A\T n A) > 0. Again since T is measure preserving, we conclude that (A \ T k A) > 0, where k D n m > 0. Repeating this argument for the iterates A; T m A; T 2m A; : : :, for all m 2 N, we easily deduce that (A \ T n A) > 0 for infinitely many n 2 N. We remark that the above argument actually shows that 1 (A \ T n A) > 0 for some n [ (A) ] C 1. Theorem 2 (Poincaré Recurrence for Points) Let (X; B; ; T) be a measure preserving system and A 2 B. Then for almost every x 2 A we have that T n x 2 A for infinitely many n 2 N. Proof Let B be the set of x 2 A such that T n x … A for all S n 2 N. Notice that B D A n n2N T n A; in particular, B is measurable. Since the iterates B; T 1 B; T 2 B; : : : are pairwise essentially disjoint, we conclude (as in the proof of Theorem 1) that (B) D 0. This shows that for almost every x 2 A we have that T n x 2 A for some n 2 N. Repeating this argument for the transformation T m in place of T for all m 2 N, we easily deduce the advertised statement. Next we give a variation of Poincaré recurrence for measure preserving systems endowed with a compatible metric: Theorem 3 (Poincaré Recurrence for Metric Systems) Let (X; B; ; T) be a measure preserving system, and suppose that X is endowed with a -compatible metric. Then for almost every x 2 X we have lim inf d(x; T n x) D 0 : n!1
The proof of this result is similar to the proof of Theorem 2 (see p. 61 in [35]). Applying this result to the doubling map Tx D 2x on T , we get that for almost every x 2 X, every string of zeros and ones in the dyadic expansion of x occurs infinitely often.
Ergodic Theory: Recurrence
n x 2 X the set R S x D fn 2 N : T x 2 Ag has well defined density and d(S x ) d(x) D (A). Furthermore, for ergodic measure preserving systems we have d(S x ) D (A) a. e.
We remark that all three formulations of the Poincaré Recurrence Theorem that we have given hold for conservative systems as well. See, e. g., [2] for details. This article is structured as follows. In Sect. “Quantitative Poincaré Recurrence” we give a few quantitative versions of the previously mentioned qualitative results. In Sects. “Subsequence Recurrence” and “Multiple Recurrence” we give several refinements of the Poincaré recurrence theorem, by restricting the scope of the return time n, and by considering multiple intersections (for simplicity we focus on Z-actions). In Sect. “Connections with Combinatorics and Number Theory” we give various implications of the recurrence results in combinatorics and number theory ( Ergodic Theory: Interactions with Combinatorics and Number Theory). Lastly, in Sect. “Future Directions” we give several open problems related to the material presented in Sects. “Subsequence Recurrence” to “Connections with Combinatorics and Number Theory”.
Theorem 6 (Kac [51]) Let (X; B; ; T) be an ergodic measure preserving system and A 2 B with (A) > 0. For x 2 X define R A (x) D minfn 2 N : T n x 2 Ag. RThen for x 2 A the expected value of R A (x) is 1/(A), i. e. A R A (x) d D 1.
Quantitative Poincaré Recurrence
More Recent Results
Early Results
As we mentioned in the previous section, if the space X is endowed with a -compatible metric d, then for almost every x 2 X we have that lim infn!1 d(x; T n x) D 0. A natural question is, how much iteration is needed to come back within a small distance of a given typical point? Under some additional hypothesis on the metric d we have the following answer:
For applications it is desirable to have quantitative versions of the results mentioned in the previous section. For example one would like to know how large (A \ T n A) can be made and for how many n. Theorem 4 (Khintchine [55]) Let (X; B; ; T) be a measure preserving system and A 2 B. Then for every " > 0 we have (A \ T n A) > (A)2 " for a set of n 2 N that has bounded gaps. By considering the doubling map Tx D 2x on T and letting A D 1[0;1/2) , it is easy to check that the lower bound of the previous result cannot be improved. We also remark that it is not possible to estimate the size of the gap by a function of (A) alone. One can see this by considering the rotations R k x D x C 1/k for k 2 N, defined on T , and letting A D 1[0;1/3] . Concerning the second version of the Poincaré recurrence theorem, it is natural to ask whether for almost every x 2 X the set of return times S x D fn 2 N : T n x 2 Ag has bounded gaps. This is not the case, as one can see by considering the doubling map Tx D 2x on T with the Lebesgue measure, and letting A D 1[0;1/2) . Since Lebesgue almost every x 2 T contains arbitrarily large blocks of ones in its dyadic expansion, the set S x has unbounded gaps. Nevertheless, as an easy consequence of the Birkhoff ergodic theorem [19], one has the following: Theorem 5 Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then for almost every
Another question that arises naturally is, given a set A with positive measure and an x 2 A, how long should one wait until some iterate T n x of x hits A? By considering an irrational rotation R˛ on T , where ˛ is very near to, but not 1 , and letting A D 1[0;1/2] , one can see that less than, 100 the first return time is a member of the set f1; 50; 51g. So it may come as a surprise that the average first return time does not depend on the system (as long as it is ergodic), but only on the measure of the set A.
Theorem 7 (Boshernitzan [20]) Let (X; B; ; T) be a measure preserving system endowed with a -compatible metric d. Assume that the Hausdorff a-measure H a (X) of X is -finite (i. e., X is a countable union of sets X i with H a (X i ) < 1). Then for almost every x 2 X, ˚ 1 lim inf n a d(x; T n x) < 1 : n!1
Furthermore, if H a (X) D 0, then for almost every x 2 X, ˚ 1 lim inf n a d(x; T n x) D 0 : n!1
One can see from rotations by “badly approximable” vectors ˛ 2 T k that the exponent 1/a in the previous theorem cannot be improved. Several applications of Theorem 7 to billiard flows, dyadic transformations, symbolic flows and interval exchange transformations are given in [20]. For a related result dealing with mean values of the limits in Theorem 7 see [67]. An interesting connection between rates of recurrence and entropy of an ergodic measure preserving system was established by Ornstein and Weiss [62], following earlier work of Wyner and Ziv [73]:
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Theorem 8 (Ornstein and Weiss [62]) Let (X; B; ; T) be an ergodic measure preserving system and P be a finite partition of X. Let Pn (x) be the element of the partition Wn1 i Tn1 i (i) : P(i) 2 P ; 0 i < ng iD0 T P D f iD0 T P that contains x. Then for almost every x 2 X, the first return time R n (x) of x to Pn (x) is asymptotically equivalent to e h(T;P )n , where h(T; P ) denotes the entropy of the system with respect to the partition P . More precisely, lim
n!1
log R n (x) D h(T; P ) : n
An extension of the above result to some classes of infinite measure preserving systems was given in [42]. Another connection of recurrence rates, this time with the local dimension of an invariant measure, is given by the next result: Theorem 9 (Barreira [4]) Let (X; B; ; T) be an ergodic measure preserving system. Define the upper and lower recurrence rates log r (x) and log r log r (x) R(x) D lim sup ; log r r!0
R(x) D lim inf r!0
where r (x) is the first return time of T k x to B(x; r), and the upper and lower pointwise dimensions log (B(x; r)) and log r log (B(x; r)) d (x) D lim sup : log r r!0
d (x) D lim inf r!0
Then for almost every x 2 X, we have R(x) d (x) and
R(x) d (x) :
Roughly speaking, this theorem asserts that for typical x 2 X and for small r, the first return time of x to B(x; r) is at most rd (x) . Since d (x) H a (X) for almost every x 2 X, we can conclude the first part of Theorem 7 from Theorem 9. For related results the interested reader should consult the survey [5] and the bibliography therein. We also remark that the previous results and related concepts have been applied to estimate the dimension of certain strange attractors (see [49] and the references therein) and the entropy of some Gibbsian systems [25]. We end this section with a result that connects “wandering rates” of sets in infinite measure preserving systems with their “recurrence rates”. The next theorem follows easily from a result about lower bounds on ergodic aver-
ages for measure preserving systems due to Leibman [57]; a weaker form for conservative, ergodic systems can be found in Aaronson [1]. Theorem 10 Let (X; B; ; T) be an infinite measure preserving system, and A 2 B with (A) < 1. Then for all N 2 N, 0 S N1 1 n A N1 T X nD0 1 @ (A \ T n A)A ((A))2 : N 2 nD0
Subsequence Recurrence In this section we discuss what restrictions one can impose on the set of return times in the various versions of the Poincaré recurrence theorem. We start with: Definition 11 Let R Z. Then R is a set of: (a) Recurrence if for any invertible measure preserving system (X; B; ; T), and A 2 B with (A) > 0, there is some nonzero n 2 R such that (A \ T n A) > 0. (b) Topological recurrence if for every compact metric space (X; d), continuous transformation T : X ! X and every " > 0, there are x 2 X and nonzero n 2 R such that d(x; T n x) < ". It is easy to check that the existence of a single n 2 R satisfying the previous recurrence conditions actually guarantees the existence of infinitely many n 2 R satisfying the same conditions. Moreover, if R is a set of recurrence then one can see from existence of some T-invariant measure that R is also a set of topological recurrence. A (complicated) example showing that the converse is not true was given by Kriz [56]. Before giving a list of examples of sets of (topological) recurrence, we discuss some necessary conditions: A set of topological recurrence must contain infinitely many multiples of every positive integer, as one can see by considering rotations on Zd , d 2 N . Hence, the sets f2n C 1; n 2 Ng, fn2 C 1; n 2 Ng, fp C 2; p primeg are not good for (topological) recurrence. If (s n )n2N is a lacunary sequence (meaning lim infn!1 (s nC1 /s n ) D > 1), then one can construct an irrational number ˛ such that fs n ˛g 2 [ı; 1 ı] for all large n 2 N, where ı > 0 depends on (see [54], for example). As a consequence, the sequence (s n )n2N is not good for (topological) recurrence. Lastly, we mention that by considering product systems, one can immediately show that any set of (topological) recurrence R is partition regular, meaning that if R is partitioned into finitely many pieces then at least one of these pieces must still be a set of (topological) recur-
Ergodic Theory: Recurrence
rence. Using this observation, one concludes for example that any union of finitely many lacunary sequences is not a set of recurrence. We present now some examples of sets of recurrence: Theorem 12 The following are sets of recurrence: S (i) Any set of the form n2N fa n ; 2a n ; : : : ; na n g where a n 2 N. (ii) Any IP-set, meaning a set that consists of all finite sums of the members of some infinite set. (iii) Any difference set S S, meaning a set that consists of all possible differences of the members of some some infinite set S. (iv) The set fp(n); n 2 Ng where p is any nonconstant integer polynomial with p(0) D 0 [35,66] (In fact we only have to assume that the range of the polynomial contains multiples of an arbitrary positive integer [53]). (v) The set fp(n); n 2 Sg, where p is an integer polynomial with p(0) D 0 and S is any IP-set [12]. (vi) The set of values of an admissible generalized polynomial (this class contains in particular the smallest function algebra G containing all integer polynomials having zero constant term and such that if g1 ; : : : ; g k 2 G and c1 ; : : : ; c k 2 R then P [[ kiD1 c i g i ]] 2 G, where [[x]] D [x C 12 ] denotes the integer nearest to x) [13]. (vii) The set of shifted primes fp 1; p primeg, and the set fp C 1; p primeg [66]. (viii) The set of values of a random non-lacunary sequence. (Pick n 2 N independently with probability b n where 0 b n 1 and limn!1 nb n D 1. The resulting set is almost surely a set of recurrence. If lim supn!1 nb n < 1 then the resulting set is almost surely a finite union of sets, each of which is the range of some lacunary sequence, hence is not a set of recurrence). Follows from [22]. Showing that the first three sets are good for recurrence is a straightforward modification of the argument used to prove Theorem 1. Examples (iv) (viii) require more work. A criterion of Kamae and Mendés-France [53] provides a powerful tool that may be used in many instances to establish that a set R is a set of recurrence. We mention a variation of their result: Theorem 13 (Kamae and Mendés-France [53]) Suppose that R D fa1 < a2 < : : :g is a subset of N such that: (i) The sequence fa n ˛gn2N is uniformly distributed in T for every irrational ˛.
(ii) The set R m D fn 2 N : mja n g has positive density for every m 2 N. Then R is a set of recurrence. We sketch a proof for this result. First, recall Herglotz’s theorem: if (a n )n2Z is a positive definite sequence, then there isR a unique measure on the torus T such that 2 i nt Ra n D e n d (t). The case of interest to us is a n D T f (x) f (T x) d, where T is measure preserving and f 2 L1 (); (a n ) is positive definite, and we call D f the spectral measure of f . Let now (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Putting f D 1A , one has lim
N!1
N Z 1 X f (x) f (T a n x) d N nD1
Z D
lim
T N!1
N 1 X 2 i a n t e N nD1
! d f (t) :
(1)
For t irrational the limit inside the integral is zero (by condition (i)), so the last integral can be taken over the rational points in T . Since the spectral measure of a function orthogonal to the subspace H D f f 2 L2 () : there exists k 2 N with T k f D f g
(2) has no rational point masses, we can easily deduce that when computing the first limit in (1), we can replace the function f by its orthogonal projection g onto the subspace H (g is again nonnegative and g ¤ 0). To complete the argument, we approximate g by a function g 0 such that T m g 0 D g 0 for some appropriately chosen m, and use condition (ii) to deduce that the limit of the average (1) is positive. In order to apply Theorem 13, one uses the standard machinery of uniform distribution. Recall Weyl’s criterion: a real-valued sequence (x n )n2N is uniformly distributed mod 1 if for every non-zero k 2 Z, lim
N!1
N 1 X 2 i kx n e D0: N nD1
This criterion becomes especially useful when paired with van der Corput’s so-called third principal property: if, for every h 2 N, (x nCh x n )n2N is uniformly distributed mod 1, then (x n )n2N is uniformly distributed mod 1. Using the foregoing criteria and some standard (albeit nontrivial) exponential sum estimates, one can verify for example that the sets (iv) and (vii) in Theorem 12 are good for recurrence.
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In light of the connection elucidated above between uniform distribution mod 1 and recurrence, it is not surprising that van der Corput’s method has been adapted by modern ergodic theorists for use in establishing recurrence properties directly.
pute
N 1 X hx nCm ; x n i N!1 N nD1
lim
D lim
N!1
Theorem 14 (Bergelson [7]) Let (x n )n2N be a bounded sequence in a Hilbert space. If D-lim
m!1
! N 1 X hx nCm ; x n i D 0 ; lim N!1 N nD1
then N 1 X xn D 0 : lim N!1 N nD1
Let us illustrate how one uses this “van der Corput trick” by showing that S D fn2 : n 2 Ng is a set of recurrence. We will actually establish the following stronger fact: If (X; B; ; T) is a measure preserving system and f 2 L1 () is nonnegative and f ¤ 0 then lim inf N!1
1 N
N Z X
2
f (x) f (T n x) d > 0 :
(3)
nD1
Then our result follows by setting f D 1A for some A 2 B with (A) > 0. The main idea is one that occurs frequently in ergodic theory; split the function f into two components, one of which contributes zero to the limit appearing in (3), and the other one being much easier to handle than f . To do this consider the T-invariant subspace of L2 (X) defined by H D f f 2 L2 () : there exists k 2 N with T k f D f g :
(4) Write f D g C h where g 2 H and h?H , and expand the average in (3) into a sum of four averages involving the functions g and h. Two of these averages vanish because iterates of g are orthogonal to iterates of h. So in order to show that the only contribution comes from the average that involves the function g alone, it suffices to establish that N 1 X n2 lim T h D0: (5) 2 N!1 N nD1 L ()
To show this we will apply the Hilbert space van der Cor2 put lemma. For given h 2 N, we let x n D T n h and com-
N Z 1 X 2 2 2 T n C2nmCm h T n h d N nD1
N Z 1 X 2 D lim T 2nm (T m h) h d : N!1 N nD1
Applying the ergodic theorem to the transformation T 2m and using the fact that h?H , we get that the last limit is 0. This implies (5). Thus far we have shown that in order to compute the limit in (3) we can assume that f D g 2 H (g is also nonnegative and g ¤ 0). By the definition of H , given any " > 0, there exists a function f 0 2 H such that T k f 0 D f 0 for some k 2 N and k f f 0 kL 2 () ". Then the limit in (3) is at least 1/k times the limit N Z 1 X 2 f (x) f (T (kn) x) d: lim inf N!1 N nD1 Applying the triangle inequality twice we get that this is greater or equal than N Z 1 X 2 lim f 0 (x) f 0 (T (kn) x) d c " N!1 N nD1 Z D ( f 0 (x))2 d 2" Z
2 f 0 (x) d c ";
for some constant c that does not depend on " (we used that T k f 0 D f 0 and the Cauchy–Schwartz inequality). Choosing " small enough we conclude that the last quantity is positive, completing the proof. Multiple Recurrence Simultaneous multiple returns of positive measure sets to themselves were first considered by H. Furstenberg [34], who gave a new proof of Szemerédi’s theorem [69] on arithmetic progressions by deriving it from the following theorem: Theorem 15 (Furstenberg [34]) Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then for every k 2 N, there is some n 2 N such that (A \ T n A \ \ T kn A) > 0 :
(6)
Furstenberg’s proof came by means of a new structure theorem allowing one to decompose an arbitrary measure preserving system into component elements exhibiting
Ergodic Theory: Recurrence
one of two extreme types of behavior: compactness, characterized by regular, “almost periodic” trajectories, and weak mixing, characterized by irregular, “quasi-random” trajectories. On T , these types of behavior are exemplified by rotations and by the doubling map, respectively. To see the point, imagine trying to predict the initial digit of the dyadic expansion of T n x given knowledge of the initial digits of T i x, 1 i < n. We use the case k D 2 to illustrate the basic idea. It suffices to show that if f 2 L1 () is nonnegative and f ¤ 0, one has N Z 1 X f (x) f (T n x) f (T 2n x) d > 0 : (7) lim inf N!1 N nD1 An ergodic decomposition argument enables us to assume that our system is ergodic. As in the earlier case of the squares, we split f into “almost periodic” and “quasi-random” components. Let K be the closure in L2 of the subspace spanned by the eigenfunctions of T, i. e. the functions f 2 L2 () that satisfy f (Tx) D e 2 i˛ f (x) for some ˛ 2 R. We write f D g C h, where g 2 K and h?K . It can be shown that g; h 2 L1 () and g is again nonnegative with g ¤ 0. We expand the average in (7) into a sum of eight averages involving the functions g and h. In order to show that the only non-zero contribution to the limit comes from the term involving g alone, it suffices to establish that N 1 X n 2n lim T g T h D0; (8) 2 N!1 N nD1
L ()
(and similarly with h and g interchanged, and with g D h, which is similar). To establish (8), we use the Hilbert space van der Corput lemma on x n D T n g T 2n h. Some routine computations and a use of the ergodic theorem reduce the task to showing that Z D-lim h(x) h(T 2m x) d D 0 : m!1
But this is well known for h?K (in virtue of the fact that for h?K the spectral measure h is continuous, for example). We are left with the average (7) when f D g 2 K . In this case f can be approximated arbitrarily well by a linear combination of eigenfunctions, which easily implies that given " > 0 one has kT n f f kL 2 () " for a set of n 2 N with bounded gaps. Using this fact and the triangle inequality, one finds that for a set of n 2 N with bounded gaps, Z 3 Z f (x) f (T n x) f (T 2n x) d f d c "
for a constant c that is independent of ". Choosing " small enough, we get (7). The new techniques developed for the proof of Theorem 15 have led to a number of extensions, many of which have to date only ergodic proofs. To expedite discussion of some of these developments, we introduce a definition: Definition 16 Let R Z and k 2 N. Then R is a set of k-recurrence if for every invertible measure preserving system (X; B; ; T) and A 2 B with (A) > 0, there is some nonzero n 2 R such that (A \ T n A \ \ T kn A) > 0 : The notions of k-recurrence are distinct for different values of k. An example of a difference set that is a set of 1-recurrence but not a set of 2-recurrence was given in [34]; sets of k-recurrence that are not sets of (k C 1)recurrencepfor general k were given in [31] (R k D fn 2 N : fn kC1 2g 2 [1/4; 3/4]g is such). Aside from difference sets, the sets of (1-)recurrence given in Theorem 12 may well be sets of k-recurrence for every k 2 N, though this has not been verified in all cases. Let us summarize the current state of knowledge. The following are sets of k-recurrence for every k: Sets of S the form n2N fa n ; 2a n ; : : : ; na n g where a n 2 N (this follows from a uniform version of Theorem 15 that can be found in [15]). Every IP-set [37]. The set fp(n); n 2 Ng where p is any nonconstant integer polynomial with p(0) D 0 [16], and more generally, when the range of the polynomial contains multiples of an arbitrary integer [33]. The set fp(n); n 2 Sg where p is an integer polynomial with p(0) D 0 and S is any IP-set [17]. The set of values of an admissible generalized polynomial [60]. Moreover, the set of shifted primes fp 1; p primeg, and the set fp C 1; p primeg are sets of 2-recurrence [32]. More generally, one would like to know for which sequences of integers a1 (n); : : : ; a k (n) it is the case that for every invertible measure preserving system (X; B; ; T) and A 2 B with (A) > 0, there is some nonzero n 2 N such that (A \ T a 1 (n) A \ \ T a k (n) A) > 0 :
(9)
Unfortunately, a criterion analogous to the one given in Theorem 13 for 1-recurrence is not yet available for k-recurrence when k > 1. Nevertheless, there have been some notable positive results, such as the following: Theorem 17 (Bergelson and Leibman [16])
Let (X;
B; ; T) be an invertible measure preserving system and
p1 (n); : : : ; p k (n) be integer polynomials with zero constant term. Then for every A 2 B with (A) > 0, there is some
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n 2 N such that (A \ T p 1 (n) A \ \ T p k (n) A) > 0 :
(10)
Furthermore, it has been shown that the n in (10) can be chosen from any IP set [17], and the polynomials p1 ; : : : ; p k can be chosen to belong to the more general class of admissible generalized polynomials [60]. Very recently, a new boost in the area of multiple recurrence was given by a breakthrough of Host and Kra [50]. Building on work of Conze and Lesigne [26,27] and Furstenberg and Weiss [41] (see also the excellent survey [52], exploring close parallels with [45] and the seminal paper of Gowers [43]), they isolated the structured component (or factor) of a measure preserving system that one needs to analyze in order to prove various multiple recurrence and convergence results. This allowed them, in particular, to prove existence of L2 limits for the so-called PN Qk in “Furstenberg ergodic averages” N1 nD1 iD0 f (T x), which had been a major open problem since the original ergodic proof of Szemerédi’s theorem. Subsequently Ziegler in [75] gave a new proof of the aforementioned limit theorem and established minimality of the factor in question. It turns out that this minimal component admits of a purely algebraic characterization; it is a nilsystem, i. e. a rotation on a homogeneous space of a nilpotent Lie group. This fact, coupled with some recent results about nilsystems (see [58,59] for example), makes the analysis of some otherwise intractable multiple recurrence problems much more manageable. For example, these developments have made it possible to estimate the size of the multiple intersection in (6) for k D 2; 3 (the case k D 1 is Theorem 4): Theorem 18 (Bergelson, Host and Kra [14]) Let (X; B; ; T) be an ergodic measure preserving system and A 2 B. Then for k D 2; 3 and for every " > 0, (A \ T n A \ \ T kn A) > kC1 (A) "
(11)
for a set of n 2 N with bounded gaps. Based on work of Ruzsa that appears as an appendix to the paper, it is also shown in [14] that a similar estimate fails for ergodic systems (with any power of (A) on the right hand side) when k 4. Moreover, when the system is nonergodic it also fails for k D 2; 3, as can be seen with the help of an example in [6]. Again considering the doubling map Tx D 2x and the set A D [0; 1/2], one sees that the positive results for k 3 are sharp. When the polynomials n; 2n; : : : ; kn are replaced by linearly independent polynomials p1 ; p2 ; : : : ; p k with zero constant term, similar lower bounds hold for every k 2 N without assuming
ergodicity [30]. The case where the polynomials n; 2n; 3n are replaced with general polynomials p1 ; p2 ; p3 with zero constant term is treated in [33]. Connections with Combinatorics and Number Theory The combinatorial ramifications of ergodic-theoretic recurrence were first observed by Furstenberg, who perceived a correspondence between recurrence properties of measure preserving systems and the existence of structures in sets of integers having positive upper density. This gave rise to the field of ergodic Ramsey theory, in which problems in combinatorial number theory are treated using techniques from ergodic theory. The following formulation is from [8]. Theorem 19 Let be a subset of the integers. There exists an invertible measure preserving system (X; B; ; T) and a set A 2 B with (A) D d() such that d( \ ( n1 ) \ : : : \ ( n k )) (A \ T n 1 A \ \ T n k A) ;
(12)
for all k 2 N and n1 ; : : : ; n k 2 Z. Proof The space X will be taken to be the sequence space f0; 1gZ , B is the Borel -algebra, while T is the shift map defined by (Tx)(n) D x(n C 1) for x 2 f0; 1gZ , and A is the set of sequences x with x(0) D 1. So the only thing that depends on is the measure which we now define. For m 2 N set 0 D Z n and 1 D . Using a diagonal argument we can find an increasing sequence of integers (N m )m2N such that limm!1 j \ [1; N m ]j/N m D d() and such that ˇ ˇ ˇ ( i 1 n1 ) \ ( i 2 n2 ) \ ˇ ˇ ˇ ˇ ˇ \( i r nr ) \ [1; N m ] (13) lim m!1 Nm exists for every n1 ; : : : ; nr 2 Z, and i1 ; : : : ; i r 2 f0; 1g. For n1 ; n2 ; : : : ; nr 2 Z, and i1 ; i2 ; : : : ; i r 2 f0; 1g, we define the measure of the cylinder set fx(n1 ) D i1 ; x(n2 ) D i2 ; : : : ; x(nr ) D i r g to be the limit (13). Thus defined, extends to a premeasure on the algebra of sets generated by cylinder sets and hence by Carathéodory’s extension theorem [24] to a probability measure on B. It is easy to check that (A) D d(), the shift transformation T preserves the measure and (12) holds. Using this principle for k D 1, one may check that any set of recurrence is intersective, that is intersects E E for every set E of positive density. Using it for n1 D n; n2 D
Ergodic Theory: Recurrence
2n; : : : ; n k D kn, together with Theorem 15, one gets an ergodic proof of Szemerédi’s theorem [69], stating that every subset of the integers with positive upper density contains arbitrarily long arithmetic progressions (conversely, one can easily deduce Theorem 15 from Szemerédi’s theorem, and that intersective sets are sets of recurrence). Making the choice n1 D n2 and using part (iv) of Theorem 13, we get an ergodic proof of the surprising result of Sárközy [66] stating that every subset of the integers with positive upper density contains two elements whose difference is a perfect square. More generally, using Theorem 19, one can translate all of the recurrence results of the previous two sections to results in combinatorics. (This is not straightforward for Theorem 18 because of the ergodicity assumption made there. We refer the reader to [14] for the combinatorial consequence of this result). We mention explicitly only the combinatorial consequence of Theorem 17: Theorem 20 (Bergelson and Leibman [16]) Let Z with d() > 0, and p1 ; : : : ; p k be integer polynomials with zero constant term. Then contains infinitely many configurations of the form fx; x C p1 (n); : : : ; x C p k (n)g. The ergodic proof is the only one known for this result, even for patterns of the form fx; xCn2 ; xC2n2 g or fx; xC n; x C n2 g. Ergodic-theoretic contributions to the field of geometric Ramsey theory were made by Furstenberg, Katznelson, and Weiss [40], who showed that if E is a positive upper density subset of R2 then: (i) E contains points with any large enough distance (see also [21] and [29]), (ii) Every ı-neighborhood of E contains three points forming a triangle congruent to any given large enough dilation of a given triangle (in [21] it is shown that if the three points lie on a straight line one cannot always find three points with this property in E itself). Recently, a generalization of property (ii) to arbitrary finite configurations of Rm was obtained by Ziegler [76]. It is also worth mentioning some recent exciting connections of multiple recurrence with some structural properties of the set of prime numbers. The first one is in the work of Green and Tao [45], where the existence of arbitrarily long arithmetic progressions of primes was demonstrated, the authors, in addition to using Szemerédi’s theorem outright, use several ideas from its ergodic-theoretic proofs, as appearing in [34] and [39]. The second one is in the recent work of Tao and Ziegler [70], where a quantitative version of Theorem 17 was used to prove that the primes contain arbitrarily long polynomial progressions. Furthermore, several recent results in ergodic theory, related to the structure of the minimal characteristic fac-
tors of certain multiple ergodic averages, play an important role in the ongoing attempts of Green and Tao to get asymptotic formulas for the number of k-term arithmetic progressions of primes up to x (see for example [46] and [47]). This project has been completed for k D 3, thus verifying an interesting special case of the Hardy– Littlewood k-tuple conjecture predicting the asymptotic growth rate of N a 1 ;:::;a k (x) D the number of configurations of primes having the form fp; p C a1 ; : : : ; p C a k g with p x. Finally, we remark that in this article we have restricted attention to multiple recurrence and Furstenberg correspondence for Z actions, while in fact there is a wealth of literature on extensions of these results to general commutative, amenable and even non-amenable groups. For an excellent exposition of these and other recent developments the reader is referred to the surveys [9] and [11]. Here, we give just one notable combinatorial corollary to some work of this kind, a density version of the classical Hales–Jewett coloring theorem [48]. Theorem 21 (Furstenberg and Katznelson [38]) Let Wn (A) denote the set of words of length n with letters in the alphabet A D fa1 ; : : : ; a k g. For every " > 0 there exists N0 D N0 ("; k) such that if n N0 then any subset S of Wn (A) with jSj "k n contains a combinatorial line, i. e., a set consisting of k n-letter words, having fixed letters in l positions, for some 0 l < n, the remaining n l positions being occupied by a variable letter x, for x D a1 ; : : : ; a k . (For example, in W4 (A) the sets f(a1 ; x; a2 ; x) : x 2 Ag and f(x; x; x; x); : x 2 Ag are combinatorial lines). At first glance, the uninitiated reader may not appreciate the importance of this “master” density result, so it is instructive to derive at least one of its immediate consequences. Let A D f0; 1; : : : ; k 1g and interpret Wn (A) as integers in base k having at most n digits. Then a combinatorial line in Wn (A) is an arithmetic progression of length k – for example, the line f(a1 ; x; a2 ; x) : x 2 Ag corresponds to the progression fm; m C n; m C 2n; m C 3ng, where m D a1 C a2 d 2 and n D d C d 3 . This allows one to deduce Szemerédi’s theorem. Similarly, one can deduce from Theorem 21 multidimensional and IP extensions of Szemerédi’s theorem [36,37], and some related results about vector spaces over finite fields [37]. Again, the only known proof for the density version of the Hales– Jewett theorem relies heavily on ergodic theory. Future Directions In this section we formulate a few open problems relating to the material in the previous three sections. It should be
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Ergodic Theory: Recurrence
noted that this selection reflects the authors’ interests, and does not strive for completeness. We start with an intriguing question of Katznelson [54] about sets of topological recurrence. A set S N is a set of Bohr recurrence if for every ˛1 ; : : : ; ˛ k 2 R and " > 0 there exists s 2 S such that fs˛ i g 2 [0; "] [ [1 "; 1) for i D 1; : : : ; k. Problem 1 Is every set of Bohr recurrence a set of topological recurrence? Background for this problem and evidence for a positive answer can be found in [54,72]. As we mentioned in Sect. “Subsequence Recurrence”, there exists a set of topological recurrence (and hence Bohr recurrence) that is not a set of recurrence. Problem 2 Is the set S D fl!2m 3n : l; m; n 2 Ng a set of recurrence? Is it a set of k-recurrence for every k 2 N? It can be shown that S is a set of Bohr recurrence. Theorem 13 cannot be applied since the uniform distribution condition fails for some irrational numbers ˛. A relevant question was asked by Bergelson in [9]: “Is the set S D f2m 3n : m; n 2 Ng good for single recurrence for weakly mixing systems?” As we mentioned in Sect. “Multiple Recurrence”, the set of primes shifted by 1 (or 1) is a set of 2-recurrence [32]. Problem 3 Show that the sets P 1 and P C 1, where P is the set of primes, are sets of k-recurrence for every k 2 N. As remarked in [32], a positive answer to this question will follow if some uniformity conjectures of Green and Tao [47] are verified. We mentioned in Sect. “Subsequence Recurrence” that random non-lacunary sequences (see definition there) are almost surely sets of recurrence. Problem 4 Show that random non-lacunary sequences are almost surely sets of k-recurrence for every k 2 N. The answer is not known even for k D 2, though, in unpublished work, Wierdl and Lesigne have shown that the answer is positive for random sequences with at most quadratic growth. We refer the reader to the survey [65] for a nice exposition of the argument used by Bourgain [22] to handle the case k D 1. It was shown in [31] that if S is a set of 2-recurrence then the set of its squares is a set of recurrence for circle rotations. The same method shows that it is actually a set of Bohr recurrence. Problem 5 If S Z is a set of 2-recurrence, is it true that S 2 D fs 2 : s 2 Sg is a set of recurrence?
A similar question was asked in [23]: “If S is a set of k-recurrence for every k, is the same true of S2 ?”. One would like to find a criterion that would allow one to deduce that a sequence is good for double (or higher order) recurrence from some uniform distribution properties of this sequence. Problem 6 Find necessary conditions for double recurrence similar to the one given in Theorem 13. It is now well understood that such a criterion should involve uniform distribution properties of some generalized polynomials or 2-step nilsequences. We mentioned in Sect. “Connections with Combinatorics and Number Theory” that every positive density subset of R2 contains points with any large enough distance. Bourgain [21] constructed a positive density subset E of R2 , a triangle T, and numbers t n ! 1, such that E does not contain congruent copies of all t n -dilations of T. But the triangle T used in this construction is degenerate, which leaves the following question open: Problem 7 Is it true that every positive density subset of R2 contains a triangle congruent to any large enough dilation of a given non-degenerate triangle? For further discussion on this question the reader can consult the survey [44]. The following question of Aaronson and Nakada [1] is related to a classical question of Erd˝os concerning whether P every K N such that n2K 1/n D 1 contains arbitrarily long arithmetic progressions: Problem 8 Suppose that (X; B; ; T) is a f1/ng-conservative ergodic measure preserving system. Is it true that for every A 2 B with (A) > 0 and k 2 N we have (A \ T n A \ \ T kn A) > 0 for some n 2 N? The answer is positive for the class of Markov shifts, and it is remarked in [1] that if the Erd˝os conjecture is true then the answer will be positive in general. The converse is not known to be true. For a related result showing that multiple recurrence is preserved by extensions of infinite measure preserving systems see [61]. Our next problem is motivated by the question whether Theorem 21 has a polynomial version (for a precise formulation of the general conjecture see [9]). Not even this most special consequence of it is known to hold. Problem 9 Let " > 0. Does there exist N D N(") having the property that every family P of subsets of f1; : : : ; Ng2 2 satisfying jPj "2 N contains a configuration fA; A [ (
)g, where A f1; : : : ; Ng2 and f1; : : : ; Ng with A \ ( ) D ;?
Ergodic Theory: Recurrence
A measure preserving action of a general countably infinite group G is a function g ! Tg from G into the space of measure preserving transformations of a probability space X such that Tg h D Tg Th . It is easy to show that a version of Khintchine’s recurrence theorem holds for such actions: if (A) > 0 and " > 0 then fg : (A \ Tg A) > ((A))2 "g is syndetic. However it is unknown whether the following ergodic version of Roth’s theorem holds. Problem 10 Let (Tg ) and (S g ) be measure preserving G-actions of a probability space X that commute in the sense that Tg S h D S h Tg for all g; h 2 G. Is it true that for all positive measure sets A, the set of g such that (A \ Tg A \ S g A) > 0 is syndetic? We remark that for general (possibly amenable) groups G not containing arbitrarily large finite subgroups nor elements of infinite order, it is not known whether one can find a single such g ¤ e. On the other hand, the answer is known to be positive for general G in case (Tg1 S g ) is a G-action [18]; even under such strictures, however, it is unknown whether a triple recurrence theorem holds. Bibliography 1. Aaronson J (1981) The asymptotic distribution behavior of transformations preserving infinite measures. J Analyse Math 39:203–234 2. Aaronson J (1997) An introduction to infinite ergodic theory. Mathematical Surveys and Monographs 50. American Mathematical Society, Providence 3. Aaronson J, Nakada H (2000) Multiple recurrence of Markov shifts and other infinite measure preserving transformations. Israel J Math 117:285–310 4. Barreira L (2001) Hausdorff dimension of measures via Poincaré recurrence. Comm Math Phys 219:443–463 5. Barreira L (2005) Poincaré recurrence: old and new. XIVth International Congress on Mathematical Physics, World Sci Publ Hackensack, NJ, pp 415–422 6. Behrend F (1946) On sets of integers which contain no three in arithmetic progression. Proc Nat Acad Sci 23:331–332 7. Bergelson V (1987) Weakly mixing PET. Ergod Theory Dynam Syst 7:337–349 8. Bergelson V (1987) Ergodic Ramsey Theory. In: Simpson S (ed) Logic and Combinatorics. Contemporary Mathematics 65. American Math Soc, Providence, pp 63–87 9. Bergelson V (1996) Ergodic Ramsey Theory – an update. In: Pollicot M, Schmidt K (eds) Ergodic theory of Zd -actions. Lecture Note Series 228. London Math Soc, London, pp 1–61 10. Bergelson V (2000) The multifarious Poincaré recurrence theorem. In: Foreman M, Kechris A, Louveau A, Weiss B (eds) Descriptive set theory and dynamical systems. Lecture Note Series 277. London Math Soc, London, pp 31–57 11. Bergelson V (2005) Combinatorial and diophantine applications of ergodic theory. In: Hasselblatt B, Katok A (eds) Handbook of dynamical systems, vol 1B. Elsevier, pp 745–841 12. Bergelson V, Furstenberg H, McCutcheon R (1996) IP-sets and polynomial recurrence. Ergod Theory Dynam Syst 16:963–974
13. Bergelson V, Håland-Knutson I, McCutcheon R (2006) IP Systems, generalized polynomials and recurrence. Ergod Theory Dynam Syst 26:999–1019 14. Bergelson V, Host B, Kra B (2005) Multiple recurrence and nilsequences. Inventiones Math 160(2):261–303 15. Bergelson V, Host B, McCutcheon R, Parreau F (2000) Aspects of uniformity in recurrence. Colloq Math 84/85(2):549–576 16. Bergelson V, Leibman A (1996) Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J Amer Math Soc 9:725–753 17. Bergelson V, McCutcheon R (2000) An ergodic IP polynomial Szemerédi theorem. Mem Amer Math Soc 146:viii–106 18. Bergelson V, McCutcheon R (2007) Central sets and a noncommutative Roth theorem. Amer J Math 129:1251–1275 19. Birkhoff G (1931) A proof of the ergodic theorem. Proc Nat Acad Sci 17:656–660 20. Boshernitzan M (1993) Quantitative recurrence results. Invent Math 113:617–631 21. Bourgain J (1986) A Szemerédi type theorem for sets of positive density in Rk . Israel J Math 54(3):307–316 22. Bourgain J (1988) On the maximal ergodic theorem for certain subsets of the positive integers. Israel J Math 61:39–72 23. Brown T, Graham R, Landman B (1999) On the set of common differences in van der Waerden’s theorem on arithmetic progressions. Canad Math Bull 42:25–36 24. Carathéodory C (1968) Vorlesungen über reelle Funktionen, 3rd edn. Chelsea Publishing Co, New York 25. Chazottes J, Ugalde E (2005) Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources. Discrete Contin Dyn Syst Ser B 5(3):565–586 26. Conze J, Lesigne E (1984) Théorèmes ergodiques pour des mesures diagonales. Bull Soc Math France 112(2):143–175 27. Conze J, Lesigne E (1988) Sur un théorème ergodique pour des mesures diagonales. Probabilités, Publ Inst Rech Math Rennes 1987-1, Univ Rennes I, Rennes, pp 1–31 28. Evans D, Searles D (2002) The fluctuation theorem. Adv Phys 51:1529–1585 29. Falconer K, Marstrand J (1986) Plane sets with positive density at infinity contain all large distances. Bull Lond Math Soc 18:471–474 30. Frantzikinakis N, Kra B (2006) Ergodic averages for independent polynomials and applications. J Lond Math Soc 74(1): 131–142 31. Frantzikinakis N, Lesigne E, Wierdl M (2006) Sets of k-recurrence but not (k C 1)-recurrence. Annales de l’Institut Fourier 56(4):839–849 32. Frantzikinakis N, Host B, Kra B (2007) Multiple recurrence and convergence for sets related to the primes. J Reine Angew Math 611:131–144. Available at http://arxiv.org/abs/ math/0607637 33. Frantzikinakis N (2008) Multiple ergodic averages for three polynomials and applications. Trans Am Math Soc 360(10):5435–5475. Available at http://arxiv.org/abs/math/ 0606567 34. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Anal Math 71:204–256 35. Furstenberg H (1981) Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton
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36. Furstenberg H, Katznelson Y (1979) An ergodic Szemerédi theorem for commuting transformations. J Analyse Math 34:275– 291 37. Furstenberg H, Katznelson Y (1985) An ergodic Szemerédi theorem for IP-systems and combinatorial theory. J Analyse Math 45:117–168 38. Furstenberg H, Katznelson Y (1991) A density version of the Hales–Jewett theorem. J Analyse Math 57:64–119 39. Furstenberg H, Katznelson Y, Ornstein D (1982) The ergodic theoretical proof of Szemerédi’s theorem. Bull Amer Math Soc (NS) 7(3):527–552 40. Furstenberg H, Katznelson Y, Weiss B (1990) Ergodic theory and configurations in sets of positive density. Mathematics of Ramsey theory. Algorithms Combin 5. Springer, Berlin, pp 184–198 41. Furstenberg H, Weiss B (1996) A mean ergodic theorem for PN 2 (1/N) nD1 f (T n x)g(T n x). Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ Math Res Inst Publ 5, de Gruyter, Berlin, pp 193–227 42. Galatolo S, Kim DH, Park KK (2006) The recurrence time for ergodic systems with infinite invariant measures. Nonlinearity 19:2567–2580 43. Gowers W (2001) A new proof of Szemerédi’s theorem. Geom Funct Anal 11:465–588 44. Graham RL (1994) Recent trends in Euclidean Ramsey theory. Trends in discrete mathematics. Discret Math 136(1–3):119– 127 45. Green B, Tao T (2008) The primes contain arbitrarily long arithmetic progressions. Ann Math 167:481–547. Available at http://arxiv.org/abs/math/0404188 46. Green B, Tao T (to appear) Quadratic uniformity of the Möbius function. Annales de l’Institut Fourier. Available at http://arxiv. org/abs/math.NT/0606087 47. Green B, Tao T () Linear equations in primes. Ann Math (to appear). Available at http://arxiv.org/abs/math.NT/0606088 48. Hales A, Jewett R (1963) Regularity and positional games. Trans Amer Math Soc 106:222–229 49. Hasley T, Jensen M (2004) Hurricanes and butterflies. Nature 428:127–128 50. Host B, Kra B (2005) Nonconventional ergodic averages and nilmanifolds. Ann Math 161:397–488 51. Kac M (1947) On the notion of recurrence in discrete stochastic processes. Bull Amer Math Soc 53:1002–10010 52. Kra B (2006) The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view. Bull Amer Math Soc (NS) 43:3–23 53. Kamae T, Mendés-France M (1978) Van der Corput’s difference theorem. Israel J Math 31:335–342 54. Katznelson Y (2001) Chromatic numbers of Cayley graphs on ˝ and his mathematics (Budapest, Z and recurrence. Paul Erdos 1999). Combinatorica 21(2):211–219 55. Khintchine A (1934) Eine Verschärfung des Poincaréschen “Wiederkehrsatzes”. Comp Math 1:177–179 56. Kriz I (1987) Large independent sets in shift invariant graphs. Solution of Bergelson’s problem. Graphs Combinatorics 3:145– 158
57. Leibman A (2002) Lower bounds for ergodic averages. Ergod Theory Dynam. Syst 22:863–872 58. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold. Ergod Theory Dynam Syst 25:201–213 59. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial actions of Zd by translations on a nilmanifold. Ergod Theory Dynam Syst 25:215–225 60. McCutcheon R (2005) FVIP systems and multiple recurrence. Israel J Math 146:157–188 61. Meyerovitch T (2007) Extensions and multiple recurrence of infinite measure preserving systems. Preprint. Available at http://arxiv.org/abs/math/0703914 62. Ornstein D, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inform Theory 39:78–83 63. Petersen K (1989) Ergodic theory. Cambridge Studies in Advanced Mathematics 2. Cambridge University Press, Cambridge 64. Poincaré H (1890) Sur le problème des trois corps et les équations de la dynamique. Acta Math 13:1–270 65. Rosenblatt J, Wierdl M (1995) Pointwise ergodic theorems via harmonic analysis. Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc, Lecture Note Series 205, Cambridge University Press, Cambridge, pp 3–151 66. Sárközy A (1978) On difference sets of integers III. Acta Math Acad Sci Hungar 31:125–149 67. Shkredov I (2002) Recurrence in the mean. Mat Zametki 72(4):625–632; translation in Math Notes 72(3–4):576–582 68. Sklar L (2004) Philosophy of Statistical Mechanics. In: Zalta EN (ed) The Stanford Encyclopedia of Philosophy (Summer 2004 edn), http://plato.stanford.edu/archives/sum2004/ entries/statphys-statmech/ 69. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:299–345 70. Tao T, Ziegler T () The primes contain arbitrarily long polynomial progressions. Acta Math (to appear). Available at http:// www.arxiv.org/abs/math.DS/0610050 71. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, Berlin 72. Weiss B (2000) Single orbit dynamics. CBMS Regional Conference Series in Mathematics 95. American Mathematical Society, Providence 73. Wyner A, Ziv J (1989) Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression. IEEE Trans Inform Theory 35:1250–1258 74. Zermelo E (1896) Über einen Satz der Dynamik und die mechanische Wärmetheorie. Annalen der Physik 57:485–94; English translation, On a theorem of dynamics and the mechanical theory of heat. In: Brush SG (ed) Kinetic Theory. Oxford, 1966, II, pp 208–17 75. Ziegler T (2007) Universal characteristic factors and Furstenberg averages. J Amer Math Soc 20:53–97 76. Ziegler T (2006) Nilfactors of Rm -actions and configurations in sets of positive upper density in Rm . J Anal Math 99:249– 266
Ergodic Theory: Rigidity
Ergodic Theory: Rigidity ˘ 1,2 VIOREL N I TIC ¸ A 1 West Chester University, West Chester, USA 2 Institute of Mathematics, Bucharest, Romania
Article Outline Glossary Definition of the Subject Introduction Basic Definitions and Examples Differentiable Rigidity Local Rigidity Global Rigidity Measure Rigidity Future Directions Acknowledgment Bibliography Glossary Differentiable rigidity Differentiable rigidity refers to finding invariants to the differentiable conjugacy of dynamical systems, and, more general, group actions. Local rigidity Local rigidity refers to the study of perturbations of homomorphisms from discrete or continuous groups into diffeomorphism groups. Global rigidity Global rigidity refers to the classification of all group actions on manifolds satisfying certain conditions. Measure rigidity Measure rigidity refers to the study of invariant measures for actions of abelian groups and semigroups. Lattice A lattice in a Lie group is a discrete subgroup of finite covolume. Conjugacy Two elements g1 ; g2 in a group G are said to be conjugated if there exists an element h 2 G such that g1 D h1 g2 h. The element h is called conjugacy. Ck Conjugacy Two diffeomorphisms 1 ; 2 acting on the same manifold M are said to be Ck -conjugated if there exists a Ck diffeomorphism h of M such that 1 D h1 ı 2 ı h. The diffeomorphism h is called Ck conjugacy. Definition of the Subject As one can see from this volume, chaotic behavior of complex dynamical systems is prevalent in nature and in large classes of transformations. Rigidity theory can be viewed as the counterpart to the generic theory of dynamical systems which often investigates chaotic dynamics for a typical transformation belonging to a large class. In rigid-
ity one is interested in finding obstructions to chaotic, or generic, behavior. This often leads to rather unexpected classification results. As such, rigidity in dynamics and ergodic theory is difficult to define precisely and the best approach to this subject is to study various results and themes that developed so far. A classification is offered below in local, global, differentiable and measurable rigidity. One should note that all branches are strongly intertwined and, at this stage of the development of the subject, it is difficult to separate them. Rigidity is a well developed and prominent topic in modern mathematics. Historically, rigidity has two main origins, one coming from the study of lattices in semi-simple Lie groups, and one coming from the theory of hyperbolic dynamical systems. From the start, ergodic theory was an important tool used to prove rigidity results, and a strong interdependence developed between these fields. Many times a result in rigidity is obtained by combining techniques from the theory of lattices in Lie groups with techniques from hyperbolic dynamical systems and ergodic theory. Among other mathematical disciplines using results and contributing to this field one can mention representation theory, smooth, continuous and measurable dynamics, harmonic and spectral analysis, partial differential equations, differential geometry, and number theory. Additional details about the appearance of rigidity in ergodic theory as well as definitions for some terminology used in the sequel can be found in the articles Ergodic Theory on Homogeneous Spaces and Metric Number Theory by Kleinbock, Ergodic Theory: Recurrence by Frantzikinakis, McCutcheon, and Ergodic Theory: Interactions with Combinatorics and Number Theory by Ward. The theory of hyperbolic dynamics is presented in the article Hyperbolic Dynamical Systems by Viana and in the article Smooth Ergodic Theory by Wilkinson. Introduction The first results about classification of lattices in semi-simple Lie groups were local, aimed at trying to understand the space of small perturbations of a given linear representation. A major contributor was Weil [110,111,112], who proved local rigidity of linear representations for large classes of groups, in particular lattices. Another breakthrough was the contribution of Kazhdan [66], who introduced property (T), allowing to show that large classes of lattices are finitely generated. Rigidity theory matured due to the remarkable global rigidity results obtained by Mostow [90] and Margulis [82], leading to a complete classification of lattices in large classes of semi-simple Lie groups.
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Briefly, a hyperbolic (Anosov) dynamical system is one that exhibits strong expansion and contraction along complementary directions. An early contribution introducing this class of objects is the paper of Smale [106], in which basic examples and techniques are introduced. A breakthrough came with the results of Anosov [1], who proved structural stability of the Anosov systems and ergodicity of the geodesic flow on a manifold of negative curvature. Motivated by questions arising in mathematical physics, chaos theory and other areas, hyperbolic dynamics emerged as one of the major fields of contemporary mathematics. From the beginning, a major unsolved problem in the field was the classification of Anosov diffeomorphisms and flows. In the 80s a change in philosophy occurred, partially motivated by a program introduced by Zimmer [114]. The goal of the program was to classify the smooth actions of higher rank semi-simple Lie groups and of their (irreducible) lattices on compact manifolds. It was expected that any such lattice action that preserves a smooth volume form and is ergodic can be reduced to one of the following standard models: isometric actions, linear actions on infranilmanifolds, and left translations on compact homogeneous spaces. This original conjecture was disproved by Katok, Lewis (see [56]): by blowing up a linear nilmanifold-action at some fixed points they exhibit real-analytic, volume preserving, ergodic lattice actions on manifolds with complicated topology. Nevertheless, imposing extra assumptions on a higher rank action, for example the existence of some hyperbolicity, allows local and global classification results. The concept of Anosov action, that is, an action that contains at least one Anosov diffeomorphism, was introduced for general groups by Pugh, Shub [99]. The significant differences between the classical Z and R cases and those of higher rank lattices, or at a more basic level, of higher rank abelian groups, went unnoticed for a while. The surge of activity in the 80s allowed these differences to surface: for lattices in the work of Hurder, Katok, Lewis, Zimmer (see [43,55,56,63]); and for higher rank abelian groups in the work of Katok, Lewis [55] and Katok, Spatzier [59]. As observed in these papers, local and global rigidity are typical for such Anosov actions. This generated additional research which is summarized in Sects. “Local Rigidity” and “Global Rigidity”. Differentiable rigidity is covered in Sect. “Differentiable Rigidity”. An interesting problem is to find moduli for the Ck conjugacy, k 1, of Anosov diffeomorphisms and flows. This was tackled so far only for low dimensional cases (n D 2 for diffeomorphisms and n D 3 for flows). Another direction that can be included here refers to find-
ing obstructions for higher transverse regularity of the stable/unstable foliation of a hyperbolic system. A spin-off of the research done so far, which is of high interest by itself, and has applications to local and global rigidity, consists of results lifting the regularity of solutions of cohomological equations over hyperbolic systems. In turn, these results motivated a more careful study of analytic questions about lifting the regularity of real valued continuous functions that enjoy higher regularity along webs of foliations. We also include in this section rigidity results for cocycles over higher rank abelian actions. These are crucial to the proof of local rigidity of higher rank abelian group actions. A more detailed presentation of the material relevant to differentiable rigidity can be found in the forthcoming monograph [58]. Measure rigidity refers to the study of invariant measures under actions of abelian groups and semigroups. If the actions are hyperbolic, higher-rank, and satisfy natural algebraic and irreducibility assumptions, one expects the invariant measures to be rare. This direction was started by a question of Furstenberg, asking if any nonatomic probability measure on the circle, invariant and ergodic under multiplications by 2 and 3, is the Lebesgue measure. An early contribution is that of Rudolph [103], who answered positively if the action has an element of strictly positive entropy. Katok, Spatzier [61] extended the question to more general higher rank abelian actions, such as actions by linear automorphisms of tori and Weyl chamber flows. A related direction is the study of the invariant sets and measures under the action of horocycle flows, where important progress was made by Ratner [100,101] and earlier by Margulis [16,81,83]. An application of these results present in the last papers is the proof of the long standing Oppenheim’s conjecture, about the density of the values of the quadratic forms at integer points. Recent developments due to and Einsiedler, Katok, Lindenstrauss [20] give a partial answer to another outstanding conjecture in number theory, Littlewood’s conjecture, and emphasize measure rigidity as one of a more promising directions in rigidity. More details are shown in Sect. “Measure Rigidity”. Four other recent surveys of rigidity theory, each one with a fair amount of overlap but also complementary in part to the present one, that discuss various aspects of the field and its significance are written by Fisher [23], Lindenstrauss [68], Ni¸tic˘a, Török [95], and Spatzier [107]. Among these, [23] concentrates mostly on local and global rigidity, [95] on differentiable rigidity, [68] on measure rigidity, and [107] gives a general overview. Here is a word of caution for the reader. Many times, instead of the most general results, we present an example
Ergodic Theory: Rigidity
that contains the essence of what is available. Also, several important facts that should have been included, are left out. This is because stating complete results would require more space than allocated to this material. The limited knowledge of the author also plays a role here. He apologizes for any obvious omissions and hopes that the bibliography will help fill the gaps. Basic Definitions and Examples A detailed introduction to the theory of Anosov systems and hyperbolic dynamics is given in the monograph [51]. The proofs of the basic results for diffeomorphisms stated below can be found there. The proofs for flows are similar. Surveys about hyperbolic dynamics in this volume are the article Hyperbolic Dynamical Systems by Viana and the article Smooth Ergodic Theory by Wilkinson. Consider a compact differentiable manifold M and f : M ! M a C1 diffeomorphism. Let TM be the tangent bundle of M, and D f : TM ! TM be the derivative of f . The map f is said to be an Anosov diffeomorphism if there is a smooth Riemannian metric k k on M, which induces a metric dM called adapted, a number 2 (0; 1), and a continuous Df invariant splitting TM D E s ˚ E u such that kD f vk kvk ; v 2 E s ;
kD f 1 vk kvk ; v 2 E u :
For each x 2 M there is a pair of embedded C1 discs s u (x), Wloc (x), called the local stable manifold and the Wloc local unstable manifold at x, respectively, such that: s (x) D E s (x) ; T W u (x) D E u (x); 1. Tx Wloc x loc s s ( f x) ; f 1 (W u (x)) W u ( f 1 x); 2. f (Wloc (x)) Wloc loc loc 3. For any 2 (; 1), there exists a constant C > 0 such that for all n 2 N,
d M ( f n x; f n y) Cn d M (x; y) ;
for
s y 2 Wloc (x) ;
d M ( f n x; f n y) Cn d M (x; y) ;
for
u y 2 Wloc (x) :
The local stable (unstable) manifolds can be extended to global stable (unstable) manifolds W s (x) and W u (x) which are well defined and smoothly injectively immersed. These global manifolds are the leaves of global foliations W s and W u of M. In general, these foliations are only continuous, but their leaves are differentiable. Let : R M ! M be a C1 flow. The flow is said to be an Anosov flow if there is a Riemannian metric k k on M, a constant 0 < < 1, and a continuous Df invariant splitting TM D E s ˚ E 0 ˚ E u such that for all x 2 M and t > 0: 1.
d t dt j tD0
2 E xc n f0g ; dim E xc D 1,
2. kD t vk t kvk ; v 2 E s , 3. kD t vk t kvk ; v 2 E u . For each x 2 M there is a pair of embedded C1 discs s (x) ; W u (x), called the local (strong) stable manifold Wloc loc and the local (strong) unstable manifold at x, respectively, such that: s (x) D E s (x) ; T W u (x) D E u (x); 1. Tx Wloc x loc s s (x)) Wloc ( t x) ; 2. t (Wloc u (x)) W u ( t x) for t > 0; t (Wloc loc 3. For any 2 (; 1), there exists a constant C > 0 such that for all n 2 N,
d M ( t x ; t y) C t d M (x; y) ; for d M (
t
x ;
t
s y 2 Wloc (x) ; t > 0 ;
t
y) C d M (x; y) ; for
u y 2 Wloc (x) ; t > 0 :
The local stable (unstable) manifolds can be extended to global stable (unstable) manifolds W s (x) and W u (x). These global manifolds are the leaves of global foliations W s and W u of M. One can also define weak stable and weak unstable foliations with leaves given by W cs (x) D [ t2R (W s (x)) and W cu (x) D [ t2R (W u (x)), which have as tangent distributions E cs D E c ˚E s and E cu D E c ˚E s . In general, all these foliations are only continuous, but their leaves are differentiable. Any Anosov diffeomorphism is structurally stable, that is, any C1 diffeomorphism that is C1 close to an Anosov diffeomorphism is topologically conjugate to the unperturbed one via a Hölder homeomorphism. An Anosov flow is structurally stable in the orbit equivalence sense: any C1 small perturbation of an Anosov flow has the orbit foliation topologically conjugate via a Hölder homeomorphism to the orbit foliation of the unperturbed flow. Let SL(n; R) be the group of all n-dimensional square matrices with real valued entries of determinant 1. Let SL(n; Z) SL(n; R) be the subgroup with integer entries. Basic examples of Anosov diffeomorphisms are automorphisms of the n-torus T n D Rn /Z n induced by hyperbolic matrices in SL(n; Z). A hyperbolic matrix is one that has only nonzero eigenvalues, all away in absolute value from 1. A specific example of such matrix in SL(2; Z) is 2 1 : 1 1 Basic examples of Anosov flows are given by the geodesic flows of surfaces of constant negative curvature. The unitary bundle of such a surface can be realized as M D n PSL(2; R), where PSL(2; R) D SL(2; R)/f˙1g and is a cocompact lattice in PSL(2; R). The action of
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the geodesic flow on M is induced by right multiplication with elements in the diagonal one-parameter subgroup
t/2 0 e ; t 2 R : 0 et/2 A related transformation, which is not hyperbolic, but will be of interest in this presentation, is the horocycle flow induced by right multiplication on M by elements in the one parameter subgroup
1 t ;t 2 R : 0 1 Of interest in this survey are also actions of more general groups than Z and R. Typical examples of higher rank Z k Anosov actions are constructed on tori using groups of units in number fields. See [65] for more details about this construction. A particular example of Anosov Z2 -action on T 3 is induced by the hyperbolic matrices: 0 1 0 1 0 1 0 2 1 0 A D @0 0 1A ; B D @0 2 1A : 1 8 2 1 8 4 One can check, by looking at the eigenvalues, that A and B are not multiples of the same matrix. Moreover, A and B commute. Typical examples of higher rank Anosov R k -actions are given by Weyl chamber flows, which we now describe using some notions from the theory of Lie groups. A good reference for the background in Lie group theory necessary here is the book of Helgason [40]. Note that for a hyperbolic element of such an action the center distribution is k-dimensional and coincides with the tangent distribution to the orbit foliation of R k . Let G be a semi-simple connected real Lie group of the noncompact type, with Lie algebra g. Let K G be a maximal compact subgroup that gives a Cartan decomposition g D k C p, where k is the Lie algebra of K and p is the orthogonal complement of k with respect to the Killing form of g. Let a p be a maximal abelian subalgebra and A D exp a be the corresponding subgroup. The simultaneous diagonalization of adg (a) gives the decomposition X gDgC g ; g0 D a C m ;
cones in a called Weyl chambers. The faces of the Weyl chambers are called Weyl chamber walls. Let M be the centralizer of A in K. Suppose is an irreducible torsion-free cocompact lattice in G. Since A commutes with M, the action of A by right translations on n G descends to an A-action on N :D n G/M. This action is called a Weyl chamber flow. Any Weyl chamber flow is an Anosov action, that is, has an element that acts hyperbolically transversally to the orbit foliation of A. Note that all maximal connected R diagonalizable subgroups of G are conjugate and their common dimension is called the R-rank of G. If the R-rank k of G is higher than 2, then the Weyl chamber flow is a higher rank hyperbolic R k -action. An example of semi-simple Lie group is SL(n; R). Let A be the diagonal subgroup of matrices with positive entries in SL(n; R). An example of Weyl chamber flow that will be discussed in the sequel is the action of A by right translations on nSL(n; R), where is a cocompact lattice. In this case the centralizer M is trivial. The rank of this action is n 1. The picture of the Weyl chambers for n D 3 is shown in Fig. 1. The signs that appear in each chamber are the signs of half of the Lyapunov exponents of a regular element from the chamber with respect to a certain fixed basis. For this action, the Lyapunov exponents appear in pairs of opposite signs. An example of higher rank lattice Anosov action that will be discussed in the sequel is the standard action of SL(n; Z) on the torus T n , (A; x) 7! Ax; A 2 SL(n; Z); x 2 T n : SL(n; Z) is a (noncocompact!) lattice in SL(n; R). As shown in [55], this action is generated by Anosov diffeomorphisms. We describe now a class of dynamical systems more general than the hyperbolic one. A C1 diffeomorphism f of a compact differentiable manifold M is called partially
2
where is the set of restricted roots. The spaces g are called root spaces. A point H 2 a is called regular if (H) ¤ 0 for all 2 . Otherwise it is called singular. The set of regular elements consists of the complement of a union of finitely many hyperplanes. Its components are
Ergodic Theory: Rigidity, Figure 1 Weyl chambers for SL(3; R)
Ergodic Theory: Rigidity
hyperbolic if there exists a continuous invariant splitting of the tangent bundle TM D E s ˚ E 0 ˚ E u such the the derivative of f expands Eu much more than E0 , and contracts Es much more than E0 . See [9,41] and [10] for the theory of partially hyperbolic diffeomorphisms. Es and Eu are called stable, respectively unstable distributions, and are integrable. E0 is called center distribution and, in general, it is not integrable. A structural stability result proved by Hirsch, Pugh, Shub [41], that is a frequently used tool in rigidity, shows that, if E0 is integrable to a smooth foliation, then any perturbation f¯ of f is partially hyperbolic and has an integrable center foliation. Moreover, the center foliations of f¯ of f are mapped one into the other by a homeomorphism that conjugates the maps induced on the factor spaces of the center foliations by f¯ of f respectively. We review now basic facts about cocycles. These basic definitions refer to several regularity classes: measurable, continuous, or differentiable. Let G be a group acting on a set M, and denote the action G M ! M by (g; x) 7! gx; thus (g1 g2 )x D g1 (g2 x). Let be a group with unit 1 . M is usually endowed with a measurable, continuous, or differentiable structure. A cocycle ˇ over the action is a function ˇ : G M ! such that ˇ(g1 g2 ; x) D ˇ(g1 ; g2 x)ˇ(g2 ; x) ;
(1)
for all g1 ; g2 2 G; x 2 M. Note that any group representation : G ! defines a cocycle called constant cocycle. The trivial representation defines the trivial cocycle. A natural equivalence relation for the set of cocycles is given by cohomology. Two cocycles ˇ1 ; ˇ2 : G M ! are cohomologous if there exists a map P : M ! , called transfer map, such that ˇ1 (g; x) D P(gx)ˇ2 (g; x)P(x)1 ;
(2)
for all g 2 G; x 2 M. Differentiable Rigidity We start by reviewing cohomological results. Several basic notions are already defined in Sect. “Basic Definitions and Examples”. In this section we assume that the cocycles are at least continuous. A cocycle ˇ : G M ! over an action (g; x) 7! gx; g 2 G; x 2 M, is said to satisfy closing conditions if for any g 2 G and x 2 M such that gx D x, one has ˇ(g; x) D 1 . Note that closing conditions are necessary in order for a cocycle to be cohomologous to the trivial one. Since a Z-cocycle is determined by a function ˇ : M ! ; ˇ(x) :D ˇ(1; x), the closing conditions become f nx D x
implies ˇ( f n1 x) : : : ˇ(x) D 1 ;
(3)
where f : M ! M is the function that implements the Zaction. The first cohomological results for hyperbolic systems were obtained by Livshits [70,71]. Let M be a compact Riemannian manifold with a Z-action implemented by a topologically transitive Anosov C1 diffeomorphism f . Then an ˛-Hölder function ˇ : M ! R determines a cocycle cohomologous to a trivial cocycle if and only if ˇ satisfies the closing conditions (3). The transfer map is ˛-Hölder. Moreover, for each Hölder class ˛ and each finite dimensional Lie group , there is a neighborhood U of the identity in such that an ˛-Hölder function ˇ : M ! determines a cocycle cohomologous to the trivial cocycle if and only if ˇ satisfies the closing conditions (3). The transfer map is again ˛-Hölder. Similar results are true for Anosov flows. Using Fourier analysis, Veech [108] extended Livshits’s result to real valued cocycles over Z-actions induced by ergodic endomorphisms of an n-dimensional torus, not necessarily hyperbolic. For cocycles with values in abelian groups, the question of two arbitrary cocycles being cohomologous reduces to the question of an arbitrary cocycle being cohomologous to a trivial one. This is not the case for cocycles with values in nonabelian groups. Parry [98] extended Livshits’s criteria to one for cohomology of two arbitrary cocycles with values in compact Lie groups. Parry’s result was generalized by Schmidt [104] to cocycles with values in Lie groups that, in addition, satisfy a center bunching condition. Ni¸tic˘a, Török [92] extended Livshits’s result to cocycles with values in the group Diff k (M) of Ck diffeomorphism of a compact manifold M with stably trivial bundle, k 3. Examples of such manifolds are the tori and the spheres. In this case, the transfer map takes values in Diff k3 (M), and it is Hölder with respect to a natural metric on Diff k3 (M). In [92] one can also find a generalization of Livshits’s result to generic Anosov actions, that is, actions generated by families of Anosov diffeomorphisms that do not interchange the stable and unstable directions of elements in the family. An example of such an action is the standard action of SL(n; Z) on the n-dimensional torus. A question of interest is the following: if two Ck cocycles, 1 k !, over a hyperbolic action, are cohomologous through a continuous/measurable transfer map P, what can be said about the higher regularity of P? For real valued cocycles the question can be reduced to one about cohomologically trivial cocycles. Livshits showed that for a real valued C1 cocycle cohomologous to a constant the transfer map is C1 . He also obtained C 1 regularity results if the action is given by hyperbolic automorphisms of a torus. After preliminary results by Guillemin and Kazh-
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dan for geodesic flows on surfaces of negative curvature, for general hyperbolic systems the question was answered positively by de la Llave, Marco, Moriyon [76] in the C 1 case and by de la Llave [74] in the real analytic case. Ni¸tic˘a, Török [93] considered the lift of regularity for a transfer map between two cohomologous cocycles with values in a Lie group or a diffeomorphism group. In contrast to the case of cocycles cohomologous to trivial ones, here one needs to require for the transfer map a certain amount of Hölder regularity that depends on the ratio between the expansion/contraction that appears in the base and the expansion/contraction introduced by the cocycle in the fiber. This assumption is essential, as follows from a counterexample found by de la Llave’s [73]. Useful tools in this development have been results from analysis that lift the regularity of a continuous real valued function which is assumed to have higher regularity along pairs of transverse Hölder foliations. Many times the foliations are the stable and unstable ones associated to a hyperbolic system. Journé [46] proved the C n;˛ regularity of a continuous function that is C n;˛ along two transverse continuous foliations with C n;˛ leaves. If one is interested only in C 1 regularity, a convenient alternative is a result of Hurder, Katok [44]. This has a simpler proof and can be applied to the more general situation in which the function is regular along a web of transverse foliations. A real analytic regularity result along these lines belongs to de la Llave [74]. In certain problems, for example when working with Weyl chamber flows, it is difficult to control the regularity in enough directions to span the whole tangent space. Nevertheless, the tangent space can be generated if one consider higher brackets of good directions. A C 1 regularity result for this case belongs to Katok, Spatzier [60]. In order to apply this result, the foliations need to be C 1 not only along the leaves, but also transversally. An application of the above regularity results is to questions about transverse regularity of the stable and unstable foliations of the geodesic flow on certain C 1 surfaces of nonpositive curvature. For compact negatively curved C 1 surfaces, E. Hopf showed that these foliations are C1 , and it follows from the work of Anosov that individual leaves are C 1 . Hurder, Katok [44] showed that once the weak-stable and weak-unstable foliations of a volume-preserving Anosov flow on a compact 3-manifold are C2 , they are C 1 . Another application of the regularity results is to the study of invariants for Ck conjugacy of hyperbolic systems. By structural stability, a small C1 perturbation of a hyperbolic system is C0 conjugate to the unperturbed one. The conjugacy, in general, is only Hölder. If the conjugacy is
C1 then it preserves the eigenvalues of the derivative at the periodic orbit. The following two results describe the invariants of smooth and real analytic conjugacy of low dimensional hyperbolic systems. They are proved in a series of papers written in various combinations by de la Llave, Marco, Moryion [72,74,75,79,80]. Let X; Y be two C 1 (C ! ) transitive Anosov vector fields on a compact three-dimensional manifold. If they are C0 conjugate and the eigenvalues of the derivative at the corresponding periodic orbits are the same, then the conjugating homeomorphism is C 1 (C! ). In particular, any C1 conjugacy is C 1 (C! ). Assume now that f ; g are two C 1 (C! ) Anosov diffeomorphisms on a compact two dimensional manifold. If they are C0 conjugate and the eigenvalues of the derivative at the corresponding periodic orbits are the same, then the conjugating diffeomorphism is C 1 (C! ). In particular, any C1 conjugacy is C 1 (C! ). An important direction was initiated by Katok, Spatzier [59] who studied cohomological results over hyperbolic Z k or R k -actions, k 2. They show that real valued smooth/Hölder cocycles over typical classes of hyperbolic Z k or R k , k 2, actions are smoothly/Hölder cohomologous to constants. These results cover, in particular, actions by hyperbolic automorphisms of a torus, and Weyl chamber flows. The proofs rely on harmonic analysis techniques, such as Fourier transform and group representations for semi-simple Lie groups. A geometric method for cocycle rigidity was developed in [64]. One constructs a differentiable form using invariant structures along stable/unstable foliations, and the commutativity of the action. The form is exact if and only if the cocycle is cohomologous to a constant one. The method covers actions on nilmanifolds satisfying a condition called TNS (Totally Non-Symplectic). This condition means that the action is higher rank abelian hyperbolic, and that the tangent space is a direct sum of invariant distributions, with each pair of these included in the stable distribution of a hyperbolic element of the action. The method was also applied to small (i. e. close to identity on a set of generators) Lie group valued cocycles. A related paper is [96] which contains rigidity results for cocycles over (TNS) actions with values in compact Lie groups. In this situation the number of cohomology classes is finite. An example of (TNS) action is given by the action of a maximal diagonalizable subgroup of SL(n; Z) on T n . Recently Damjanovi´c, Katok [14] developed a new method that was applied to the action of the matrix diagonal group on n SL(n; R). They use techniques from [54], where one finds cohomology invariants for cocycles over partially hyperbolic actions that satisfy accessibility prop-
Ergodic Theory: Rigidity
erty. Accessibility means that one can connect any two points from the manifold supporting the partially hyperbolic dynamical system by transverse piecewise smooth paths included in stable/unstable leaves. This notion was introduced by Brin, Pesin [9] and it is playing a crucial role in the recent surge of activity in the field of partially hyperbolic diffeomorphisms. See [10] for a recent survey of the subject. The cohomology invariants described in [54] are heights of the cocycle over cycles constructed in the base out of pieces inside stable/unstable leaves. They provide a complete set of obstructions for solving the cohomology equation. A new tool introduced in [14] is algebraic K-theory [88]. The method can be extended to cocycles with non-abelian range. In [57] one finds related results for small cocycles with values in a Lie group or the diffeomorphism group of a compact manifold. The equivalent of the Livshits theorem in the higherrank setting appears to be a description of the highest cohomology rather than the first cohomology. Indeed, for higher rank partially hyperbolic actions of the torus, the intermediate cohomologies are trivial, while for the highest one the closing conditions characterize the cohomology classes. This behavior provides a generalization of Veech cohomological result and of Katok, Spatzier cohomological result for toral automorphisms, and was discovered by A. Katok, S. Katok [52,53]. Flaminio, Forni [27] studied the cohomological equation over the horocycle flow. It is shown that there are infinitely many obstructions to the existence of a smooth solution. Moreover, if these obstructions vanish, then one can solve the cohomological equation. In [28] it is shown a similar result for cocycles over area preserving flows on compact higher-genus surfaces under certain assumptions that hold generically. Mieczkowski [87] extended these techniques and studied the cohomology of parabolic higher rank abelian actions. All these results rely on noncommutative Fourier analysis, more specifically representation theory of SL(2; R) and SL(2; C). Local Rigidity Let be a finitely (or compactly) generated group, G a topological group, and : ! G a homomorphism. The target of local rigidity theory is to understand the space of perturbations of various homomorphisms . Trivial perturbations of a homomorphism arise from conjugation by an arbitrary element of G. In order to rule them out, one says that is locally rigid if any nearby homomorphism 0 , (that is, 0 close to on a finite or compact set of generators of ), is conjugate to by an element g 2 G, that is, ( ) D g 0 ( )g 1 for all 2 . If G
is path-wise connected, one can also consider deformation rigidity, meaning that any nearby continuous path of homomorphisms is conjugated to the initial one via a continuous path of elements in G that has an end in the identity. Initial results on local rigidity are about embeddings of lattices into semi-simple Lie groups. The main results belong to Weil [110,111,112]. He showed that if G is a semisimple Lie group that is not locally isomorphic to SL(2; R) and if G is an irreducible cocompact lattice, then the natural embedding of into G is locally rigid. Earlier results were obtained by Selberg [105], Calabi, Vesentini [12], and Calabi [11]. Selberg proved the local rigidity of the natural embedding of cocompact lattices into SL(n; R). His proof used the dynamics of iterates of matrices, in particular the existence of singular directions, or walls of Weyl chambers, in the maximal diagonalizable subgroups of SL(n; R). Selberg’s approach inspired Mostow [90] to use the boundaries at infinity in his proof of strong rigidity of lattices, which in turn was crucial to the development of superrigidity due to Margulis [82]. See Sect. “Global Rigidity” for more details. Recall that hyperbolic dynamical systems are structurally stable. Thus they are, in a certain sense, locally rigid. We introduce now a precise definition of local rigidity in the infinite-dimensional setup. The fact that for general group actions one needs to consider different regularities for the actions, perturbations and conjugacies is apparent from the description of structural stability for Anosov systems. A Ck action ˛ of a finitely generated discrete group on a manifold M, that is, a homomorphism ˛ : ! Diff k (M), is said to be C k;l ;p;r locally rigid if any Cl perturbation ˛˜ which is Cp close to ˛ on a family of generators, is Cr conjugate to ˛, i.e there exists a Cr diffeomorphism H : M ! M which conjugates ˛˜ to ˛, that is, ˜ H ı ˛(g) D ˛(g) ı H for all g 2 . Note that for Anosov Z-actions, C 1;1;1;0 rigidity is known as structural stability. One can also introduce the notion of deformation rigidity if the initial action and the perturbation are conjugate by a continuous path of diffeomorphisms that has an end coinciding to the identity. A weaker notion of local rigidity can be defined in the presence of invariant foliations for the initial group action and for the perturbation. The map H is now required to preserve the leaves of the foliations and to conjugate only after factorization by the invariant foliations. The importance of this notion is apparent from the leafwise conjugacy structural stability theorem of Hirsch, Pugh, Shub [41]. See Sect. “Basic Definitions and Examples”. Moreover, for Anosov flows this is the natural notion of structural stability, and appears by taking the invariant fo-
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liation to be the one-dimensional orbit foliation. For more general actions, of lattices or higher rank abelian groups, this property is often used in combination to cocycle rigidity in order to show local rigidity. We discuss more about this when we review local rigidity results for partially hyperbolic actions. We summarize now several developments in local rigidity that emerged in the 80s. Initial results [67,115] were about infinitesimal rigidity, that is, a weaker version of local rigidity suitable for discrete groups representations in infinite-dimensional spaces of smooth vector fields. Then Hurder [43] proved C 1;1;1;1 deformation rigidity and Katok, Lewis, Zimmer [55,56,63] proved C 1;1;1;1 local rigidity of the standard action of SL(n; Z); n 3, on the n-dimensional torus. In these results crucial use was made of the presence of an Anosov element in the action. Due to the uniqueness of the conjugacy coming from structural stability, one has a continuous candidate for the conjugacy between the actions. Margulis, Qian [85] used the existence of a spanning family of directions that are hyperbolic for certain elements of the action to show local rigidity of partially hyperbolic actions that are not hyperbolic. Another important tool present in many proofs is Margulis and Zimmer superrigidity. These results allow one to produce a measurable conjugacy for the perturbation. Then one shows that the conjugacy has higher regularity using the presence of hyperbolicity. Having enough directions to span the whole tangent space is essential to lift the regularity. A cocycle to which superrigidity can be applied is the derivative cocycle. The study of local rigidity of partially hyperbolic actions that contain a compact factor was initiated by Ni¸tic˘a, Török [92,94]. Let n 3 and d 1. Let be the action of SL(n; Z) on T nCd D T n T d given by (A)(x; y) D (Ax; y) ; x 2 T n ; y 2 T d ; A 2 SL(n; Z). Then, for K 1, [92] shows that is C 1;1;5;K1 deformation rigid. The proof is based on three results in hyperbolic dynamics: the generalization of Livshits’s cohomological results to cocycles with values in diffeomorphism groups, the extension of Livshits’s result to general Anosov actions, and a version of the Hirsch, Pugh, Shub structural stability theorem improving the regularity of the conjugacy. Assume now n 3 and K 1. If is the action of SL(n; Z) on T nC1 D T n T given by (A)(x; y) D (Ax; y) ; x 2 T n ; y 2 T ; A 2 SL(n; Z), [94] shows that the action is C 1;1;2;K1 locally rigid. Ingredients in the proof are two rigidity results, one about TNS actions, and one about actions of property (T) groups. A locally compact group has property (T) if the trivial representation is isolated in the Fell topology. This means that if G acts on a Hilbert space unitarily and it has almost invari-
ant vectors, then it has invariant vectors. Hirsch–Pugh– Shub theorem implies that perturbations of abelian partially hyperbolic actions of product type are conjugated to skew-products of abelian Anosov actions via cocycles with values in diffeomorphism groups. In addition, the TNS property implies that the sum of the stable and unstable distributions of any regular element of the perturbation is integrable. The leaves of the integral foliation are closed, covering the base simply. Thus one obtains a conjugacy between the perturbation and a product action. Property (T) is used to show that the conjugacy reduces the perturbed action to a family of perturbations of hyperbolic actions. But the last ones are already known to be conjugate to the hyperbolic action in the base. Recent important progress in the question of local rigidity of lattice actions was made by Fisher, Margulis [24, 25,26]. Their proofs are modeled along the proof of Weil’s local rigidity result [112] and use an analog of Hamilton’s [39] hard implicit function theorem. Let G be a connected semi-simple Lie group with all simple factors of rank at least two, and G a lattice. The main result shows that a volume preserving affine action of G or on a compact smooth manifold X is C 1;1;1;1 locally rigid. Lower regularity results are also available. A component of the proof shows that if is a group with property (T), X a compact smooth manifold, and a smooth action of on X by Riemannian isometries, then is C 1;1;1;1 locally rigid. An earlier local rigidity result for this type of actions by cocompact lattices was obtained by Benveniste [5]. Many lattices act naturally on “boundaries” of type G/P, where G is a semi-simple algebraic Lie group and P is a parabolic subgroup. An example is given by G D SL(2; R) and P the subgroup in G consisting of upper triangular matrices. Local rigidity results for this type of actions were found by Ghys [34], Kanai [50] and Katok, Spatzier [62]. Starting with the work of Katok and Lewis, a related direction was the study of local rigidity for higher rank abelian actions. They prove in [55] the C 1;1;1;1 local rigidity of the action of a Zn maximal diagonalizable (over R) subgroup of SL(n C 1; Z); n 2, acting on the torus T nC1 . These type of results were later pushed forward by Katok, Spatzier [62]. Using the theory of nonstationary normal forms developed in [38] and [37] by Katok, Guysinsky, they proved several local rigidity results. The first one assumes that G is a semisimple Lie group with all simple factors of rank at least two, a lattice in G, N a nilpotent Lie group and a lattice in N. Then any Anosov affine action of on N/ is C 1;1;1;1 locally rigid. Second, let Zd be a group of affine transformations
Ergodic Theory: Rigidity
of N/ for which the derivatives are simultaneously diagonalizable over R with no eigenvalues on the unit circle. Then the Zd -action on N/ is C 1;1;1;1 locally rigid. A related result for continuous groups is the C 1;1;1;1 local rigidity (after factorization by the orbit foliation) of the action of a maximal abelian R-split subgroup in an R-split semi-simple Lie group of real rank at least two on G/, where is a cocompact lattice in G. One can also study rigidity of higher rank abelian partially hyperbolic actions that are not hyperbolic. Natural examples appear as automorphisms of tori and as variants of Weyl chamber flows. For the case of ergodic actions by automorphisms of a torus, this was investigated using a version of the KAM (Kolmogorov, Arnold, Moser) method by Damianovi´c, Katok [15]. As usual in the KAM method, one starts with a linearization of the conjugacy equation. At each step of the iterative KAM scheme, some twisted cohomological equations are solved. The existence of the solutions is forced by the ergodicity of the action and the higher rank assumptions. Diophantine conditions present in this case allow to control the fixed loss of regularity which is necessary for the convergence of these solutions to a conjugacy. Global Rigidity The first remarkable result in global rigidity belongs to Mostow [90]. For G a connected non-compact semi-simple Lie group not locally isomorphic to SL(2; R), and two irreducible cocompact lattices 1 ; 2 G, Mostow showed that any isomorphism from 1 into 2 extends to an isomorphism of G into itself. G has an involution whose fixed set is a maximal compact subgroup K. One constructs the symmetric Riemannian space X D G/K. To each chamber of X corresponds a parabolic group and these parabolic groups are endowed with a Tits geometry similar to the projective geometry of lines, planes etc. formed in the classical case when G D PGL(n; R). The proof of Mostow’s result starts by building a -equivariant pseudo-isometric map : G/K1 ! G/K2 . The map induces an incidence preserving -equivariant isomorphism 0 of the Tits geometries. By Tits’ generalized fundamental theorem of projective geometry, 0 is induced by an isomorphism of G. Finally, ( ) D 0 01 gives the desired conclusion. The next remarkable result in global rigidity is Margulis’ superrigidity theorem. An account of this development can be found in the monograph [82]. For large classes of irreducible lattices in semi-simple Lie groups, this result classifies all finite dimensional representations. Let G be a semi-simple simply connected Lie group of rank
higher than two and < G an irreducible lattice. Then any linear representation of is almost the restriction of a linear representation of G. That is, there exists a linear representation 1 of G and a bounded image representation 2 of such that D 1 2 . The possible representations 2 are also classified by Margulis up to some facts concerning finite image representations. As in the case of Mostow’s result, the proof involved the analysis of a map defined on the boundary at infinity. In this case the map is studied using deep results from dynamics like the multiplicativity ergodic theorem of Oseledec [97] or the theory of random walks on groups developed by Furstenberg [31]. An important consequence of Margulis superrigidity result is the arithmeticity of irreducible lattices in connected semi-simple Lie groups of rank higher than two. A basic example of arithmetic lattice can be obtained by taking the integer points in a semi-simple Lie group that is a matrix group, like taking SL(n; Z) inside SL(n; R). Special cases of superrigidity theorems were proved by Corlette [13] and Gromov, Schoen [36] for the rank one groups Sp(1; n) and respectively F 4 using the theory of harmonic maps. A consequence is the arithmeticity of lattices in these groups. Some of these results are put into differential geometric setting in [89]. Margulis supperrigidity result was extended to cocycles by Zimmer. A detailed exposition, including a self contained presentation of several rigidity results of Margulis, can be found in the monograph [113]. We mention here a version of this result that can be found in [24]. Let M be a compact manifold, H a matrix group, P D M H, and a lattice in a simply connected, semi-simple Lie group with all factors of rank higher that two. Assume that acts on M and H in a way that makes the projection from P to M equivariant. Moreover, the action of on P is measure preserving and ergodic. Then there exists a measurable map s : M ! H, a representation : G ! H, a compact subgroup K < H which commute with (G) and a measurable map k : M ! K such that s(m) D k( ; m) ( ) s( m). One can easily check from the last equation that k is a cocycle. So, up to a measurable change of coordinates given by the map s, the action of on P is a compact extension via a cocycle of a linear representation of G. Developing further the method of Mostow for studying the Tits building associated to a symmetric space of non-positive curvature led Ballman, Brin, Eberlein, Spatzier [2,3] to a number of characterizations of symmetric spaces. In particular, they showed that if M is a complete Riemannian manifold of non-positive curvature, finite volume, with simply connected cover, irreducible and of rank at least two, then M is isometric to a symmetric
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space with the connected component of Isom(M) having no compact factors. A topological rigidity theorem has been proved by Farrell, Jones [21]. They showed that if N is a complete connected Riemannian manifold whose sectional curvature lies in a closed interval included in (1; 0], and M is a topological manifold of dimension greater than 5, then any proper homotopy equivalence f : M ! N is properly homotopic to a homeomorphism. In particular, if M and N are both compact connected negatively curved Riemannian manifolds with isomorphic fundamental groups, then M and N are homeomorphic. Likewise to the case of local rigidity, a source of inspiration for results in global rigidity was the theory of hyperbolic systems, in particular their classification. The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. Moreover, any Anosov diffeomorphism on an infranilmanifold is topologically conjugate to a hyperbolic automorphism [29,78]. It is conjectured that any Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism of an infranilmanifold. Partial results are obtained in [91], where the conjecture is proved for Anosov diffeomorphisms with codimension one stable/unstable foliation. The proof of the general conjecture eluded efforts done so far. It is not even known if any Anosov diffeomorphism is topologically transitive, that is, if it has a dense orbit. A few positive results are available. Let M be a C 1 compact manifold endowed with a C 1 affine connection. Let f be a topologically transitive Anosov diffeomorphism preserving the connection and such that the stable and unstable distributions are C 1 . Then Benoist, Labourie [4] proved that f is C 1 conjugate to a hyperbolic automorphism of an infranilmanifold. The situation for Anosov flows is somehow different. As shown in [30], there exist Anosov flows that are not topologically transitive, so a general analog of the conjecture is false. Nevertheless, for the case of codimension one stable or unstable foliation, it is conjectured in [109] that any Anosov flow on a manifold of dimension greater than three admits a global cross-section. This would imply that the flow is topologically conjugate to the suspension of a linear automorphism of a torus. For actions of groups larger than Z, or R, global classification results are more abundant. A useful strategy in these type of results, which are quite technical, is to start by obtaining a measurable description of the action, most of the time using Margulis–Zimmer superrigidity results, and then use extra assumptions on the action, such as the presence of a hyperbolic element, or the presence of an invariant geometric structure, or both, in order to show that
the measurable model is actually continuous or even differentiable. For actions of higher rank Lie groups and their lattices some representative papers are by Katok, Lewis, Zimmer [63] and Goetze, Spatzier [35]. For actions of higher rank abelian groups see Kalinin, Spatzier [49]. Measure Rigidity Measure rigidity is the study of invariants measures for actions of one parameter and multiparameter abelian groups and semigroups acting on manifolds. Typical situations when interesting rigidity phenomena appear are for one parameter unipotent actions and higher rank hyperbolic actions, discrete or continuous. A unipotent matrix is one all of whose eigenvalues are one. An important case where the action of a unipotent flow appears is that of the horocycle flow. The invariant measures for it were studied by Furstenberg [33], who showed that the horocycle flow on a compact surface is uniquely ergodic, that is, the ergodic measure is unique. Dani and Smillie [17] extended this result to the case of non-compact surfaces, with the only other ergodic measures appearing being those supported on compact horocycles. An important breakthrough is the work of Margulis [81], who solved a long standing question in number theory, Oppenheim’s conjecture. The conjecture is about the density properties of the values of an indefinite quadratic form in three or more variables, provided the form is not proportional to a rational form. The proof of the conjecture is based on the study of the orbits for unipotent flows acting by translations on the homogenous space SL(n; Z) n SL(n; R). All these results were special cases of the Raghunathan conjecture about the structure of the orbits of the actions of unipotent flows on homogenous spaces. Raghunathan’s conjecture was proved in full generality by Ratner [100,101]. Borel, Prasad [8] raised the question of an analog of Raghunathan’s conjecture for S-algebraic groups. S-algebraic groups are products of real and p-adic algebraic groups. This was answered independently by Margulis, Tomanov [86] and Ratner [102]. A basic example of higher rank abelian hyperbolic action is given by the action of S m;n , the multiplicative semigroup of endomorphisms generated by the multiplication by m and n, two nontrivial integers, on the one dimensional torus T 1 . In a pioneering paper [32] Furstenberg showed that for m; n that are not powers of the same integer the action of S m;n has a unique closed, infinite invariant set, namely T 1 itself. Since there are many closed, infinite invariant sets for multiplication by m, and by n, this result shows a remarkable rigidity property of the joint action. Furstenberg’s result was generalized later by Berend
Ergodic Theory: Rigidity
for other group actions on higher dimensional tori and on other compact abelian groups in [6] and [7]. Furstenberg further opened the field by raising the following question: Conjecture 1 Let be a S m;n -invariant and ergodic probability measure on T 1 . Then is either an atomic measure supported on a finite union of (rational) periodic orbits or is the Lebesgue measure. While the statement appears to be simple, proving it has been elusive. The first partial result was given by Lyons [77] under the strong additional assumption that the measure makes one of the endomorphisms generating the action exact. Later Rudolph [103] and Johnson [45] weaken the exactness assumption and proved that must be the Lebesgue measure provided that multiplication by m (or multiplication by n) has positive entropy with respect to . Their results have been proved again using slightly different methods by Feldman [22]. A further extension was given by Host [42]. Katok proposed another example for which measure rigidity can be fruitfully tested, the Z2 -action induced by two commuting hyperbolic automorphisms on the torus T 3 . An example of such action is shown in Sect. “Basic Definitions and Examples”. One can also consider the action induced by a Z n1 maximal abelian group of hyperbolic automorphisms acting on the torus T n . Katok, Spatzier [61] developed a more geometric technique allowing to prove measure rigidity for these actions if they have an element acting with positive entropy. Their technique can be applied in higher generality if the action is irreducible in a strong sense and, in addition, it has individual ergodic elements or it is TNS. See also [47]. This method is based on the study of conditional measures induced by the invariant measure on certain invariant foliations that appear naturally in the presence of a hyperbolic action. Besides stable and unstable foliations, one can also consider various intersections of them. Einsiedler, Lindenstrauss [19] were able to eliminate the ergodicity and TNS assumptions. Yet another interesting example of higher rank abelian action is given by Weyl chamber flows. These do not satisfy the TNS condition. Einsiedler, Katok [18] proved that if G is SL(n; R); G is a lattice, H is the subgroup of positive diagonal matrices in G, and a H-invariant and ergodic measure on G/ such that the entropy of with respect to all one parameter subgroups of H is positive, then is the G invariant measure on G/ . These results are useful in the investigation of several deep questions in number theory. Define X D SL(3; Z) n SL(3; R) the diagonal subgroup of matrices with posi-
tive entries in SL(3; R). This space is not compact but is endowed with a unique translation invariant probability measure. The diagonal subgroup 80 s 9 1 0 0 < e = H D @ 0 et 0 A : s; t 2 R : ; 0 0 est acts naturally on X by right translations. It was conjectured by Margulis [83] that any compact H-invariant subset of X is a union of compact H-orbits. A positive solution to this conjecture implies a long standing conjecture of Littlewood: Conjecture 2 Let kxk denote the distance between the real number x and the closest integer. Then lim inf nkn˛kknˇk D 0 n!1
(4)
for any real numbers ˛ and ˇ. A partial result was obtained by Einsiedler, Katok, Lindenstrauss [20] who proved that the set of pairs (˛; ˇ) 2 R2 for which (4) is not satisfied has Hausdorff dimension zero. Applications of these techniques to questions in quantum ergodicity were found by Lindenstrauss [69]. A current direction in measure rigidity is attempting to classify the invariant measures under rather weak assumptions about the higher rank abelian action, like the homotopical data for the action. Kalinin and Katok [48] proved that any Z k ; k 2, smooth action ˛ on a k C 1dimensional torus whose elements are homotopic to the corresponding elements of an action by hyperbolic automorphisms preserves an absolutely continuous measure. Future Directions An important open problem in differential rigidity is to find invariants for the Ck conjugacy of the perturbations of higher dimensional hyperbolic systems. For Anosov diffeomorphisms, de la Llave counterexample [73] shows that this extension is not possible for a four dimensional example that appears as a direct product of two dimensional Anosov diffeomorphisms. Indeed, there are C 1 perturbations of the product that are only Ck conjugate to the unperturbed system for any k 1. In the positive direction, Katok conjectured that generalizations are possible for the diffeomorphism induced by an irreducible hyperbolic automorphism of a torus. One can also investigate this question for Anosov flows. Examples of higher rank Lie groups can be obtained by taking products of rank one Lie groups. Many actions of irreducible lattices in these type of groups are believed
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to be locally rigid, but the techniques available so far cannot be applied. A related problem is to study local rigidity in low regularity classes, for example the local rigidity of homomorphisms from higher rank lattices into homeomorphism groups. More problems related to local rigidity can be found in [23]. An important problem in global rigidity, emphasized by Katok and Spatzier, is to show that, up to differentiable conjugacy, any higher rank Anosov or partially hyperbolic action is algebraic under the assumption that it is sufficiently irreducible. The irreducibility assumption is needed in order to exclude actions obtained by successive application of products, extensions, restrictions and time changes from basic ingredients which include some rank one actions. Another problem of current research in measure rigidity is to develop a counterpart of Ratner’s theory for the case of actions by hyperbolic higher rank abelian groups on homogenous spaces. It was conjectured by Katok, Spatzier [61] that the invariant measures for such actions given by toral automorphisms or Weyl chamber flows are essentially algebraic, that is, supported on closed orbits of connected subgroups. Margulis in [84] extended this conjecture to a rather general setup addressing both the topological and measurable aspects of the problem. More details about actions on homogenous spaces, as well as connections to diophantine approximation, can be found in the article Ergodic Theory on Homogeneous Spaces and Metric Number Theory by Kleinbock. Acknowledgment This research was supported in part by NSF Grant DMS0500832. Bibliography Primary Literature 1. Anosov DV (1967) Geodesic flows on closed Riemannian manifolds with negative curvature. Proc Stek Inst 90:1–235 2. Ballman W, Brin M, Eberlein P (1985) Structure of manifolds of non-negative curvature. I. Ann Math 122:171–203 3. Ballman W, Brin M, Spatzier R (1985) Structure of manifolds of non-negative curvature. II. Ann Math 122:205–235 4. Benoist Y, Labourie F (1993) Flots d’Anosov á distribuitions stable et instable différentiables. Invent Math 111:285–308 5. Benveniste EJ (2000) Rigidity of isometric lattice actions on compact Riemannian manifolds. Geom Func Anal 10:516–542 6. Berend D (1983) Multi-invariant sets on tori. Trans AMS 280:509–532 7. Berend D (1984) Multi-invariant sets on compact abelian groups. Trans AMS 286:505–535 8. Borel A, Prasad G (1992) Values of isotropic quadratic forms at S-integral points. Compositio Math 83:347–372
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56. Katok A, Lewis J (1996) Global rigidity results for lattice actions on tori and new examples of vol preserving actions. Isr J Math 93:253–280 57. Katok A, Ni¸tic˘a V (2007) Rigidity of higher rank abelian cocycles with values in diffeomorphism groups. Geometriae Dedicata 124:109–131 58. Katok A, Ni¸tic˘a V () Differentiable rigidity of abelian group actions. Cambridge University Press (to appear) 59. Katok A, Spatzier R (1994) First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ Math IHES 79:131–156 60. Katok A, Spatzier R (1994) Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math Res Lett 1:193–202 61. Katok A, Spatzier R (1996) Invariant measures for higherrank abelian actions. Erg Theor Dyn Syst 16:751–778; Katok A, Spatzier R (1998) Corrections to: Invariant measures for higher-rank abelian actions.; (1996) Erg Theor Dyn Syst 16:751–778; Erg Theor Dyn Syst 18:503–507 62. Katok A, Spatzier R (1997) Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Trudy Mat Inst Stek 216:292–319 63. Katok A, Lewis J, Zimmer R (1996) Cocycle superrigidity and rigidity for lattice actions on tori. Topology 35:27–38 64. Katok A, Ni¸tic˘a V, Török A (2000) Non-abelian cohomology of abelian Anosov actions. Erg Theor Dyn Syst 2:259–288 65. Katok A, Katok S, Schmidt K (2002) Rigidity of measurable structure for Zd -actions by automorphisms of a torus. Comm Math Helv 77:718–745 66. Kazhdan DA (1967) On the connection of a dual space of a group with the structure of its closed subgroups. Funkc Anal Prilozen 1:71–74 67. Lewis J (1991) Infinitezimal rigidity for the action of SLn (Z) on T n . Trans AMS 324:421–445 68. Lindenstrauss E (2005) Rigidity of multiparameter actions. Isr Math J 149:199–226 69. Lindenstrauss E (2006) Invariant measures and arithmetic quantum unique ergodicity. Ann Math 163:165–219 70. Livshits A (1971) Homology properties of Y-systems. Math Zametki 10:758–763 71. Livshits A (1972) Cohomology of dynamical systems. Izvestia 6:1278–1301 he Livšic cohomology equation. Ann Math 123:537–611 72. de la Llave R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm Math Phys 109:369–378 73. de la Llave R (1992) Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic dynamical systems. Comm Math Phys 150:289–320 74. de la Llave R (1997) Analytic regularity of solutions of Livshits’s cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems. Erg Theor Dyn Syst 17:649–662 75. de la Llave R, Moriyon R (1988) Invariants for smooth conjugacy of hyperbolic dynamical systems. IV. Comm Math Phys 116:185–192 76. de la Llave R, Marco JM, Moriyon R (1986) Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann Math 123:537–611 77. Lyons R (1988) On measures simultaneously 2- and 3-invariant. Isr J Math 61:219–224
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Books and Reviews de la Harpe P, Valette A (1989) La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 Feres R (1998) Dynamical systems and semi-simple groups: An introduction. Cambridge Tracts in Mathematics, vol 126. Cambridge University Press, Cambridge Feres R, Katok A (2002) Ergodic theory and dynamics of G-spaces. In: Handbook in Dynamical Systems, 1A. Elsevier, Amsterdam, pp 665–763 Gromov M (1988) Rigid transformation groups. In: Bernard D, Choquet-Bruhat Y (eds) Géométrie Différentielle (Paris, 1986). Hermann, Paris, pp 65–139; Travaux en Cours. 33 Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook in Dynamical Systems, 1A. Elsevier, Amsterdam, pp 813–930 Knapp A (2002) Lie groups beyond an introduction, 2nd edn. Progress in Mathematics, 140. Birkhäuser, Boston Raghunathan MS (1972) Discrete subgroups of Lie groups. Springer, Berlin Witte MD (2005) Ratner’s theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago
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Ergodic Theory on Homogeneous Spaces and Metric Number Theory DMITRY KLEINBOCK Department of Mathematics, Brandeis University, Waltham, USA Article Outline Glossary Definition of the Subject Introduction Basic Facts Connection with Dynamics on the Space of Lattices Diophantine Approximation with Dependent Quantities: The Set-Up Further Results Future Directions Acknowledgment Bibliography Glossary Diophantine approximation Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector-valued) functions attain values close to integers. Metric number theory Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties. Homogeneous spaces A homogeneous space G/ of a group G by its subgroup is the space of cosets fg g. When G is a Lie group and is a discrete subgroup, the space G/ is a smooth manifold and locally looks like G itself. Lattice; unimodular lattice A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1. Ergodic theory The study of statistical properties of orbits in abstract models of dynamical systems. Hausdorff dimension A nonnegative number attached to a metric space and extending the notion of topological dimension of “sufficiently regular” sets, such as smooth submanifolds of real Euclidean spaces. Definition of the Subject The theory of Diophantine approximation, named after Diophantus of Alexandria, in its simplest set-up deals with the approximation of real numbers by rational num-
bers. Various higher-dimensional generalizations involve studying values of linear or polynomial maps at integer points. Often a certain “approximation property” is fixed, and one wants to characterize the set of numbers (vectors, matrices) which share this property, by means of certain measures (Lebesgue, or Hausdorff, or some other interesting measures). This is usually referred to as metric Diophantine approximation. The starting point for the theory is an elementary fact that Q, the set of rational numbers, is dense in R, the reals. In other words, every real number can be approximated by rationals: for any y 2 R and any " > 0 there exists p/q 2 Q with jy p/qj < " :
(1)
To answer questions like “how well can various real numbers be approximated by rational numbers? i. e., how small can " in (1) be chosen for varying p/q 2 Q?”, a natural approach has been to compare the accuracy of the approximation of y by p/q to the “complexity” of the latter, which can be measured by the size of its denominator q in its reduced form. This seemingly simple set-up has led to introducing many important Diophantine approximation properties of numbers/vectors/matrices, which show up in various fields of mathematics and physics, such as differential equations, KAM theory, transcendental number theory. Introduction As the first example of refining the statement about the density of Q in R, consider a theorem by Kronecker stating that for any y 2 R and any c > 0, there exist infinitely many q 2 Z such that jy p/qj < c/jqj
i. e. jqy pj < c
(2)
for some p 2 Z. A comparison of (1) and (2) shows that it makes sense to multiply both sides of (1) by q, since in the right hand side of (2) one would still be able to get very small numbers. In other words, approximation of y by p/q translates into approximating integers by integer multiples of y. Also, if y is irrational, (p; q) can be chosen to be relatively prime, i. e. one gets infinitely many different rational numbers p/q satisfying (2). However if y 2 Q the latter is no longer true for small enough c. Thus it seems to be more convenient to talk about pairs (p; q) rather than p/q 2 Q, avoiding a necessity to consider the two cases separately. At this point it is convenient to introduce the following central definition: if is a function N ! RC and y 2 R, say that y is -approximable (notation: y 2 W ( )) if
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
there exist infinitely many q 2 N such that jqy pj
0 there exist q D (q1 ; : : : ; q n ) 2 Z n n f0g and p D (p1 ; : : : ; p m ) 2 Z m satisfying the following system of inequalities: kYq pk < et/m
and kqk e t/n :
(5)
From this it easily follows that Wm;n ( 1;n/m ) D M m;n . In fact, it is this paper of Dirichlet which gave rise to his box principle. Later another proof of the same result was given by Minkowski. The constant c D 1 is not optimal: p the smallest value of c for which W1;1 ( c;1 ) D R is 1/ 5, and the optimal constants are not known in higher dimensions, although some estimates can be given [80]. Systems of linear forms which do not belong to W m;n ( c;n/m ) for some positive c are called badly approximable; that is, we set def
BA m;n D M m;n n [c>0 Wm;n (
c;n/m )
:
Their existence in arbitrary dimensions was shown by Perron. Note that a real number y (m D n D 1) is badly approximable if and only if its continued fraction coefficients are uniformly bounded. It was proved by Jarnik [46] in the
303
304
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
case m D n D 1 and by Schmidt in the general case [78] that badly approximable matrices form a set of full Hausdorff dimension: that is, dim(BA m;n ) D mn. On the other hand, it can be shown that each of the sets W m;n ( c;n/m ) for any c > 0 has full Lebesgue measure, and hence the complement BA m;n to their intersection has measure zero. This is a special case of a theorem due to Khintchine [48] in the case n D 1 and to Groshev [42] in full generality, which gives the precise condition on the function under which the set of -approximable matrices has full measure. Namely, if is non-increasing (this assumption can be removed in higher dimensions but not for n D 1, see [29]), then -almost no (resp. -almost every) Y 2 M m;n is -approximable, provided the sum 1 X
k n1 (k)m
(6)
kD1
converges (resp. diverges). (Here and hereafter stands for Lebesgue measure). This statement is usually referred to as the Khintchine–Groshev Theorem. The convergence case of this theorem follows in a straightforward manner from the Borel–Cantelli Lemma, but the divergence case is harder. It was reproved and sharpened in 1960 by Schmidt [76], who showed that if the sum (6) diverges, then for almost all Y the number of solutions to (4) with kqk N is asymptotic to the partial sum of the series (6) (up to a constant), and also gave an estimate for the error term. A special case of the convergence part of the theorem shows that Wm;n ( 1;v ) has measure zero whenever v > n/m. Y is said to be very well approximable if it belongs to W m;n ( 1;v ) for some v > n/m. That is, def
VWAm;n D [v>n/m Wm;n (
1;v )
:
More specifically, let us define the Diophantine exponent !(Y) of Y (sometimes called “the exact order” of Y) to be the supremum of v > 0 for which Y 2 Wm;n ( 1;v ). Then !(Y) is always not less than n/m, and is equal to n/m for Lebesgue-a.e. Y; in fact, VWA m;n D fY 2 M m;n : !(Y) > n/mg. The Hausdorff dimension of the null sets Wm;n ( 1;v ) was computed independently by Besicovitch [14] and Jarnik [45] in the one-dimensional case and by Dodson [26] in general: when v > n/m, one has dim Wm;n (
1;v )
D (n 1)m C
mCn : vC1
(7)
See [27] for a nice exposition of ideas involved in the proof of both the aforementioned formula and the Khintchine– Groshev Theorem.
Note that it follows from (7) that the null set VWA m;n has full Hausdorff dimension. Matrices contained in the intersection \ W m;n ( 1;v ) D fY 2 M m;n : !(Y) D 1g v
are called Liouville and form a set of Hausdorff dimension (n 1)m, that is, to the dimension of Y for which Yq 2 Z for some q 2 Zn n f0g (the latter belong to Wm;n ( ) for any positive ). Note also that the aforementioned properties behave nicely with respect to transposition; this is described by the so-called Khintchine’s Transference Principle (Chap. V in [17]). For example, Y 2 BA m;n if and only if Y T 2 BA n;m , and Y 2 VWA m;n if and only if Y T 2 VWA n;m . In particular, many problems related to approximation properties of vectors (n D 1) and linear forms (m D 1) reduce to one another. We refer the readers to [43] and [8] for very detailed and comprehensive recent accounts of various further aspects of the theory. Connection with Dynamics on the Space of Lattices General references for this section: [4,86]. Interactions between Diophantine approximation and the theory of dynamical systems has a long history. Already in Kronecker’s Theorem one can see a connection. Indeed, the statement of the theorem can be rephrased as follows: the points on the orbit of 0 under the rotation of the circle R/Z by y approach the initial point 0 arbitrarily closely. This is a special case of the Poincare Recurrence Theorem in measurable dynamics. And, likewise, all the aforementioned properties of Y 2 M m;n can be restated in terms of recurrence properties of the Z n -action on the m-dimensional torus Rm /Z m given by x 7! Yx mod Z m . In other words, fixing Y gives rise to a dynamical system in which approximation properties of Y show up. However the theme of this section is a different dynamical system, whose phase space is (essentially) the space of parameters Y, and which can be used to read the properties of Y from the behavior of the associated trajectory. It has been known for a long time (see [81] for a historical account) that Diophantine properties of real numbers can be coded by the behavior of geodesics on the quotient of the hyperbolic plane by SL2 (Z). In fact, the latter flow can be viewed as the suspension flow of the Gauss map mentioned at the end of Sect. “Introduction”. There have been many attempts to construct a higher-dimensional analogue of the Gauss map so that it captures all the fea-
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
tures of simultaneous approximation, see [47,63,65] and references therein. On the other hand, it seems to be more natural and efficient to generalize the suspension flow itself, and this is where one needs higher rank homogeneous dynamics. As was mentioned above, in the basic set-up of simultaneous Diophantine approximation one takes a system of m linear forms Y1 ; : : : ; Ym on Rn and looks at the values of jYi (q) C p i j; p i 2 Z, when q D (q1 ; : : : ; q n ) 2 Z n is far from 0. The trick is to put together Y1 (q) C p1 ; : : : ; Ym (q) C p m
and
q1 ; : : : ; q n ;
fg t LY Z k : t 2 RC g
and consider the collection of vectors
ˇ Yq C p ˇˇ m n D LY Z k ; q 2 Z p 2 Z ˇ q where k D m C n and Y def I LY D m ; Y 2 M m;n : 0 In
(8)
This collection is a unimodular lattice in R k , that is, a discrete subgroup of R k with covolume 1. Our goal is to keep track of vectors in such a lattice having small projections onto the first m components of R k and big projections onto the last n components. This is where dynamics comes into the picture. Denote by g t the one-parameter subgroup of SL k (R) given by g t D diag(et/m ; : : : ; e t/m ; et/n ; : : : ; et/n ) : „ ƒ‚ … „ ƒ‚ … mtimes
(9)
ntimes
The vectors in the lattice LY Z k are moved by the action of g t , t > 0, and a special role is played by the moment t when the “small” and “big” projections equalize. That is, one is led to consider a new dynamical system. Its phase space is the space of unimodular lattices in R k , which can be naturally identified with the homogeneous space def
˝ k D G/; where G D SL k (R) and D SL k (Z) ;
Z k nf0gg such that g i (v i ) ! 0 as i ! 1. Equivalently, for " > 0 consider a subset K " of ˝ k consisting of lattices with no nonzero vectors of norm less than "; then all the sets K " are compact, and every compact subset of ˝ k is contained in one of them. Moreover, one can choose a metric on ˝ k such that dist(; Z k ) is, up to a uniform multiplicative constant, equal to log minv2 nf0g kvk (see [25]); then the length of the smallest nonzero vector in a lattice will determine how far away is this lattice in the “cusp” of ˝ k . Using Mahler’s Criterion, it is not hard to show that Y 2 BA m;n if and only if the trajectory
(10)
and the action is given by left multiplication by elements of the subgroup (9) of G, or perhaps other subgroups H G. Study of such systems has a rich history; for example, they are known to be ergodic and mixing whenever H is unbounded [74]. What is important in this particular case is that the space ˝ k happens to be noncompact, and its structure at infinity is described via Mahler’s Compactness Criterion, see Chap. V in [4]: a sequence of lattices g i Z k goes to infinity in ˝ k () there exists a sequence fv i 2
(11)
is bounded in ˝ k . This was proved by Dani [20] in 1985, and later generalized in [57] to produce a criterion for Y to be -approximable for any non-increasing function . An important special case is a criterion for a system of linear forms to be very well approximable: Y 2 VWAm;n if and only if the trajectory (11) has linear growth, that is, there exists a positive such that dist(g t LY Z k ; Z k ) > t for an unbounded set of t > 0. This correspondence allows one to link various Diophantine and dynamical phenomena. For example, from the results of [55] on abundance of bounded orbits on homogeneous spaces one can deduce the aforementioned theorem of Schmidt [78]: the set BAm;n has full Hausdorff dimension. And a dynamical Borel–Cantelli Lemma established in [57] can be used for an alternative proof of the Khintchine–Groshev Theorem; see also [87] for an earlier geometric approach. Note that both proofs are based on the following two properties of the g t -action: mixing, which forces points to return to compact subsets and makes preimages of cusp neighborhoods quasi-independent, and hyperbolicity, which implies that the behavior of points on unstable leaves is generic. The latter is important since the orbits of the group fLY Z k : Y 2 M m;n g are precisely the unstable leaves with respect to the g t -action. We note that other types of Diophantine problems, such as conjectures of Oppenheim and Littlewood mentioned in the previous section, can be reduced to statements involving ˝ k by means of the same principle: Mahler’s Criterion is used to relate small values of some function at integer points to excursions to infinity in ˝ k of orbit of the stabilizer of this function. Other important and useful recent applications of homogeneous dynamics to metric Diophantine approximation are related to the circle of ideas roughly called “Diophantine approximation with dependent quantities” (terminology borrowed from [84]), to be surveyed in the next two sections.
305
306
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
Diophantine Approximation with Dependent Quantities: The Set-Up General references for this section: [12,84]. Here we restrict ourselves to Diophantine properties of vectors in Rn . In particular, we will look more closely at the set of very well approximable vectors, which we will simply denote by VWA, dropping the subscripts. In many cases it does not matter whether one works with row or column vectors, in view of the duality remark made at the end of Sect. “Basic Facts”. We begin with a non-example of an application of dynamics to Diophantine approximation: a celebrated and difficult theorem which currently, to the best of the author’s knowledge, has no dynamical proof. Suppose that y D (y1 ; : : : ; y n ) 2 Rn is such that each yi is algebraic and 1; y1 ; : : : ; y n are linearly independent over Q. It was established by Roth for n D 1 [75] and then generalized to arbitrary n by Schmidt [79], that y as above necessarily belongs to the complement of VWA. In other words, vectors with very special algebraic properties happen to follow the behavior of a generic vector in Rn . We would like to view the above example as a special case of a general class of problems. Namely, suppose we are given a Radon measure on Rn . Let us say that is extremal [85] if -a.e. y 2 Rn is not very well approximable. Further, define the Diophantine exponent !() of to be the -essential supremum of the function !(); in other words, ˚ def !() D sup vj W ( 1;v ) > 0 : Clearly it only depends on the measure class of . If is naturally associated with a subset M of Rn supporting (for example, if M is a smooth submanifold of Rn and is the measure class of the Riemannian volume on M, or, equivalently, the pushforward f of by a smooth map f parametrizing M), one defines the Diophantine exponent !(M) of M to be equal to that of , and says that M is extremal if f(x) is not very well approximable for -a.e. x. Then !() n for any , and !() D !(Rn ) is equal to n. The latter justifies the use of the word “extremal”: is extremal if !() is equal to n, i. e. attains the smallest possible value. The aforementioned results of Roth and Schmidt then can be interpreted as the extremality of atomic measures supported on algebraic vectors without rational dependence relations. Historically, the first measure (other than ) to be considered in the set-up described above was the pushforward of by the map f(x) D (x; x 2 ; : : : ; x n ) :
(12)
The extremality of f for f as above was conjectured in 1932 by K. Mahler [67] and proved in 1964 by Sprindžuk [82,83]. It was important for Mahler’s study of transcendental numbers: this result, roughly speaking, says that almost all transcendental numbers are “not very algebraic”. At about the same time Schmidt [77] proved the extremality of f when f : I ! R2 , I R, is C3 and satisfies ˇ 0 ˇ ˇ f (x) f 0 (x) ˇ 2 ˇ 1 ˇ ˇ ˇ ¤ 0 for -a.e. x 2 I ; ˇ f 00 (x) f 00 (x)ˇ 1
2
in other words, the curve parametrized by f has nonzero curvature at almost all points. Since then, a lot of attention has been devoted to showing that measures f are extremal for other smooth maps f. To describe a broader class of examples, recall the following definition. Let x 2 Rd and let f D ( f1 ; : : : ; f n ) be a Ck map from a neighborhood of x to Rn . Say that f is nondegenerate at x if Rn is spanned by partial derivatives of f at x up to some order. Say that f is nondegenerate if it is nondegenerate at -a.e. x. It was conjectured by Sprindžuk [84] in 1980 that f for real analytic nondegenerate f are extremal. Many special cases were established since then (see [12] for a detailed exposition of the theory and many related results), but the general case stood open until the mid-1990s [56], when Sprindžuk’s conjecture was proved using the dynamical approach (later Beresnevich [6] succeeded in establishing and extending this result without use of dynamics). The proof in [56] uses the correspondence outlined in the previous section plus a measure estimate for flows on the space of lattices which is described below. In the subsequent work the method of [56] was adapted to a much broader class of measures. To define it we need to introduce some more notation and definitions. If x 2 Rd and r > 0, denote by B(x; r) the open ball of radius r centered at x. If B D B(x; r) and c > 0, cB will denote the ball B(x; cr). For B Rd and a real-valued function f on B, let def
k f kB D sup j f (x)j : x2B
If is a measure on Rd such that (B) > 0, define k f k;B def Dk f kB \ supp ; this is the same as the L1 ()-norm of f j B if f is continuous and B is open. If D > 0 and U Rd is an open subset, let us say that is D-Federer on U if for any ball B U centered at supp one has (3B)/(B) < D whenever 3B U. This condition is often called “doubling” in the literature. See [54,72] for examples and references. is called Federer if for -a.e. x 2 Rd there exist a
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
neighborhood U of x and D > 0 such that is D-Federer on U. Given C; ˛ > 0, open U Rd and a measure on U, a function f : U ! R is called (C; ˛)-good on U with respect to if for any ball B U centered in supp and any " > 0 one has ˛ " fx 2 B : j f (x)j < "g C (B) : (13) k f k;B This condition was formally introduced in [56] for being Lebesgue measure, and in [54] for arbitrary . A basic example is given by polynomials, and the upshot of the above definition is the formalization of a property needed for the proof of several basic facts [19,21,68] about polynomial maps into the space of lattices. In [54] a strengthening of this property was considered: f was called absolutely (C; ˛)-good on U with respect to if for B and " as above one has ˛ " fx 2 B : j f (x)j < "g C (B) : (14) k f kB There is no difference between (13) and (14) when has full support, but it turns out to be useful for describing measures supported on proper (e. g. fractal) subsets of Rd . Now suppose that we are given a measure on Rd , an open U Rd with (U) > 0 and a map f D ( f1 ; : : : ; f n ) : Rd ! Rn . Following [62], say that a pair (f; ) is (absolutely) good on U if any linear combination of 1; f1 ; : : : ; f n is (absolutely) (C; ˛)-good on U with respect to . If for -a.e. X there exists a neighborhood U of X and C; ˛ > 0 such that is (absolutely) (C; ˛)-good on U, we will say that the pair (f; ) is (absolutely) good. Another relevant notion is the nonplanarity of (f; ). Namely, (f; ) is said to be nonplanar if whenever B is a ball with (B) > 0, the restrictions of 1; f1 ; : : : ; f n to B \ supp are linearly independent over R; in other words, f(B \ supp ) is not contained in any proper affine subspace of Rn . Note that absolutely good implies both good and nonplanar, but the converse is in general not true. Many examples of (absolutely) good and nonplanar pairs (f; ) can be found in the literature. Already the case n D d and f D Id is very interesting. A measure on Rn is said to be friendly (resp., absolutely friendly) if and only if it is Federer and the pair (Id; ) is good and nonplanar (resp., absolutely good). See [54,88,89] for many examples. An important class of measures is given by limit measures of irreducible system of self-similar or self-conformal contractions satisfying the Open Set Condition [44]; those are shown to be absolutely friendly in [54]. The prime example is the middle-third Cantor set on the real line. The term
“friendly” was cooked up as a loose abbreviation for “Federer, nonplanar and decaying”, and later proved to be particularly friendly in dealing with problems arising in metric number theory, see e. g. [36]. Also let us say that a pair (f; ) is nondegenerate if f is nondegenerate at -a.e. X. When is Lebesgue measure on Rd , it is proved in Proposition 3.4 in [56], that a nondegenerate (f; ) is good and nonplanar. The same conclusion is derived in Proposition 7.3 in [54], assuming that is absolutely friendly. Thus volume measures on smooth nondegenerate manifolds are friendly, but not absolutely friendly. It turns out that all the aforementioned examples of measures can be proved to be extremal by a generalization of the argument from [56]. Specifically, let be a Federer measure on Rd , U an open subset of Rd , and f : U ! Rn a continuous map such that the pair (f; ) is good and nonplanar; then f is extremal. This can be derived from the Borel–Cantelli Lemma, the correspondence described in the previous section, and the following measure estimate: if , U and f are as above, then for -a.e. x0 2 U there ex˜ ˛ > 0 such that for ists a ball B U centered at x0 and C; any t 2 RC and any " > 0, ˚ ˜ ˛: (15) x 2 B : g t Lf(x) Z nC1 … K" < C" Here g t is as in (9) with m D 1 (assuming that the row vector viewpoint is adopted). This is a quantitative way of saying that for fixed t, the “flow” x 7! g t Lf(x) Z nC1 , B ! ˝nC1 , cannot diverge, and in fact must spend a big (uniformly in t) proportion of time inside compact sets K " . The inequality (15) is derived from a general “quantitative non-divergence” estimate, which can be thought of a substantial generalization of theorems of Margulis and Dani [19,21,68] on non-divergence of unipotent flows on homogeneous spaces. One of its most general versions [54] deals with a measure on Rd , a continuous map h: e B ! G, where e B is a ball in Rd centered at supp and G is as in (10). To describe the assumptions on h, one needs to employ the combinatorial structure of lattices in R k , and it will be convenient to use the following notation: if V is a nonzero rational subspace of R k and g 2 G, define `V (g) to be the covolume of g(V \ Z k ) in gV . Then, given positive constants C; D; ˛, there exists C1 D C1 (d; k; C; ˛; D) > 0 with the following property. Suppose is D-Federer on e B, 0 < 1, and h is such that for each rational V R k (i) `V ı h is (C; ˛)-good on B˜ with respect to , and ˜ Then (ii) k`V ı hk;B , where B D 3(k1) B. (iii) for any positive " , one has ˚ x 2 B : h(x)Z k … K" C1 (" )˛ (B) : (16)
307
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Ergodic Theory on Homogeneous Spaces and Metric Number Theory
Taking h(x) D g t Lf(x) and unwinding the definitions of good and nonplanar pairs, one can show that (i) and (ii) can be verified for some balls B centered at -almost every point, and derive (15) from (16). Further Results The approach to metric Diophantine approximation using quantitative non-divergence, that is, the implication (i) + (ii) ) (iii), is not omnipotent. In particular, it is difficult to use when more precise results are needed, such as for example computing/estimating the Hausdorff dimension of the set of 1;v -approximable vectors on a manifold. See [9,10] for such results. On the other hand, the dynamical approach can often treat much more general objects that its classical counterpart, and also can be perturbed in a lot of directions, producing many generalizations and modifications of the main theorems from the preceding section. One of the most important of them is the so-called multiplicative version of the set-up of Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”. def Q def Namely, define functions ˘ (x) D i jx i j and ˘C (x) D Q : N ! RC , one i max(jx i j; 1) Then, given a function says that Y 2 M m;n is multiplicatively -approximable ( )) if there are infinitely many (notation: Y 2 Wm;n q 2 Z n such that ˘ (Yq C p)1/m ˘C (q)1/n (17) for some p 2 Z m . Since ˘ (x) ˘C (x) kxk k for x 2 R k , any -approximable Y is multiplicatively -approximable; but the converse is in general not true, see e. g. [37]. However if one, as before, considers the family f 1;v g, the critical parameter for which the drop from full measure to measure zero occurs is again n/m. That is, if one defines the multiplicative Diophantine exponent def ( ! (Y) of Y by ! (Y) D supfv : Y 2 Wm;n 1;v )g, then clearly ! (Y) !(Y) for all Y, and yet ! (Y) D n/m for -a.e. Y 2 M m;n . Now specialize to Rn (by the same duality principle as before, it does not matter whether to think in terms of row or column vectors, but we will adopt the row vector set-up), and define the multiplicative˚ exponent ! () of def n a measure on R by ! () D sup vj W ( 1;v ) > 0 ; then ! () D n. Following Sprindžuk [85], say that is strongly extremal if ! () D n. It turns out that all the results mentioned in the previous section have their multiplicative analogues; that is, the measures described there happen to be strongly extremal. This was conjectured by A. Baker [1] for the curve (12), and then by Sprindžuk in 1980 [85] for analytic nondegenerate manifolds. (We re-
mark that only very few results in this set-up can be obtained by the standard methods, see e. g. [10]). The proof of this stronger statement is based on using the multi-parameter action of gt D diag(e t1 CCt n ; et1 ; : : : ; et n ) ; where t D (t1 ; : : : ; t n ) instead of g t considered in the previous section. One can show that the choice h(x) D gt Lf(x) allows one to verify n , and the proof is fin(i) and (ii) uniformly in t 2 RC ished by applying a multi-parameter version of the correspondence described in Sect. “Connection with Dynamics on the Space of Lattices”. Namely, one can show that k : y 2 VWA 1;n if and only if the trajectory fg t Ly Z n t 2 RC g grows linearly, that is, for some > 0 one has dist(gt LY Z nC1 ; Z nC1 ) > ktk for an unbounded n . A similar correspondence was recently set of t 2 RC used in [30] to prove that the set of exceptions to Littlewood’s Conjecture, which, using the terminology introduced above, can be called badly multiplicatively approximable vectors: def n [ BA W n;1 ( c;1/n ) n;1 D R n
D y:
c>0
inf
q2Znf0g; p2Z n
jqj ˘ (qy p) > 0
; (18)
has Hausdorff dimension zero. This was done using a measure rigidity result for the action of the group of diagonal matrices on the space of lattices. See [18] for an implicit description of this correspondence and [32,66,70] for more detail. The dynamical approach also turned out to be fruitful in studying Diophantine properties of pairs (f; ) for which the nonplanarity condition fails. Note that obvious examples of non-extremal measures are provided by proper affine subspaces of Rn whose coefficients are rational or are well enough approximable by rational numbers. On the other hand, it is clear from a Fubini argument that almost all translates of any given subspace are extremal. In [51] the method of [56] was pushed further to produce criteria for the extremality, as well as the strong extremality, of arbitrary affine subspaces L of Rn . Further, it was shown that if L is extremal (resp. strongly extremal), then so is any smooth submanifold of L which is nondegenerate in L at a.e. point. (The latter property is a straightforward generalization of the definition of nondegeneracy in Rn : a map f is nondegenerate in L at x if the linear part of L is spanned by partial derivatives of f at x). In other words, extremality and strong extremality pass from affine subspaces to their nondegenerate submanifolds.
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
A more precise analysis makes it possible to study Diophantine exponents of measures with supports contained in arbitrary proper affine subspaces of Rn . Namely, in [53] it is shown how to compute !(L) for any L, and furthermore proved that if is a Federer measure on Rd , U an open subset of Rd , and f : U ! Rn a continuous map such that the pair (f; ) is good and nonplanar in L, then !(f ) D !(L). Here we say, generalizing the definition from Sect.“Diophantine Approximation with Dependent Quantities: The Set-Up”, that (f; ) is nonplanar in L if for any ball B with (B) > 0, the f-image of B \ supp is not contained in any proper affine subspace of L. (It is easy to see that for a smooth map f : U ! L, (f; ) is good and nonplanar in L whenever f is nondegenerate in L at a.e. point). It is worthwhile to point out that these new applications require a strengthening of the measure estimate described at the end of Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”: it was shown in [53] that (i) and (ii) would still imply (iii) if in (ii) is replaced by dim V . Another application concerns badly approximable vectors. Using the dynamical description of the set BA
Rn due to Dani [20], it turns out to be possible to find badly approximable vectors inside supports of certain measures on Rn . Namely, if a subset K of Rn supports an absolutely friendly measure, then BA \ K has Hausdorff dimension not less than the Hausdorff dimension of this measure. In particular, it proves that limit measures of irreducible system of self-similar/self-conformal contractions satisfying the Open Set Condition, such as e. g. the middle-third Cantor set on the real line, contain subsets of full Hausdorff dimension consisting of badly approximable vectors. This was established in [60] and later independently in [64] using a different approach. See also [36] for a stronger result. The proof in [60] uses quantitative nondivergence estimates and an iterative procedure, which requires the measure in question to be absolutely friendly and not just friendly. A similar question for even the simplest not-absolutely friendly measures is completely open. For example, it is not known whether there exist uncountably many badly approximable pairs of the form (x; x 2 ). An analogous problem for atomic measures supported on algebraic numbers, that is, a “badly approximable” version of Roth’s Theorem, is currently beyond reach as well — there are no known badly approximable (or, for that matter, well approximable) algebraic numbers of degree bigger than two. It has been recently understood that the quantitative nondivergence method can be applied to the question of improvement to Dirichlet’s Theorem (see the beginning of Sect. “Basic Facts”). Given a positive " < 1, let us say
that Dirichlet’s Theorem can be "-improved for Y 2 M m;n , writing Y 2 DIm;n ("), if for every sufficiently large t the system kYq pk < "et/m
and kqk < "e t/n
(19)
(that is, (5) with the right hand side terms multiplied by ") has a nontrivial integer solution (p; q). of It is a theorem Davenport and Schmidt [24] that DIm;n (") D 0 for any " < 1; in other words, Dirichlet’s Theorem cannot be improved for Lebesgue-generic systems of linear forms. By a modification of the correspondence between dynamics and approximation, (19) is easily seen to be equivalent to g t LY Z k 2 K" , and since the complement to K " has nonempty interior for any " < 1, the result of Davenport and Schmidt follows from the ergodicity of the g t -action on ˝ k . Similar questions with replaced by f for some specific smooth maps f were considered in [2,3,15,23]. For example, [15], Theorem 7, provides an explicitly computable constant "0 D "0 (n) such that for f as in (12), f DI1;n (") D 0 for " < "0 : This had been previously done in [23] for n D 2 and in [2] for n D 3. In [62] this is extended to a much broader class of measures using estimates described in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”. In particular, almost every point of any nondegenerate smooth manifold is proved not to lie in DI(") for small enough " depending only on the manifold. Earlier this was done in [61] for the set of singular vectors, defined as the intersection of DI(") over all positive "; those correspond to divergent g t -trajectories. As before, the advantage of the method is allowing a multiplicative generalization of the Dirichlet-improvement set-up; see [62] for more detail. It is also worthwhile to mention that a generalization of the measure estimate discussed in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” was used in [13] to estimate the measure of the set of points x in a ball B Rd for which the system 8 ˆ <jf(x) q C pj < " jf0 (x) qj < ı ˆ : jq i j < Q i ; i D 1; : : : ; n; where f is a smooth nondegenerate map B ! Rn , has a nonzero integer solution. For that, Lf(x) as in (15) has to be replaced by the matrix 1 0 1 0 f(x) @0 1 f0 (x)A ; 0 0 In
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and therefore (i) and (ii) turn into more complicated conditions, which nevertheless can be checked when f is smooth and nondegenerate and is Lebesgue measure. This has resulted in proving the convergence case of Khintchine–Groshev Theorem for nondegenerate manifolds [13], in both standard and multiplicative versions. The aforementioned estimate was also used in [7] for the proof of the divergence case, and in [38,40] for establishing the convergence case of Khintchine–Groshev theorem for affine hyperplanes and their nondegenerate submanifolds. This generalized results obtained by standard methods for the curve (12) by Bernik and Beresnevich [6,11]. Finally, let us note that in many of the problems mentioned above, the ground field R can be replaced by Q p , and in fact several fields can be taken simultaneously, thus giving rise to the S-arithmetic setting where S D fp1 ; : : : ; p s g is a finite set of normalized valuations of Q, which may or may not include the infinite valuation (cf. [83,90]). The space of lattices in RnC1 is replaced there by the space of lattices in QnC1 , where QS is the prodS uct of the fields R and Q p 1 ; : : : ; Q p s . This is the subject of the paper [59], where S-arithmetic analogues of many results reviewed in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” have been established. Similarly one can consider versions of the above theorems over local fields of positive characteristic [39]. See also [52] where Sprindžuk’s solution [83] of the complex case of Mahler’s Conjecture has been generalized (the latter involves studying small values of linear forms with coefficients in C at real integer points), and [31] which establishes a p-adic analogue of the result of [30] on the set of exceptions to Littlewood’s Conjecture. Future Directions Interactions between ergodic theory and number theory have been rapidly expanding during the last two decades, and the author has no doubts that new applications of dynamics to Diophantine approximation will emerge in the near future. Specializing to the topics discussed in the present paper, it is fair to say that the list of “further results” contained in the previous section is by no means complete, and many even further results are currently in preparation. This includes: extending proofs of extremality and strong extremality of certain measures to the set-up of systems of linear forms (namely, with min(m; n) > 1; this was mentioned as work in progress in [56]); proving Khintchine-type theorems (both convergence and divergence parts) for p-adic and S-arithmetic nondegenerate manifolds, see [73] for results in this direction; extending [38,40] to establish Khintchine-type theo-
rems for submanifolds of arbitrary affine subspaces. The work of Dru¸tu [28], who used ergodic theory on homogeneous spaces to compute the Hausdorff dimension of the intersection of Wn;1 ( 1;v ), v > 1, with some rational quadratic hypersurfaces in Rn deserves a special mention; it is plausible that using this method one can treat more general situations. Several other interesting open directions are listed in [41], Section 9, in the final sections of papers [7,54], in the book [15], and in surveys by Frantzikinakis–McCutcheon, Nitica and Ward in this volume. Acknowledgment The work on this paper was supported in part by NSF Grant DMS-0239463. Bibliography 1. Baker A (1975) Transcendental number theory. Cambridge University Press, London 2. Baker RC (1976) Metric diophantine approximation on manifolds. J Lond Math Soc 14:43–48 3. Baker RC (1978) Dirichlet’s theorem on diophantine approximation. Math Proc Cambridge Phil Soc 83:37–59 4. Bekka M, Mayer M (2000) Ergodic theory and topological dynamics of group actions on homogeneous spaces. Cambridge University Press, Cambridge 5. Beresnevich V (1999) On approximation of real numbers by real algebraic numbers. Acta Arith 90:97–112 6. Beresnevich V (2002) A Groshev type theorem for convergence on manifolds. Acta Mathematica Hungarica 94:99–130 7. Beresnevich V, Bernik VI, Kleinbock D, Margulis GA (2002) Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds. Moscow Math J 2:203–225 8. Beresnevich V, Dickinson H, Velani S (2006) Measure theoretic laws for lim sup sets. Mem Amer Math Soc:179:1–91 9. Beresnevich V, Dickinson H, Velani S (2007) Diophantine approximation on planar curves and the distribution of rational points. Ann Math 166:367–426 10. Beresnevich V, Velani S (2007) A note on simultaneous Diophantine approximation on planar curves. Ann Math 337: 769–796 11. Bernik VI (1984) A proof of Baker’s conjecture in the metric theory of transcendental numbers. Dokl Akad Nauk SSSR 277:1036–1039 12. Bernik VI, Dodson MM (1999) Metric Diophantine approximation on manifolds. Cambridge University Press, Cambridge 13. Bernik VI, Kleinbock D, Margulis GA (2001) Khintchine-type theorems on manifolds: convergence case for standard and multiplicative versions. Int Math Res Notices 2001:453–486 14. Besicovitch AS (1929) On linear sets of points of fractional dimensions. Ann Math 101:161–193 15. Bugeaud Y (2002) Approximation by algebraic integers and Hausdorff dimension. J London Math Soc 65:547–559 16. Bugeaud Y (2004) Approximation by algebraic numbers. Cambridge University Press, Cambridge 17. Cassels JWS (1957) An introduction to Diophantine approximation. Cambridge University Press, New York
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18. Cassels JWS, Swinnerton-Dyer H (1955) On the product of three homogeneous linear forms and the indefinite ternary quadratic forms. Philos Trans Roy Soc London Ser A 248:73– 96 19. Dani SG (1979) On invariant measures, minimal sets and a lemma of Margulis. Invent Math 51:239–260 20. Dani SG (1985) Divergent trajectories of flows on homogeneous spaces and diophantine approximation. J Reine Angew Math 359:55–89 21. Dani SG (1986) On orbits of unipotent flows on homogeneous spaces. II. Ergodic Theory Dynam Systems 6:167–182 22. Dani SG, Margulis GA (1993) Limit distributions of orbits of unipotent flows and values of quadratic forms. In: IM Gelfand Seminar. American Mathematical Society, Providence, pp 91–137 23. Davenport H, Schmidt WM (1970) Dirichlet’s theorem on diophantine approximation. In: Symposia Mathematica. INDAM, Rome, pp 113–132 24. Davenport H, Schmidt WM (1969/1970) Dirichlet’s theorem on diophantine approximation. II. Acta Arith 16:413–424 25. Ding J (1994) A proof of a conjecture of C. L. Siegel. J Number Theory 46:1–11 26. Dodson MM (1992) Hausdorff dimension, lower order and Khintchine’s theorem in metric Diophantine approximation. J Reine Angew Math 432:69–76 27. Dodson MM (1993) Geometric and probabilistic ideas in the metric theory of Diophantine approximations. Uspekhi Mat Nauk 48:77–106 28. Dru¸tu C (2005) Diophantine approximation on rational quadrics. Ann Math 333:405–469 29. Duffin RJ, Schaeffer AC (1941) Khintchine’s problem in metric Diophantine approximation. Duke Math J 8:243–255 30. Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann Math 164:513–560 31. Einsiedler M, Kleinbock D (2007) Measure rigidity and p-adic Littlewood-type problems. Compositio Math 143:689–702 32. Einsiedler M, Lindenstrauss E (2006) Diagonalizable flows on locally homogeneous spaces and number theory. In: Proceedings of the International Congress of Mathematicians. Eur Math Soc, Zürich, pp 1731–1759 33. Eskin A (1998) Counting problems and semisimple groups. In: Proceedings of the International Congress of Mathematicians. Doc Math, Berlin, pp 539–552 34. Eskin A, Margulis GA, Mozes S (1998) Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann Math 147:93–141 35. Eskin A, Margulis GA, Mozes S (2005) Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-tori. Ann Math 161:679–725 36. Fishman L (2006) Schmidt’s games on certain fractals. Israel S Math (to appear) 37. Gallagher P (1962) Metric simultaneous diophantine approximation. J London Math Soc 37:387–390 38. Ghosh A (2005) A Khintchine type theorem for hyperplanes. J London Math Soc 72:293–304 39. Ghosh A (2007) Metric Diophantine approximation over a local field of positive characteristic. J Number Theor 124:454–469 40. Ghosh A (2006) Dynamics on homogeneous spaces and Diophantine approximation on manifolds. Ph D Thesis, Brandeis University, Walthum
41. Gorodnik A (2007) Open problems in dynamics and related fields. J Mod Dyn 1:1–35 42. Groshev AV (1938) Une théorème sur les systèmes des formes linéaires. Dokl Akad Nauk SSSR 9:151–152 43. Harman G (1998) Metric number theory. Clarendon Press, Oxford University Press, New York 44. Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30:713–747 45. Jarnik V (1928-9) Zur metrischen Theorie der diophantischen Approximationen. Prace Mat-Fiz 36:91–106 46. Jarnik V (1929) Diophantischen Approximationen und Hausdorffsches Mass. Mat Sb 36:371–382 47. Khanin K, Lopes-Dias L, Marklof J (2007) Multidimensional continued fractions, dynamical renormalization and KAM theory. Comm Math Phys 270:197–231 48. Khintchine A (1924) Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math Ann 92:115–125 49. Khintchine A (1963) Continued fractions. P Noordhoff Ltd, Groningen 50. Kleinbock D (2001) Some applications of homogeneous dynamics to number theory. In: Smooth ergodic theory and its applications. American Mathematical Society, Providence, pp 639–660 51. Kleinbock D (2003) Extremal subspaces and their submanifolds. Geom Funct Anal 13:437–466 52. Kleinbock D (2004) Baker–Sprindžuk conjectures for complex analytic manifolds. In: Algebraic groups and Arithmetic. Tata Inst Fund Res, Mumbai, pp 539–553 53. Kleinbock D (2008) An extension of quantitative nondivergence and applications to Diophantine exponents. Trans AMS, to appear 54. Kleinbock D, Lindenstrauss E, Weiss B (2004) On fractal measures and diophantine approximation. Selecta Math 10: 479–523 55. Kleinbock D, Margulis GA (1996) Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In: Sina˘ı’s Moscow Seminar on Dynamical Systems. American Mathematical Society, Providence, pp 141–172 56. Kleinbock D, Margulis GA (1998) Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann Math 148:339–360 57. Kleinbock D, Margulis GA (1999) Logarithm laws for flows on homogeneous spaces. Invent Math 138:451–494 58. Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook on Dynamical Systems, vol 1A. Elsevier Science, North Holland, pp 813–930 59. Kleinbock D, Tomanov G (2007) Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comm Math Helv 82:519–581 60. Kleinbock D, Weiss B (2005) Badly approximable vectors on fractals. Israel J Math 149:137–170 61. Kleinbock D, Weiss B (2005) Friendly measures, homogeneous flows and singular vectors. In: Algebraic and Topological Dynamics. American Mathematical Society, Providence, pp 281–292 62. Kleinbock D, Weiss B (2008) Dirichlet’s theorem on diophantine approximation and homogeneous flows. J Mod Dyn 2:43– 62
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Ergodic Theory, Introduction to
Ergodic Theory, Introduction to BRYNA KRA Northwestern University, Evanston, USA Ergodic theory lies at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Problems, techniques, and results are related to many other areas of mathematics, and ergodic theory has had applications both within mathematics and to numerous other branches of science. Ergodic theory has particularly strong overlap with other branches of dynamical systems; to clarify what distinguishes it from other areas of dynamics, we start with a quick overview of dynamical systems. Dynamical systems is the study of systems that evolve with time. The evolution of a dynamical system is given by some fixed rule that determines the states of the system a short time into the future, given only the present states. Reflecting the origins of the subject in celestial mechanics, the set of states through which the system evolves with time is called an orbit. Many important concepts in dynamical systems are related to understanding the orbits in the system: Do the orbits fill out the entire space? Do orbits collapse? Do orbits return to themselves? What are statistical properties of the orbits? Are orbits stable under perturbation? For simple dynamical systems, knowing the individual orbits is often sufficient to answer such questions. However, in most dynamical systems it is impossible to write down explicit formulae for orbits, and even when one can, many systems are too complicated to be understood just in terms of individual orbits. The orbits may only be known approximately, some orbits may appear to be random while others exhibit regular behavior, and varying the parameters defining the system may give rise to qualitatively different behaviors. The various branches of dynamical systems have been developed for understanding long term properties of the orbits. To make the notion of a dynamical system more precise, let X denote the collection of all states of the system. The evolution of these states is given by some fixed rule T : X ! X, dictating where each state x 2 X is mapped. An application of the transformation T : X ! X corresponds to the passage of a unit of time and for a positive integer n, the map T n D T ı T ı : : : ı T denotes the composition of T with itself taken n times. Given a state x 2 X, the orbit of the point x under the transformation T is the collection of iterates x; Tx; T 2 x; : : : of the state x. Thus the single transformation T generates a semigroup of transformations acting on X, by considering the
powers T n . More generally, one can consider a family of transformations fTt : t 2 Rg with each Tt : X ! X. Assuming that T0 (x) D x and that TtCs (x) D Tt (Ts (x)) for all states x 2 X and all real t and s, this models the evolution of continuous time in a system. Autonomous differential equations are examples of such continuous time systems. In almost all cases of interest, the space X has some underlying structure which is preserved by the transformation T. Different underlying structures X and different properties of the transformation T give rise to different branches of dynamical systems. When X is a smooth manifold and T : X ! X is a differentiable mapping, one is in the framework of differentiable dynamics. When X is a topological space and T : X ! X is a continuous map, one is in the framework of topological dynamics. When X is a measure space and T : X ! X is a measure preserving map, one is in the framework of ergodic theory. These categories are not mutually exclusive, and the relations among them are deep and interesting. Some of these relations are explored in the articles Topological Dynamics, Symbolic Dynamics, and Smooth Ergodic Theory. To further explain the role of ergodic theory, a few definitions are needed. The state space X is assumed to be a measure space, endowed with a -algebra B of measurable sets and a measure . The measure assigns each set B 2 B a non-negative number (its measure), and usually one assumes that (X) D 1 (thus is a probability measure). The transformation T : X ! X is assumed to be a measurable and measure preserving map: For all B 2 B, (T 1 B) D (B). The quadruple (X; B; ; T) is called a measure preserving system. For the precise definitions and background on measure preserving transformations, see the article Measure Preserving Systems. An extensive discussion of examples of measure preserving systems and basic constructions used in ergodic theory is given in Ergodic Theory: Basic Examples and Constructions. The origins of ergodic theory are in the nineteenth century work of Boltzmann on the foundations of statistical mechanics. Boltzmann hypothesized that for large systems of interacting particles in equilibrium, the “time average” is equal to the “space average”. This question can be reformulated in the context of modern terminology. Assume that (X; B; ; T) is a measure preserving system and that f : X ! R is some measurement taken on the system. Thus if x 2 X is a state, evaluating the sequence f (x) ; f (Tx) ; f (T 2 x); : : : can be viewed as successive values of this measurement. Boltzmann’s question can be phrased as: Under what conditions is the time mean equal to the space mean? In short,
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when does lim
N!1
N1 1 X f (T n x) D N nD0
Z f d ? X
Boltzmann hypothesized that if orbits went “everywhere” in the space, then such a conclusion would hold. The study of the equality of space and time averages has been a major direction of research in ergodic theory. The long term behavior of the average lim
N!1
N1 1 X f (T n x) ; N nD0
and especially the existence of this limit, is a basic question. Roughly speaking, the ergodic theorem states that starting at almost any initial point, the distribution of its iterates obeys some asymptotic law. This, and more general convergence questions, are addressed in the article Ergodic Theorems. Perhaps the earliest result in ergodic theory is the Poincaré Recurrence Theorem: In a finite measure space, some iterate of any set with positive measure intersects the set itself in a set of positive measure. More generally, the qualitative behavior of orbits is used to understand conditions under which the time average is equal to the space average of the system (see the article Ergodic Theory: Recurrence). If the time average is equal almost everywhere to the space average, then the system is said to be ergodic. Ergodicity is a key notion, giving a simple expression for the time average of an arbitrary function. Moreover, using the Ergodic Decomposition Theorem, the study of arbitrary measure preserving systems can be reduced to ergodic ones. Ergodicity and related properties of a system are discussed in the article Ergodicity and Mixing Properties. Another central problem in ergodic theory is the classification of measure preserving systems. There are various notions of equivalence, and a classical approach to checking if systems are equivalent is finding invariants that are preserved under the equivalence. This subject, including an introduction to Ornstein Theory, is covered in the article Isomorphism Theory in Ergodic Theory. A map T : X ! X determines an associated unitary operator U D U T defined on L2 (X) by U T f (x) D f (Tx) : There are also numerical invariants that can be assigned to
a system, for example entropy, and this is discussed in the article Entropy in Ergodic Theory. When two systems are not equivalent, one would like to understand what properties they do have in common. An essential tool in such a classification is the notion of joinings (see the article Joinings in Ergodic Theory). Roughly speaking, a joining is a way of embedding two systems in the same space. When this can be done in a nontrivial manner, one obtains information on properties shared by the systems. Some systems have predictable behavior and can be classified according to the behavior of individual points and their iterates. Others have behavior that is too complex or unpredictable to be understood on the level of orbits. Ergodic theory provides a statistical understanding of such systems and this is discussed in the article Chaos and Ergodic Theory. A prominent role in chaotic dynamical systems is played by one dimensional Gibbs measure and by equilibrium states and (see the article Pressure and Equilibrium States in Ergodic Theory). Rigidity theory addresses the opposite case, studying what kinds of properties in a system are obstructions to general chaotic behavior. The role of ergodic theory in this area is discussed in the article Ergodic Theory: Rigidity. Another important class of systems arises when one relaxes the condition that the transformation T preserves the measure of sets on X, only requiring that the transformation preserve the negligible sets. Such systems are discussed in Joinings in Ergodic Theory. Ergodic theory has seen a burst of recent activity, and most of this activity comes from interaction with other fields. Historically, ergodic theory has interacted with numerous fields, including other areas of dynamics, probability, statistical mechanics, and harmonic analysis. More recently, ergodic theory and its techniques have been imported into number theory and combinatorics, proving new results that have yet to be proved by other methods, and in turn, combinatorial problems have given rise to new areas of research within ergodic theory itself. Problems related to Diophantine approximation are discussed in Ergodic Theory on Homogeneous Spaces and Metric Number Theory and Ergodic Theory: Rigidity and ones related to combinatorial problems are addressed in Ergodic Theory: Interactions with Combinatorics and Number Theory and in Ergodic Theory: Recurrence. Interaction with problems that are geometric in nature, in particular dimension theory, is discussed in Ergodic Theory: Fractal Geometry.
Ergodicity and Mixing Properties
Ergodicity and Mixing Properties ANTHONY QUAS Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Article Outline Glossary Definition of the Subject Introduction Basics and Examples Ergodicity Ergodic Decomposition Mixing Hyperbolicity and Decay of Correlations Future Directions Bibliography Glossary Bernoulli shift Mathematical abstraction of the scenario in statistics or probability in which one performs repeated independent identical experiments. Markov chain A probability model describing a sequence of observations made at regularly spaced time intervals such that at each time, the probability distribution of the subsequent observation depends only on the current observation and not on prior observations. Measure-preserving transformation A map from a measure space to itself such that for each measurable subset of the space, it has the same measure as its inverse image under the map. Measure-theoretic entropy A non-negative (possibly infinite) real number describing the complexity of a measure-preserving transformation. Product transformation Given a pair of measure-preserving transformations: T of X and S of Y, the product transformation is the map of X Y given by (T S)(x; y) D (T(x); S(y)). Definition of the Subject Many physical phenomena in equilibrium can be modeled as measure-preserving transformations. Ergodic theory is the abstract study of these transformations, dealing in particular with their long term average behavior. One of the basic steps in analyzing a measure-preserving transformation is to break it down into its simplest possible components. These simplest components are its ergodic components, and on each of these components, the system enjoys the ergodic property: the long-term time
average of any measurement as the system evolves is equal to the average over the component. Ergodic decomposition gives a precise description of the manner in which a system can be split into ergodic components. A related (stronger) property of a measure-preserving transformation is mixing. Here one is investigating the correlation between the state of the system at different times. The system is mixing if the states are asymptotically independent: as the times between the measurements increase to infinity, the observed values of the measurements at those times become independent. Introduction The term ergodic was introduced by Boltzmann [8,9] in his work on statistical mechanics, where he was studying Hamiltonian systems with large numbers of particles. The system is described at any time by a point of phase space, a subset of R6N where N is the number of particles. The configuration describes the 3-dimensional position and velocity of each of the N particles. It has long been known that the Hamiltonian (i. e. the overall energy of the system) is invariant over time in these systems. Thus, given a starting configuration, all future configurations as the system evolves lie on the same energy surface as the initial one. Boltzmann’s ergodic hypothesis was that the trajectory of the configuration in phase space would fill out the entire energy surface. The term ergodic is thus an amalgamation of the Greek words for work and path. This hypothesis then allowed Boltzmann to conclude that the long-term average of a quantity as the system evolves would be equal to its average value over the phase space. Subsequently, it was realized that this hypothesis is rarely satisfied. The ergodic hypothesis was replaced in 1911 by the quasi-ergodic hypothesis of the Ehrenfests [17] which stated instead that each trajectory is dense in the energy surface, rather than filling out the entire energy surface. The modern notion of ergodicity (to be defined below) is due to Birkhoff and Smith [7]. Koopman [44] suggested studying a measure-preserving transformation by means of the associated isometry on Hilbert space, U T : L2 (X) ! L2 (X) defined by U T ( f ) D f ı T. This point of view was used by von Neumann [91] in his proof of the mean ergodic theorem. This was followed closely by Birkhoff [6] proving the pointwise ergodic theorem. An ergodic measure-preserving transformation enjoys the property that Boltzmann first intended to deduce from his hypothesis: that long-term averages of an observable quantity coincide with the integral of that quantity over the phase space.
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These theorems allow one to deduce a form of independence on the average: given two sets of configurations A and B, one can consider the volume of the phase space consisting of points that are in A at time 0 and in B at time t. In an ergodic measure-preserving transformation, if one computes the average of the volumes of these regions over time, the ergodic theorems mentioned above allow one to deduce that the limit is simply the product of the volume of A and the volume of B. This is the weakest mixing-type property. In this article, we will outline a rather full range of mixing properties with ergodicity at the weakest end and the Bernoulli property at the strongest end. We will set out in some detail the various mixing properties, basing our study on a number of concrete examples sitting at various points of this hierarchy. Many of the mixing properties may be characterized in terms of the Koopman operators operators mentioned above (i. e. they are spectral properties), but we will see that the strongest mixing properties are not spectral in nature. We shall also see that there are connections between the range of mixing properties that we discuss and measure-theoretic entropy. In measure-preserving transformations that arise in practice, there is a correlation between strong mixing properties and positive entropy, although many of these properties are logically independent. One important issue for which many questions remain open is that of higher-order mixing. Here, the issue is if instead of asking that the observations at two times separated by a large time T be approximately independent, one asks whether if one makes observations at more times, each pair suitably separated, the results can be expected to be approximately independent. This issue has an analogue in probability theory, where it is well-known that it is possible to have a collection of random variables that are pairwise independent, but not mutually independent. Basics and Examples In this article, except where otherwise stated, the measurepreserving transformations that we consider will be defined on probability spaces. More specifically, given a measurable space (X; B) and a probability measure defined on B, a measure-preserving transformation of (X; B; ) is a B-measurable map T : X ! X such that (T 1 B) D (B) for all B 2 B. While this definition makes sense for arbitrary measures, not simply probability measures, most of the results and definitions below only make sense in the probability measure case. Sometimes it will be helpful to make the assumption that the underlying probability space is
a Lebesgue space (that is, the space together with its completed -algebra agrees up to a measure-preserving bijection with the unit interval with Lebesgue measure and the usual -algebra of Lebesgue measurable sets). Although this sounds like a strong restriction, in practice it is barely a restriction at all, as almost all of the spaces that appear in the theory (and all of those that appear in this article) turn out to be Lebesgue spaces. For a detailed treatment of the theory of Lebesgue spaces, the reader is referred to Rudolph’s book [76]. The reader is referred also to the chapter on Measure Preserving Systems. While many of the definitions that we shall present are valid for both invertible and non-invertible measure-preserving transformations, the strongest mixing conditions are most useful in the case of invertible transformations. It will be helpful to present a selection of simple examples, relative to which we will be able to explore ergodicity and the various notions of mixing. These examples and the lemmas necessary to show that they are measure-preserving transformations as claimed may be found in the books of Petersen [64], Rudolph [76] and Walters [92]. More details on these examples can also be found in the chapter on Ergodic Theory: Basic Examples and Constructions. Example 1 (Rotation on the circle) Let ˛ 2 R. Let R˛ : [0; 1) ! [0; 1) be defined by R˛ (x) D x C ˛ mod 1. It is straightforward to verify that R˛ preserves the restriction of Lebesgue measure to [0; 1) (it is sufficient to check that (R˛1 (J)) D (J) for an interval J). Example 2 (Doubling Map) Let M2 : [0; 1) ! [0; 1) be defined by M2 (x) D 2x mod 1. Again, Lebesgue measure is invariant under M 2 (to see this, one observes that for an interval J, M21 (J) consists of two intervals, each of half the length of J). This may be generalized in the obvious way to a map M k for any integer k 2. Example 3 (Interval Exchange Transformation) The class of interval exchange transformations was introduced by Sinai [85]. An interval exchange transformation is the map obtained by cutting the interval into a finite number of pieces and permuting them in such a way that the resulting map is invertible, and restricted to each interval is an order-preserving isometry. More formally, one takes a sequence of positive lengths `1 ; `2 ; : : : ; ` k summing to 1 and a permutation P of f1; : : : ; kg and defines a i D j 0), the eigenvector is unique. Given the pair (P; ), one defines the measure of a cylinder set by ([a m : : : a n ]nm ) D a m Pa m a mC1 : : : Pa n1 a n and extends as before to a probability measure on AN or AZ . Example 6 (Hard Sphere Gases and Billiards) We wish to model the behavior of a gas in a bounded region. We make the assumption that the gas consists of a large number N
We will need to make use of the concept of measure-theoretic isomorphism. Two measure-preserving transformations T of (X; B; ) and S of (Y; F ; ) are measure-theoretically isomorphic (or just isomorphic) if there exist measurable maps g : X ! Y and h : Y ! X such that
Measure-theoretic isomorphism is the basic notion of ‘sameness’ in ergodic theory. It is in some sense quite weak, so that systems may be isomorphic that feel very different (for example, as we discuss later, the time one map of a geodesic flow is isomorphic to a Bernoulli shift). For comparison, the notion of sameness in topological dynamical systems (topological conjugacy) is far stronger. As an example of measure-theoretic isomorphism, it may be seen that the doubling map is isomorphic to the one-sided Bernoulli shift on f0; 1g with p0 D p1 D 1/2 (the map g takes an x 2 [0; 1) to the sequence of 0’s and 1’s in its binary expansion (choosing the sequence ending with 0’s, for example, if x is of the form p/2n ) and the inverse map h takes a sequence of 0’s and 1’s to the point in [0; 1) with that binary expansion.)
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Given a measure-preserving transformation T of a probability space (X; B; ), T is associated to an isometry of L2 (X; B; ) by U T ( f ) D f ı T. This operator is known as the Koopman Operator. In the case where T is invertible, the operator U T is unitary. Two measure-preserving transformations T and S of (X; B; ) and (Y; F ; ) are spectrally isomorphic if there is a Hilbert space isomorphism ‚ from L2 (X; B; ) to L2 (Y; F ; ) such that ‚ ı U T D U S ı ‚. As we shall see below, spectral isomorphism is a strictly weaker property than measure-theoretic isomorphism. Since in ergodic theory, measure-theoretic isomorphism is the basic notion of sameness, all properties that are used to describe measure-preserving systems are required to be invariant under measure-theoretic isomorphism (i. e. if two measure-preserving transformations are measure-theoretically isomorphic, the first has a given property if and only if the second does). On the other hand, we shall see that some mixing-type properties are invariant under spectral isomorphism, while others are not. If a property is invariant under spectral isomorphism, we say that it is a spectral property. There are a number of mixing type properties that occur in the probability literature (˛-mixing, ˇ-mixing, -mixing, -mixing etc.) (see Bradley’s survey [12] for a description of these conditions). Many of these are stronger than the Bernoulli property, and are therefore not preserved under measure-theoretic isomorphism. For this reason, these properties are not widely used in ergodic theory, although ˇ-mixing turns out to be equivalent to the so-called weak Bernoulli property (which turns out to be stronger than the Bernoulli property that we discuss in this article – see Smorodinsky’s paper [87]) and ˛-mixing is equivalent to strong-mixing. A basic construction (see the article on Ergodic Theory: Basic Examples and Constructions) that we shall require in what follows is the product of a pair of measurepreserving transformations: given transformations T of (X; B; ) and S of (Y; F ; ), we define the product transformation T S : (X Y; B ˝ F ; ) by (T S)(x; y) D (Tx; Sy). One issue that we face on occasion is that it is sometimes convenient to deal with invertible measure-preserving transformations. It turns out that given a non-invertible measure-preserving transformation, there is a natural way to uniquely associate an invertible measurepreserving transformation transformation sharing almost all of the ergodic properties of the original transformation. Specifically, given a non-invertible measure-preserving transformation T of (X; B; ), one lets X D f(x0 ; x1 ; : : : ) : x n 2 X and T(x n ) D x n1 for all ng, B
be the -algebra generated by sets of the form A¯ n D ¯ A¯ n ) D (A) and T(x0 ; x1 ; : : : ) D fx¯ 2 X : x n 2 Ag, ( (T(x0 ); x0 ; x1 ; : : : ). The transformation T of (X; B; ) is called the natural extension of the transformation T of (X; B; ) (see the chapter on Ergodic Theory: Basic Examples and Constructions for more details). In situations where one wants to use invertibility, it is often possible to pass to the natural extension, work there and then derive conclusions about the original non-invertible transformation. Ergodicity Given a measure-preserving transformation T : X ! X, if T 1 A D A, then T 1 Ac D Ac also. This allows us to decompose the transformation X into two pieces A and Ac and study the transformation T separately on each. In fact the same situation holds if T 1 A and A agree up to a set of measure 0. For this reason, we call a set A invariant if (T 1 AA) D 0. Returning to Boltzmann’s ergodic hypothesis, existence of an invariant set of measure between 0 and 1 would be a bad situation as his essential idea was that the orbit of a single point would ‘see’ all of X, whereas if X were decomposed in this way, the most that a point in A could see would be all of A, and similarly the most that a point in Ac could see would be all of Ac . A measure-preserving transformation will be called ergodic if it has no non-trivial decomposition of this form. More formally, let T be a measure-preserving transformation of a probability space (X; B; ). The transformation T is said to be ergodic if for all invariant sets, either the set or its complement has measure 0. Unlike the remaining concepts that we discuss in this article, this definition of ergodicity applies also to infinite measure-preserving transformations and even to certain non-measure-preserving transformations. See Aaronson’s book [1] for more information. The following lemma is often useful: Lemma 1 Let (X; B; ) be a probability space and let T : X ! X be a measure-preserving transformation. Then T is ergodic if and only if the only measurable functions f satisfying f ı T D f (up to sets of measure 0) are constant almost everywhere. For the straightforward proof, we notice that if the condition in the lemma holds and A is an invariant set, then 1A ı T D 1A almost everywhere, so that 1A is an a. e. constant function and so A or Ac is of measure 0. Conversely, if f is an invariant function, we see that for each ˛, fx : f (x) < ˛g is an invariant set and hence of measure 0
Ergodicity and Mixing Properties
or 1. It follows that f is constant almost everywhere. We remark for future use that it is sufficient to check that the bounded measurable invariant functions are constant. The following corollary of the lemma shows that ergodicity is a spectral property. Corollary 2 Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if 1 is a simple eigenvalue of U T . The ergodic theorems mentioned earlier due to von Neumann and Birkhoff are the following (see also the chapter on Ergodic Theorems). Theorem 3 (von Neumann Mean Ergodic Theorem [91]) Let T be a measure-preserving transformation of the probability space (X; B; ). For f 2 L2 (X; B; ), let A N f D 1/N( f C f ı T C C f ı T N1 ). Then for all f 2 L2 (X; B; ), A N f converges in L2 to an invariant function f . Theorem 4 (Birkhoff Pointwise Ergodic Theorem [6]) Let T be a measure-preserving transformation of the probability space (X; B; ). Let f 2 L1 (X; B; ). Let A N f be as above. Then for -almost every x 2 X, (A N f (x)) is a convergent sequence. Of these two theorems, the pointwise ergodic theorem is the deeper result, and it is straightforward to deduce the mean ergodic theorem from the pointwise ergodic theorem. The mean ergodic theorem was reproved very concisely by Riesz [71] and it is this proof that is widely known now. Riesz’s proof is reproduced in Parry’s book [63]. There have been many different proofs given of the pointwise ergodic theorem. Notable amongst these are the argument due to Garsia [23] and a proof due to Katznelson and Weiss [40] based on work of Kamae [35], which appears in a simplified form in work of Keane and Petersen [42]. If the measure-preserving transformation T is ergodic, then by virtue of Lemma 1, the limit functions appearing in the ergodic theorems are constant. One sees that the constant is simply the integral of f with respect to , so R that in this situation A N f (x) converges to f d in norm and pointwise almost everywhere, thereby providing a justification of Boltzmann’s original claim: for ergodic measure-preserving transformations, time averages agree with spatial averages. In the case where T is not ergodic, it is also possible to identify the limit in the ergodic theorems: we have f D E( f jI ), where I is the -algebra of T-invariant sets. Note that the set on which the almost everywhere convergence in Birkhoff’s theorem takes place depends on the L1 function f that one is considering. Straightforward considerations show that there is no single full measure set
that works simultaneously for all L1 functions. In the case where X is a compact metric space, it is well known that C(X), the space of continuous functions on X with the uniform norm has a countable dense set, ( f n )n1 say. If the invariant measure is ergodic, then for each n, there is aR set Bn of measure 1 such that for all x 2 B n , A N f n (x) ! T f n d. Letting B D n B n , one obtains a full Rmeasure set such that for all n and all x 2 B, A N f n (x) ! f n d. A simple approximation argument thenRshows that for all x 2 B and all f 2 C(X), A N f (x) ! f d. A point x with this property is said to be generic for . The observations above show that for an ergodic invariant measure , we have fx : x is generic for g D 1. If T is ergodic, but T n is not ergodic for some n, then one can show that the space X splits up as A1 ; : : : ; A d for some djn in such a way that T(A i ) D A iC1 for i < d and T(A d ) D A1 with T n acting ergodically on each Ai . The transformation T is totally ergodic if T n is ergodic for all n 2 N. One can check that a non-invertible transformation T is ergodic if and only if its natural extension is ergodic. The following lemma gives an alternative characterization of ergodicity, which in particular relates it to mixing. Lemma 5 (Ergodicity as a Mixing Property) Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if for all f and g in L2 , N 1 X h f ; g ı T n i ! h f ; 1ih1; gi: N nD0
P N1 In particular, if T is ergodic, then (1/N) nD0 (A \ T n B) ! (A)(B) for all measurable sets A and B. Proof Suppose that T is ergodic. Then the left-hand side P N1 of the equality is equal to h f ; (1/N) nD0 g ı T n i. The mean ergodic theorem shows that the second term conR verges in L2 to the constant function with value g d D hg; 1i, and the equality follows. Conversely, if the equation holds for all f and g in L2 , suppose that A is an invariant set. Let f D g D 1A . Then since g ı T n D 1A for all n, the left-hand side is h1A ; 1A i D (A). On the other hand, the right-hand side is (A)2 , so that the equation yields (A) D (A)2 , and (A) is either 0 or 1 as required. Taking f D 1A and g D 1B for measurable sets A and B gives the final statement. We now examine the ergodicity of the examples presented above. Firstly, for the rotation of the circle, we claim that
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the transformation is ergodic if and only if the ‘angle’ ˛ is irrational. To see this, we argue as follows. If ˛ D p/q, then we see that f (x) D e 2 i qx is a non-constant R˛ invariant function, and hence R˛ is not ergodic. On the other hand, if ˛ is irrational, suppose f is a bounded measurable invariant function. Since f is bounded, it is an L2 function, and so f may be expressed in L2 as a Fourier seP 2 i nx . We then ries: f D n e n where e n (x) D e n2Z cP see that f ı R˛ D n2Z e 2 i n˛ c n e n . In order for f to be equal in L2 to f ı R˛ , they must have the same Fourier coefficients, so that c n D e 2 i n˛ c n for each n. Since ˛ is irrational, this forces c n D 0 for all n ¤ 0, so that f is constant as required. The doubling map and the Bernoulli shift are both ergodic, although we defer proof of this for the time being, since they in fact have the strong-mixing property. A Markov chain with matrix P and vector is ergodic if and only if for all i and j in A with i > 0 and j > 0, there exists an n 0 with Pinj > 0. This follows from the ergodic theorem for Markov chains (which is derived from the Strong Law of Large Numbers) (see [18] for details). In particular, if the underlying Markov chain is irreducible, then the measure is ergodic. In the case of interval exchange transformations, there is a simple necessary condition on the permutation for irreducibility, namely for 1 j < k, we do not have f1; : : : ; jg D f1; : : : ; jg. Under this condition, Masur [49] and Veech [88] independently showed that for almost all values of the sequence of lengths (` i )1ik , the interval exchange transformation is ergodic. (In fact they showed the stronger condition of unique ergodicity: that the transformation has no other invariant measure than Lebesgue measure. This implies that Lebesgue measure is ergodic, because if there were a non-trivial invariant set, then the restriction of Lebesgue measure to that set would be another invariant measure). For the hard sphere systems, there are no results on ergodicity in full generality. Important special cases have been studied by Sinai [84], Sinai and Chernov [86], Krámli, Simányi and Szász [45], Simányi and Szász [81], Simnyi [79,80] and Young [95]. Ergodic Decomposition We already observed that if a transformation is not ergodic, then it may be decomposed into parts. Clearly if these parts are not ergodic, they may be further decomposed. It is natural to ask whether the transformation can be decomposed into ergodic parts, and if so what form does the decomposition take? In fact such a decomposition does exist, but rather than decompose the transforma-
tion, it is necessary to decompose the measure into ergodic pieces. This is known as ergodic decomposition. The set of invariant measures for a measurable map T of a measurable space (X; B) to itself forms a simplex. General functional analytic considerations (due to Choquet [14,15] – see also Phelps’ account [66] of this theory) mean that it is possible to write any member of the simplex as an integral-convex combination of the extreme points. Further, the extreme points of the simplex may be identified as precisely the ergodic invariant measures for T. It follows that any invariant probability measure for T may be uniquely expressed in the form Z (A) D (A) dm() ; M erg (X;T)
where Merg (X; T) denotes the set of ergodic T-invariant measures on X and m is a measure on Merg (X; T). We will give a proof of this theorem in the special case of a continuous transformation of a compact space. Our proof is based on the Birkhoff ergodic theorem and the Riesz Representation Theorem identifying the dual space of the space of continuous functions on a compact space as the set of bounded signed measures on the space (see Rudin’s book [75] for details). We include it here because this special case covers many cases that arise in practice, and because few of the standard ergodic theory references include a proof of ergodic decomposition. An exception to this is Rudolph’s book [76] which gives a full proof in the case that X is a Lebesgue space. This is based on a detailed development of the theory of these spaces and builds measures using conditional exceptions. Kalikow’s notes [32] give a brief outline of a proof similar to that which follows. Oxtoby [62] also wrote a survey article containing much of the following (and much more besides). Theorem 6 Let X be a compact metric space, B be the Borel -algebra, be an invariant Borel probability measure and T be a continuous measure-preserving transformation of (X; B; ). Then for each x 2 X, there exists an invariant Borel measure x such that: R R R 1. For f 2 L1 (X; B; ), f d D f dx d(x); 2. Given f 2 L1 (X; R B; ), for -almost every x 2 X, one has A N f (x) ! f dx ; 3. The measure x is ergodic for -almost every x 2 X. Notice that conclusion (2) shows that x can be understood as the distribution on the phase space “seen” if one starts the system in an initial condition of x. This interpretation of the measures x corresponds closely with the ideas of Boltzmann and the Ehrenfests in the formulation
Ergodicity and Mixing Properties
of the ergodic and quasi-ergodic hypotheses, which can be seen as demanding that x is equal to for (almost) all x. Proof The proof will be divided into 3 main steps: defining the measures x , proving measurability with respect to x and proving ergodicity of the measures. Step 1: Definition of x Given a function f 2 L1 (X; B; ), Birkhoff’s theorem states that for -almost every x 2 X, (A N f (x)) is convergent. It will be convenient to denote the limit by f˜(x). Let f1 ; f2 ; : : : be a sequence of continuous functions that is dense in C(X). For each k, there is a set Bk of x’s measure 1 for which (A n f k (x))1 nD1 is a convergent sequence. Intersecting these gives a set B of full measure such that for x 2 B, for each k 1, A n f k (x) is convergent. A simple approximation argument shows that for x 2 B and f an arbitrary continuous function, A n f (x) is convergent. Given x 2 B, define a map L x : C(X) ! R by L x ( f ) D f˜(x). This is a continuous linear functional on C(X), and hence by the Riesz Representation Theorem there exists a Borel R measure x such that f˜(x) D f dx for each f 2 C(X) and x 2 B. Since L x ( f ) 0 when f is a non-negative function and L x (1) D 1, the measure x is a probability measure. Since L x ( f ı T) D L x ( f ) for f 2 C(X), one can check that x must be an invariant probability measure. For x 62 B, simply define x D . Since Bc is a set of measure 0, this will not affect any of the statements that we are trying to prove. Now for f continuous, we have A N f is a bounded seR to f d quence of functions withR A N f (x) converging x R almost everywhere and A N f d D f d since T is measure-preserving. It follows from the bounded convergence theorem that for f 2 C(X), Z Z Z f d D f dx d(x) : (1) Step 2: Measurability of x 7! x (A) Lemma 7 Let C 2 B satisfy (C) D 0. Then x (C) D 0 for -almost every x 2 X. Proof Using regularity of Borel probability measures (see Rudin’s book [75] for details), there exist open sets U1 U2 C with (U k ) < 1/k. There exist continuous functions g k;m with (g k;m (x))1 mD1 increasing to 1U k everywhere R(e.Rg. g k;m (x) D min(1; m d(x; U kc ))). By (1), we Rhave ( g k;m dx ) d(x) < 1/k for all k; m. Note that g k;m dx D lim n!1 A n g k;m (x) is a measurable function of x, so that using the monotone converR gence theorem (taking the limit in m), x ! 7 1 U k d x D R R x (U k ) is measurable and ( 1U k dx ) d(x) 1/k. T We now see that x 7! lim k!1 x (U k ) D x ( U k )
is measurable, and by monotone convergence we see R alsoT T x ( U k ) d(x) D 0. It follows that x ( U k ) D 0 T for -almost every x. Since U k C, the lemma follows. Given a set A 2 B, let f k be a sequence of continuous functions (uniformly bounded by 1) satisfying k f k 1A kL 1 () < 2n , so that in particular R f k (x) ! 1A (x) for -almost every x. For each k, x 7! f k dx D lim n!1 A n f k (x) is a measurable function. By Lemma 7, for -almost every x, f k ! 1A x -almost everywhere, so R that by the bounded convergence theorem limk!1 f k dx D x (A) for -almost every x. Since the limit of measurable functions is measurable, it follows that x 7! x (A) is measurable for any measurable set A 2 B. R This allows us to define a measure by (A) D x (A) R d(x). RForR a bounded measurable function f , we have f d D ( f dx ) d(x). Since this agrees with R f d for continuous functions by (1), it follows that D . Conclusion (1) of the theorem now follows easily. Given f 2 L1 (X), we let ( f k ) be a sequence of continuous functions such that k f k f kL 1 () is summable. This implies that k f k f kL 1 ( R for -almost evR x ) is summable ery x and in particular, f k dx ! f dx for almost every x. On the other hand, by the remark following the statement of Birkhoff ’s theorem, we have f˜k D E( f k jI ) so that k f˜ f˜k kL 1 () is summable and f˜k (x) ! f˜(x) for -almost every x. Combining these two statements, we see that for -almost every x, we have f˜(x) D lim f˜k (x) D lim k!1
k!1
Z
Z f k dx D
f dx :
This establishes conclusion (2) of the theorem. Step 3: Ergodicity of x We have shown how to disintegrate the invariant measure as an integral combination of x ’s, and we have interpreted the x ’s as describing the average behavior starting from x. It remains to show that the x ’s are ergodic measures. Fix for now a continuous function f and a number 0 < < 1. Since A n f (x) ! f˜(x) -almost everywhere, there exists an N such that fx : jA N f (x) f˜(x)j > /2g < 3 /8. We now claim the following: R ˚ x : x fy : j f˜(y) f dx j > g > < :
(2)
R To see this, note that fy : j f˜(y) f dx j > g
fy : j f˜(y)A N f (y)j > /2g[fy : jA N f (y) f˜(x)j > /2g, R ˜ so that if x fy : j f (y) f dx j > g > , then either
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x fy : j f˜(y) A N f (y)j > /2g > /2 or x fy : jA N f (y) f˜(x)j > /2g > /2. We show that the set of x’s satisfying each condition is small. 3 ˜ Firstly, R we have /8 > fy : j f (y) A N f (y)j > /2g D x fy : j f˜(y) A N f (y)j > /2g d(x), so that fx : x fy : j f˜(y) A N f (y)j > /2g > /2g < 2 /4 < /2. For the second term, given c 2 R, let Fc (x) D jA N f (x) cj and G(x) D F f˜(x) (x). Note that Z F f˜(x) (y) dx (y) D lim A n F f˜(x) (x) D lim A n G(x) n!1
n!1
(using the facts that y 7! F f˜(x) (y) is a continuous function and that since f˜(x) is R an invariant function, F f˜(x) (T k x) D G(T k x)). Since G(x) d(x) < 3 /8, it R follows that F f˜(x) (y) dx (y) 2 /4 except on a set of x’s of measure less than /2. Outside this bad set, we have x fy : jA N f (y) f˜(x)j > /2g < /2 so that fx : x fy : jA N f (y) f˜(x)j > /2g > /2g < /2 as required. This establishes our claim (2) above. Since > 0 is arbitrary, it follows that for each f 2 C(X), for R -almost every x, x -almost every y satisfies f˜(y) D f dx . As usual, taking a countable dense sequence ( f k ) in C(X), it ˜ is R the case that for all k and -almost every x, f k (y) D f k dx x -almost everywhere. Let the set of x’s with this property be D. We claim that for x 2 D, x is ergodic. Suppose not. Then let x 2 D and let J be an invariant set of x measure between ı and 1 ı for some ı > 0. Then by density of C(X) in L1 (x ), there exists an f k with k f k 1 J kL 1 (x ) < ı. Since 1 J is an invariant function, we have 1˜ J D 1 J . On the other hand, f˜k is a constant function. It follows that k f˜k 1˜ J kL 1 (x ) ı > k f k 1 J kL 1 (x ) . This contradicts the identification of the limit as a conditional expectation and concludes the proof of the theorem. Mixing As mentioned above, ergodicity may be seen as an independence on average property. More specifically, one wants to know whether in some sense (A \ T n B) converges to (A)(B) as n ! 1. Ergodicity is the property that there is convergence in the Césaro sense. Weak-mixing is the property that there is convergence in the strong Césaro sense. That is, a measure-preserving transformation T is weak-mixing if N1 1 X j(A\ T n B)(A)(B)j ! 0 N nD0
as
N ! 1:
In order for T to be strong-mixing, we require simply (A \ T N B) ! (A)(B) as N ! 1. It is clear that strong-mixing implies weak-mixing and weak-mixing implies ergodicity. If T d is not ergodic (so that T d A D A for some A of measure strictly between 0 and 1), then j(T nd A \ A) (A)2 j D (A)(1 (A)), so that T is not weak-mixing. An alternative characterization of weak-mixing is as follows: Lemma 8 The measure-preserving transformation T is weak-mixing if and only if for every pair of measurable sets A and B, there exists a subset J of N of density 1 (i. e. #(J \ f1; : : : ; Ng)/N ! 1) such that lim
n!1 n62 J
(A \ T n B) D (A)(B) :
(3)
By taking a countable family of measurable sets that are dense (with respect to the metric d(A; B) D (AB)) and taking a suitable intersection of the corresponding J sets, one shows that for a given weak-mixing measure-preserving transformation, there is a single set J N such that (3) holds for all measurable sets A and B (see Petersen [64] or Walters [92] for a proof). We show that an irrational rotation of the circle is not weak-mixing as follows: let ˛ 2 R n Q and let A be the interval [ 14 ; 34 ). There is a positive proportion of n’s in the natural numbers (in fact proportion 1/3) with the property that jT n ( 12 ) 12 j < 16 . For these n’s (A \ T n A) > 13 , 1 . so that in particular j(A \ T n A) (A)(A)j > 12 Clearly this precludes the required convergence to 0 in the definition of weak-mixing, so that an irrational rotation is ergodic but not weak-mixing. Since R˛n D R n˛ , the earlier argument shows that R˛n is ergodic, so that R˛ is totally ergodic. On the other hand, we show that any Bernoulli shift is strong-mixing. To see this, let A and B be arbitrary measurable sets. By standard measure-theoretic arguments, A and B may each be approximated arbitrarily closely by a finite union of cylinder sets. Since if A0 and B0 are finite unions of cylinder sets, we have that (A0 \ T n B0 ) is equal to (A0 )(B0 ) for large n, it is easy to deduce that (A \ T n B) ! (A)(B) as required. Since the doubling map is measure-theoretically isomorphic to a onesided Bernoulli shift, it follows that the doubling map is also strong-mixing. Similarly, if a Markov Chain is irreducible (i. e. for any states i and j, there exists an n 0 such that Pinj > 0) and aperiodic (there is a state i such that gcdfn : Pini > 0g D 1), then given any pair of cylinder sets A0 and B0 we have by standard theorems of Markov chains (A0 \ T n B0 ) !
Ergodicity and Mixing Properties
(A0 )(B0 ). The same argument as above then shows that an aperiodic irreducible Markov Chain is strong-mixing. On the other hand, if a Markov chain is periodic (d D gcdfn : Pini > 0g > 0), then letting A D B D fx : x0 D ig, we have that (A \ T n B) D 0 whenever d − n. It follows T d is not ergodic, so that T is not weak-mixing. Both weak- and strong-mixing have formulations in terms of functions: Lemma 9 Let T be a measure-preserving transformation of the probability space (X; B; ). 1. T is weak-mixing if and only if for every f ; g 2 L2 one has N1 1 X jh f ; g ı T n i h f ; 1ih1; gij ! 0 as N ! 1 : N nD0
2. T is strong-mixing if and only if for every f ; g 2 L2 , one has h f ; g ı T N i ! h f ; 1ih1; gi as N ! 1 : Using this, one can see that both mixing conditions are spectral properties. Lemma 10 Weak- and strong-mixing are spectral properties. Proof Suppose S is a weak-mixing transformation of (Y; F ; ) and the transformation T of (X; B; ) is spectrally isomorphic to S by the Hilbert space isomorphism ‚. Then for f ; g 2 L2 (X; B; ), h f ; g ı T n i X h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY h‚( f ); ‚(1)iY h‚(1); ‚(g)iY . Since 1 is an eigenfunction of U T with eigenvalue 1, ‚(1) is an eigenfunction of U S with an eigenvalue 1, so since S is ergodic, ‚(1) must be a constant function. Since ‚ preserves norms, ‚(1) must have a constant value of absolute value 1 and hence h f ; g ı T n i X h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY h‚( f ); 1iY h1; ‚(g)iY . It follows from Lemma 9 that T is weak-mixing. A similar proof shows that strong-mixing is a spectral property. Both weak- and strong-mixing properties are preserved by taking natural extensions. Recent work of Avila and Forni [4] shows that for interval exchange transformations of k 3 intervals with the underlying permutation satisfying the non-degeneracy condition above, almost all divisions of the interval (with respect to Lebesgue measure on the k1-dimensional simplex) lead to weak-mixing transformations. On the other
hand, work of Katok [36] shows that no interval exchange transformation is strong-mixing. It is of interest to understand the behavior of the ‘typical’ measure-preserving transformation. There are a number of Baire category results addressing this. In order to state them, one needs a set of measure-preserving transformations and a topology on them. As mentioned earlier, it is effectively no restriction to assume that a transformation is a Lebesgue-measurable map on the unit interval preserving Lebesgue measure. The classical category results are then on the collection of invertible Lebesgue-measure preserving transformations of the unit interval. One topology on these is the ‘weak’ topology, where a sub-base is given by sets of the form N(T; A; ) D fS : (S(A)T(A)) < g. With respect to this topology, Halmos [26] showed that a residual set (i. e. a dense Gı set) of invertible measurepreserving transformations is weak-mixing (see also work of Alpern [3]), while Rokhlin [72] showed that the set of strong-mixing transformations is meagre (i. e. a nowhere dense F set), allowing one to conclude that with respect to this topology, the typical transformation is weak- but not strong-mixing. As often happens in these cases, even when a certain kind of behavior is typical, it may not be simple to exhibit concrete examples. In this case, a well-known example of a transformation that is weak-mixing but not strong-mixing was given by Chacon [13]. While on the face of it the formulation of weak-mixing is considerably less natural than that of strong-mixing, the notion of weak-mixing turns out to be extremely natural from a spectral point of view. Given a measure-preserving transformation T, let U T be the Koopman operator described above. Since this operator is an isometry, any eigenvalue must lie on the unit circle. The constant function 1 is always an eigenfunction with eigenvalue 1. If T is ergodic and g and h are eigenfunctions of U T with eigenvalue , then g h¯ is an eigenfunction with eigenvalue 1, hence invariant, so that g D Kh for some constant K. We see that for ergodic transformations, up to rescaling, there is at most one eigenfunction with any given eigenvalue. If U T has a non-constant eigenfunction f , then one has jhU Tn f ; f ij D k f k2 for each n, whereas by Cauchy– Schwartz, jh f ; 1ij2 < k f k2 . It follows that jhU Tn f ; f i h f ; 1ih1; f ij c for some positive constant c, so that using Lemma 9, T is not weak-mixing. Using the spectral theorem, the converse is shown to hold. Theorem 11 The measure-preserving transformation T is weak-mixing if and only U T has no non-constant eigenfunctions.
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Of course this also shows that weak-mixing is a spectral property. Equivalently, this says that the transformation T is weak-mixing if and only if the apart from the constant eigenfunction, the operator U T has only continuous spectrum (that is, the operator has no other eigenfunctions). For a very nice and concise development of the part of spectral theory relevant to ergodic theory, the reader is referred to the Appendix in Parry’s book [63]. Using this theory, one can establish the following: Theorem 12 1. T is weak-mixing if and only if T T is ergodic; 2. If T and S are ergodic, then T S is ergodic if and only if U S and U T have no common eigenvalues other than 1. Proof The main factor in the proof is that the eigenvalues of U TS are precisely the set of ˛ˇ, where ˛ is an eigenvalue of U T and ˇ is an eigenvalue of U S . Further, the eigenfunctions of U TS with eigenvalue are spanned by eigenfunctions of the form f ˝ g, where f is an eigenfunction of U T , g is an eigenfunction of U S , and the product of the eigenvalues is . Suppose that T is weak-mixing. Then the only eigenfunction is the constant function, so that the only eigenfunction of U TT is the constant function, proving that T T is ergodic. Conversely, if U T has an eigenvalue (so that f ı T D ˛T for some non-constant f ) then f ˝ f¯ is a non-constant invariant function of T T so that T T is not ergodic. For the second part, if U S and U T have a common eigenvalue other than 1 (say f ı T D ˛ f and g ı T D ˛g), then f ˝ g¯ is a non-constant invariant function. Conversely, if T S has a non-constant invariant function h, then h can be decomposed into functions of the form f ˝g, where f and g are eigenfunctions of U T and U S respectively with eigenvalues ˛ and ˇ satisfying ˛ˇ D 1. Since the eigenvalues of S are closed under complex conjugation, we see that U T and U S have a common eigenvalue other than 1 as required. For a measure-preserving transformation T, we let K be the subspace of L2 spanned by the eigenfunctions of U T . It is a remarkable fact that K may be identified as L2 (X; B0 ; ) where B0 is a sub--algebra of B. The space K is called the Kronecker factor of T. The terminology comes from the fact that any sub--algebra F of B gives rise to a factor mapping : (X; B; ) ! (X; F ; ) with (x) D x. By construction L2 (X; B0 ; ) is the closed linear span of the eigenfunctions of T considered as a measure-preserving transformation of (X; B0 ; ). By the Discrete Spectrum Theorem of Halmos and von Neumann [27], T act-
ing on (X; B0 ; ) is measure-theoretically isomorphic to a rotation on a compact group. This allows one to split L2 (X; B; ) as L2 (X; B0 ; ) ˚ L2c (X; B; ), where, as mentioned above the first part is the discrete spectrum part, spanned by eigenfunctions, and the second part is the continuous spectrum part, consisting of functions whose spectral measure is continuous. Since we have split L2 into a discrete part and a continuous part, it is natural to ask whether the underlying transformation T can be split up in some way into a weak-mixing part and a discrete spectrum (compact group rotation) part, somewhat analogously to the ergodic decomposition. Unfortunately, there is nosuch decomposition available. However for some applications, for example to multiple recurrence (starting with the work of Furstenberg [20,21]), the decomposition of L2 (possibly into more complicated parts) plays a crucial role (see the chapters on Ergodic Theory: Recurrence and Ergodic Theory: Interactions with Combinatorics and Number Theory). For non-invertible measure-preserving transformations, the transformation is weak- or strong-mixing if and only if its natural extension has that property. The understanding of weak-mixing in terms of the discrete part of the spectrum of the operator also extends to total ergodicity. T n is ergodic if and only if T has no eigenvalues of the form e 2 i p/n other than 1. From this it follows that an ergodic measure-preserving transformation T is totally ergodic if and only if it has no rational spectrum (i. e. no eigenvalues of the form e 2 i p/q other than the simple eigenvalue 1). An intermediate mixing condition between strongand weak- mixing is that a measure-preserving transformation is mild-mixing if whenever f ı T n i ! f for an L2 function f and a sequence n i ! 1, then f is a.e. constant. Clearly mild-mixing is a spectral property. If a transformation has an eigenfunction f , then it is straightforward to find a sequence ni such that f ı T n i ! f , so we see that mild-mixing implies weak-mixing. To see that strong-mixing implies mild-mixing, suppose that T is and that f ı T n i ! f . Then we have R strong-mixing n ¯ i f ı T f ! k f k2 . On R the other hand, the strong mixing property implies that f ı T n i f¯ ! jh f ; 1ij2. The equality of these implies that f is a. e. constant. Mild-mixing has a useful reformulation in terms of ergodicity of general (not necessarily probability) measure-preserving transformations: A transformation T is mild-mixing if and only if for every conservative ergodic measure-preserving transformation S, T S is ergodic. See Furstenberg and Weiss’ article [22] for further information on mild-mixing. The strongest spectral property that we consider is that of having countable Lebesgue spectrum. While we
Ergodicity and Mixing Properties
will avoid a detailed discussion of spectral theory in this article, this is a special case that can be described simply. Specifically, let T be an invertible measure-preserving transformation. Then T has countable Lebesgue spectrum if there is a sequence of functions f1 ; f2 ; : : : such that f1g [ fU Tn f j : n 2 Z; j 2 Ng forms an orthonormal basis for L2 (X). To see that this property is stronger than strongmixing, we simply observe that it implies that hU Tt U Tn f j ; U Tm f k i ! 0 as t ! 1. Then by approximating f and g by their expansions with respect to a finite part of the basis, we deduce that hU Tn f ; gi ! h f ; 1ih1; gi as required. Since already strong-mixing is atypical from the topological point of view, it follows that countable Lebesgue spectrum has to be atypical. In fact, Yuzvinskii [96] showed that the typical invertible measure-preserving transformation has simple singular spectrum. The property of countable Lebesgue spectrum is by definition a spectral property. Since it completely describes the transformation up to spectral isomorphism, there can be no stronger spectral properties. The remaining properties that we shall examine are invariant under measuretheoretic isomorphisms only. An invertible measure-preserving transformation T of (X; B; ) is said to be K (for Kolmogorov) if there is a sub-algebra F of B such that T n F is the trivial -algebra up to sets of mea1. 1 nD1 T sure 0 (i. e. the intersection consists only of null sets and sets of full measure). W n 2. 1 nD1 T F D B (i. e. the smallest -algebra containn ing T F for all n > 0 is B). The K property has a useful reformulation in terms of entropy as follows: T is K if and only if for every non-trivial partition P of X, the entropy of T with respect to the partition P is positive: T has completely positive entropy. See the chapter on Entropy in Ergodic Theory for the relevant definitions. The equivalence of the K property and completely positive entropy was shown by Rokhlin and Sinai [74]. For a general transformation T, one can consider the collection of all subsets B of X such that with respect to the partition PB D fB; B c g, h(PB ) D 0. One can show that this is a -algebra. This -algebra is known as the Pinsker -algebra. The above reformulation allows us to say that a transformation is K if and only if it has a trivial Pinsker -algebra. The K property implies countable Lebesgue spectrum (see Parry’s book [63] for a proof). To see that K is not implied by countable Lebesgue spectrum, we point out that certain measure-preserving transformations derived from Gaussian systems (see for example the paper of Parry and
Newton [52]) have countable Lebesgue spectrum but zero entropy. The fact that (two-sided) Bernoulli shifts have the K property follows from Kolmogorov’s 0–1 law by taking W1 F D nD0 T n P , where P is the partition into cylinder sets (see Williams’s book [93] for details of the 0–1 law). Although the K property is explicitly an invertible property, it has a non-invertible counterpart, namely exactness. A transformation T of (X; B; ) is exact if T1 n B consists entirely of null sets and sets of meanD0 T sure 1. It is not hard to see that a non-invertible transformation is exact if and only if its natural extension is K. The final and strongest property in our list is that of being measure-theoretically isomorphic to a Bernoulli shift. If T is measure-theoretically isomorphic to a Bernoulli shift, we say that T has the Bernoulli property. While in principle this could apply to both invertible and non-invertible transformations, in practice the definition applies to a large class of invertible transformations, but occurs comparatively seldom for non-invertible transformations. For this reason, we will restrict ourselves to a discussion of the Bernoulli property for invertible transformations (see however work of Hoffman and Rudolph [29] and Heicklen and Hoffman [28] for work on the one-sided Bernoulli property). In the case of invertible Bernoulli shifts, Ornstein [53,58] developed in the early 1970s a powerful isomorphism theory, showing that two Bernoulli shifts are measure-theoretically isomorphic if and only if they have the same entropy. Entropy had already been identified as an invariant by Kolmogorov and Sinai [43,82], so this established that it was a complete invariant for Bernoulli shifts. Keane and Smorodinsky [41] gave a proof which showed that two Bernoulli shifts of the same entropy are isomorphic using a conjugating map that is continuous almost everywhere. With other authors, this theory was extended to show that the property of being isomorphic to a Bernoulli shift applied to a surprisingly large class of measure-preserving transformations (e. g. geodesic flows on manifolds of constant negative curvature (Ornstein and Weiss [60]), aperiodic irreducible Markov chains (Friedman and Ornstein [19]), toral automorphisms (Katznelson [39]) and more generally many Gibbs measures for hyperbolic dynamical systems (see the book of Bowen [11])). Initially, it was conjectured that the properties of being K and Bernoulli were the same, but since then a number of measure-preserving transformations that are K but not Bernoulli have been identified. The earliest was due to Ornstein [55]. Ornstein and Shields [59] then provided an uncountable family of non-isomorphic K automorphisms.
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Katok [37] gave an example of a smooth diffeomorphism that is K but not Bernoulli; and Kalikow [33] gave a very natural probabilistic example of a transformation that has this property (the T, T 1 process). While in systems that one regularly encounters there is a correlation between positive entropy and the stronger mixing properties that we have discussed, these properties are logically independent (for example taking the product of a Bernoulli shift and the identity transformation gives a positive entropy transformation that fails to be ergodic; also, the zero entropy Gaussian systems with countable Lebesgue spectrum mentioned above have relatively strong mixing properties but zero entropy). In many of the mixing criteria discussed above we have considered a pair of sets A and B and asked for asymptotic independence of A and B (so that for large n, A and T n B become independent). It is natural to ask, given a finite collection of sets A0 ; A1 ; : : : ; A k , under what conditions Q (A0 \T n 1 A1 \ \T n k A k ) converges to kjD0 (A j ). A measure-preserving transformation is said to be mixing of order k + 1 if for all measurable sets A0 ; : : : ; A k , lim
n 1 !1;n jC1n j !1
A0 \ T n 1 A1 \ \ T n k A k D
k Y
(A j ) :
jD0
An outstanding open question asked by Rokhlin [73] appearing already in Halmos’ 1956 book [27] is to determine whether mixing (i. e. mixing of order 2) implies mixing of all orders. Kalikow [34] showed that mixing implies mixing of all orders for rank 1 transformations (existence of rank one mixing transformations having been previously established by Ornstein in [54]). Later Ryzhikov [77] used joining methods to establish the result for transformations with finite rank, and Host [30] also used joining methods to establish the result for measure-preserving transformations with singular spectrum, but the general question remains open. It is not hard to show using martingale arguments that K automorphisms and hence all Bernoulli measure-preserving transformations are mixing of all orders. For weak-mixing transformations, Furstenberg [21] has established the following weak-mixing of all orders statement: if a measure-preserving transformation T is weak-mixing, then given sets A0 ; : : : ; A k , there is a subsequence J of the integers of density 0 such that
lim
n!1 n62 J
k Y A0 \ T n A1 \ \ T kn A k D (A i ) : iD0
Bergelson [5] generalized this by showing that A0 \ T p 1 (n) A1 \ \ T p k (n) A k n!1;n62 J k Y D (A i ) lim
iD0
whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n) p j (n) is unbounded for i ¤ j. The method of proof of both of these results was a Hilbert space version of the van der Corput inequality of analytic number theory. Furstenberg’s proof played a key role in his ergodic proof [20] of Szemerédi’s theorem on the existence of arbitrarily long arithmetic progressions in a subset of the integers of positive density (see the chapter on Ergodic Theory: Interactions with Combinatorics and Number Theory for more information about this direction of study). The conclusions that one draws here are much weaker than the requirement for mixing of all orders. For mixing of all orders, it was required that provided the gaps between 0; n1 ; : : : ; n k diverge to infinity, one achieves asymptotic independence, whereas for these weak-mixing results, the gaps are increasing along prescribed sequences with regular growth properties. It is interesting to note that the analogous question of whether mixing implies mixing of all orders is known to fail in higher-dimensional actions. Here, rather than a Z action, in which there is a single measure-preserving transformation (so that the integer n acts on a point x 2 X by mapping it to T n x), one takes a Zd action. For such an action, one has d commuting transformations T1 ; : : : ; Td and a vector (n1 ; : : : ; nd ) acts on a point x by sending it to n T1n 1 Td d x. Ledrappier [46] studied the following two2 dimensional action. Let X D fx 2 f0; 1gZ : xv C xvCe1 C xvCe2 D 0 (mod 2)g and let Ti (x)v D xvCe i . Since X is a compact Abelian group, it has a natural measure invariant under the group operations (the Haar measure). It is not hard to show that this system is mixing (i. e. given any measurable sets A and B, (A \ T1n 1 T2n 2 B) ! (A)(B) as k(n1 ; n2 )k ! 1). Ledrappier showed that the system fails to be 3-mixing. Subsequently Masser [48] established necessary and sufficient conditions for similar higher-dimensional algebraic actions to be mixing of order k but not order k C 1 for any given k. Hyperbolicity and Decay of Correlations One class of systems in which the stronger mixing properties are often found is the class of smooth systems possessing uniform hyperbolicity (i. e. the tangent space to the manifold at each point splits into stable and unstable
Ergodicity and Mixing Properties
subspaces E s (x) and E u (x) such that the kDTj E s (x) k a < 1 for all x and kDT 1 j E u (x) k a and DT(Es (x)) D Es (T(x)) and DT(Eu (x)) D Eu (T(x))). In some cases similar conclusions are found in systems possessing non-uniform hyperbolicity. See Katok and Hasselblatt’s book [38] for an overview of hyperbolic dynamical systems, as well as the chapter in this volume on Smooth Ergodic Theory. In the simple case of expanding piecewise continuous maps of the interval (that is, maps for which the absolute value of the derivative is uniformly bounded below by a constant greater than 1), it is known that if they are totally ergodic and topologically transitive (i. e. the forward images of any interval cover the entire interval), then provided that the map has sufficient smoothness (e. g. the map is C1 and the derivative satisfies a certain additional summability condition), the map has a unique absolutely continuous invariant measure which is exact and whose natural extension is Bernoulli (see the paper of Góra [25] for results of this type proved under some of the mildest hypotheses). These results were originally established for maps that were twice continuously differentiable, and the hypotheses were progressively weakened, approaching, but never meeting, C1 . Subsequent work of Quas [68,69] provided examples of C1 expanding maps of the interval for which Lebesgue measure was invariant, but respectively not ergodic and not weak-mixing. Some of the key tools in controlling mixing in one-dimensional expanding maps that are absent in the C1 case are bounded distortion estimates. Here, there is a constant 1 C < 1 such that given any interval I on which some power T n of T acts injectively and any sub-interval J of I, one has 1/C (jT n Jj/jT n Ij)/(jJj/jIj) C. An early place in which bounded distortion estimates appear is the work of Rényi [70]. One important class of results for expanding maps establishes an exponential decay of correlations. Here, one starts with f and g and one estiR a pair of smooth R functions R mates f g ı T n d f d g d, where is an absolutely continuous invariant measure. If is mixing, we expect this to converge to 0. In fact though, in good cases this converges to 0 at an exponential rate for each pair of functions f and g belonging to a sufficiently smooth class. In this case, the measure-preserving transformation T is said to have exponential decay of correlations. See Liverani’s article [47] for an introduction to a method of establishing this based on cones. Exponential decay of correlations implies in particular that the natural extension is Bernoulli. Hu [31] has studied the situation of maps of the interval for which the derivative is bigger than 1 everywhere except at a fixed point, where the local behavior is of the form x 7! x C x 1C˛ for 0 < ˛ < 1. In this case, rather
than exhibiting exponential decay of correlations, the map has polynomial decay of correlations with a rate depending on ˛. In Young’s survey [94], a variety of techniques are outlined for understanding the strong ergodic properties of non-uniformly hyperbolic diffeomorphisms. In her article [95], methods are introduced for studying many classes of non-uniformly hyperbolic systems by looking at suitably high powers of the map, for which the power has strong hyperbolic behavior. The article shows how to understand the ergodic behavior of these systems. These methods are applied (for example) to billiards, one-dimensional quadratic maps and Hénon maps. Future Directions Problem 1 (Mixing of all orders) Does mixing imply mixing of all orders? Can the results of Kalikow, Ryzhikov and Host be extended to larger classes of measure-preserving transformations? Thouvenot observed that it is sufficient to establish the result for measure-preserving transformations of entropy 0. This observation (whose proof is based on the Pinsker -algebra) was stated in Kalikow’s paper [34] and is reproduced as Proposition 3.2 in recent work of de la Rue [16] on the mixing of all orders problem. Problem 2 (Multiple weak-mixing) As mentioned above, Bergelson [5] showed that if T is a weak-mixing transformation, then there is a subset J of the integers of density 0 such that lim
n!1;n62 J
A0 \ T p 1 (n) A1 \ \ T p k (n) A k D
k Y
(A i )
iD0
whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n) p j (n) is unbounded for i ¤ j. It is natural to ask what is the most general class of times that can replace the sequences (p1 (n)); : : : ; (p k (n)). In unpublished notes, Bergelson and Håland considered as times the values taken by a family of integer-valued generalized polynomials (those functions of an integer variable that can be obtained by the operations of addition, multiplication, addition of or multiplication by apreal constantp and taking integer parts (e. g. g(n) D b 2b nc C b 3nc2 c)). They conjectured necessary and sufficient conditions for the analogue of Bergelson’s weakmixing polynomial ergodic theorem to hold, and proved the conjecture in certain cases.
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Ergodicity and Mixing Properties
In a recent paper of McCutcheon and Quas [50], the analogous question was addressed in the case where T is a mild-mixing transformation. Problem 3 (Pascal adic transformation) Vershik [89,90] introduced a family of transformations known as the adic transformations. The underlying spaces for these transformations are certain spaces of paths on infinite graphs, and the transformations act by taking a path to its lexicographic neighbor. Amongst the adic transformations, the so-called Pascal adic transformation (so-called because the underlying graph resembles Pascal’s triangle) has been singled out for attention in work of Petersen and others [2,10,51,65]. In particular, it is unresolved whether this transformation is weak-mixing with respect to any of its ergodic measures. Weak-mixing has been shown by Petersen and Schmidt to follow from a number-theoretic condition on the binomial coefficients [2,10]. Problem 4 (Weak Pinsker Conjecture) Pinsker [67] conjectured that in a measure-preserving transformation with positive entropy, one could express the transformation as a product of a Bernoulli shift with a system with zero entropy. This conjecture (now known as the Strong Pinsker Conjecture) was shown to be false by Ornstein [56,57]. Shields and Thouvenot [78] showed that the collection of transformations that can be written as a product of a zero entropy transformation with a Bernoulli shift ¯ is closed in the so-called d-metric that lies at the heart of Ornstein’s theory. It is, however, the case that if T : X ! X has entropy h > 0, then for all h0 h, T has a factor S with entropy h0 (this was originally proved by Sinai [83] and reproved using the Ornstein machinery by Ornstein and Weiss in [61]). The Weak Pinsker Conjecture states that if a measure-preserving transformation T has entropy h > 0, then for all > 0, T may be expressed as a product of a Bernoulli shift and a measure-preserving transformation with entropy less than . Bibliography 1. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence 2. Adams TM, Petersen K (1998) Binomial coefficient multiples of irrationals. Monatsh Math 125:269–278 3. Alpern S (1976) New proofs that weak mixing is generic. Invent Math 32:263–278 4. Avila A, Forni G (2007) Weak-mixing for interval exchange transformations and translation flows. Ann Math 165:637–664 5. Bergelson V (1987) Weakly mixing pet. Ergodic Theory Dynam Systems 7:337–349 6. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci 17:656–660
7. Birkhoff GD, Smith PA (1924) Structural analysis of surface transformations. J Math 7:345–379 8. Boltzmann L (1871) Einige allgemeine Sätze über Wärmegleichgewicht. Wiener Berichte 63:679–711 9. Boltzmann L (1909) Wissenschaftliche Abhandlungen. Akademie der Wissenschaften, Berlin 10. Boshernitzan M, Berend D, Kolesnik G (2001) Irrational dilations of Pascal’s triangle. Mathematika 48:159–168 11. Bowen R (1975) Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer, Berlin 12. Bradley RC (2005) Basic properties of strong mixing conditions. A survey and some open questions. Probab Surv 2:107–144 13. Chacon RV (1969) Weakly mixing transformations which are not strongly mixing. Proc Amer Math Soc 22:559–562 14. Choquet G (1956) Existence des représentations intégrales au moyen des points extrémaux dans les cônes convexes. C R Acad Sci Paris 243:699–702 15. Choquet G (1956) Unicité des représentations intégrales au moyen de points extrémaux dans les cônes convexes réticulés. C R Acad Sci Paris 243:555–557 16. de la Rue T (2006) 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bull Braz Math Soc (NS) 37(4):503–521 17. Ehrenfest P, Ehrenfest T (1911) Begriffliche Grundlage der statistischen Auffassung in der Mechanik. No. 4. In: Encyclopädie der mathematischen Wissenschaften. Teubner, Leipzig 18. Feller W (1950) An Introduction to Probability and its Applications. Wiley, New York 19. Friedman NA, Ornstein DS (1970) On isomorphism of weak Bernoulli transformations. Adv Math 5:365–394 20. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Analyse Math 31:204–256 21. Furstenberg H (1981) Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton 22. Furstenberg H, Weiss B (1978) The finite multipliers of infinite ergodic transformations. In: The Structure of Attractors in Dynamical Systems. Proc Conf, North Dakota State Univ, Fargo, N.D., 1977. Springer, Berlin 23. Garsia AM (1965) A simple proof of E. Hopf’s maximal ergodic theory. J Math Mech 14:381–382 24. Girsanov IV (1958) Spectra of dynamical systems generated by stationary Gaussian processes. Dokl Akad Nauk SSSR 119:851– 853 25. Góra P (1994) Properties of invariant measures for piecewise expanding one-dimensional transformations with summable oscillations of derivative. Ergodic Theory Dynam Syst 14:475– 492 26. Halmos PA (1944) In general a measure-preserving transformation is mixing. Ann Math 45:786–792 27. Halmos P (1956) Lectures on Ergodic Theory. Chelsea, New York 28. Hoffman C, Heicklen D (2002) Rational maps are d-adic bernoulli. Ann Math 156:103–114 29. Hoffman C, Rudolph DJ (2002) Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann Math 76:79–101 30. Host B (1991) Mixing of all orders and pairwise independent joinings of systems with singular spectrum. Israel J Math 76:289–298
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31. Hu H (2004) Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergodic Theory Dynam Syst 24:495–524 32. Kalikow S () Outline of ergodic theory. Notes freely available for download. See http://www.math.uvic.ca/faculty/aquas/ kalikow/kalikow.html 33. Kalikow S (1982) T; T 1 transformation is not loosely Bernoulli. Ann Math 115:393–409 34. Kalikow S (1984) Twofold mixing implies threefold mixing for rank one transformations. Ergodic Theory Dynam Syst 2:237– 259 35. Kamae T (1982) A simple proof of the ergodic theorem using non-standard analysis. Israel J Math 42:284–290 36. Katok A (1980) Interval exchange transformations and some special flows are not mixing. Israel J Math 35:301–310 37. Katok A (1980) Smooth non-Bernoulli K-automorphisms. Invent Math 61:291–299 38. Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge, Cambridge 39. Katznelson Y (1971) Ergodic automorphisms of T n are Bernoulli shifts. Israel J Math 10:186–195 40. Katznelson Y, Weiss B (1982) A simple proof of some ergodic theorems. Israel J Math 42:291–296 41. Keane M, Smorodinsky M (1979) Bernoulli schemes of the same entropy are finitarily isomorphic. Ann Math 109:397–406 42. Keane MS, Petersen KE (2006) Nearly simultaneous proofs of the ergodic theorem and maximal ergodic theorem. In: Dynamics and Stochastics: Festschrift in Honor of M.S. Keane. Institute of Mathematical Statistics, pp 248–251, Bethesda MD 43. Kolmogorov AN (1958) New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces. Dokl Russ Acad Sci 119:861–864 44. Koopman BO (1931) Hamiltonian systems and Hilbert space. Proc Nat Acad Sci 17:315–218 45. Krámli A, Simányi N, Szász D (1991) The K-property of three billiard balls. Ann Math 133:37–72 46. Ledrappier F (1978) Un champ Markovien peut être d’entropie nulle et mélangeant. C R Acad Sci Paris, Sér A-B 287:561–563 47. Liverani C (2004) Decay of correlations. Ann Math 159:1275– 1312 48. Masser DW (2004) Mixing and linear equations over groups in positive characteristic. Israel J Math 142:189–204 49. Masur H (1982) Interval exchange transformations and measured foliations. Ann Math 115:169–200 50. McCutcheon R, Quas A (2007) Generalized polynomials and mild mixing systems. Canad J Math (to appear) 51. Méla X, Petersen K (2005) Dynamical properties of the Pascal adic transformation. Ergodic Theory Dynam Syst 25:227–256 52. Newton D, Parry W (1966) On a factor automorphism of a normal dynamical system. Ann Math Statist 37:1528–1533 53. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352 54. Ornstein DS (1972) On the root problem in ergodic theory. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), vol II: Probability theory. Univ. California Press, pp 347–356 55. Ornstein DS (1973) An example of a Kolmogorov automorphism that is not a Bernoulli shift. Adv Math 10:49–62 56. Ornstein DS (1973) A K-automorphism with no square root and Pinsker’s conjecture. Adv Math 10:89–102
57. Ornstein DS (1973) A mixing transformation for which Pinsker’s conjecture fails. Adv Math 10:103–123 58. Ornstein DS (1974) Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, Newhaven 59. Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:89–102 60. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Israel J Math 14:184–198 61. Ornstein DS, Weiss B (1975) Unilateral codings of Bernoulli systems. Israel J Math 21:159–166 62. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116–136 63. Parry W (1981) Topics in Ergodic Theory. Cambridge, Cambridge 64. Petersen K (1983) Ergodic Theory. Cambridge, Cambridge 65. Petersen K, Schmidt K (1997) Symmetric Gibbs measures. Trans Amer Math Soc 349:2775–2811 66. Phelps R (1966) Lectures on Choquet’s Theorem. Van Nostrand, New York 67. Pinsker MS (1960) Dynamical systems with completely positive or zero entropy. Soviet Math Dokl 1:937–938 68. Quas A (1996) A C 1 expanding map of the circle which is not weak-mixing. Israel J Math 93:359–372 69. Quas A (1996) Non-ergodicity for C 1 expanding maps and g-measures. Ergodic Theory Dynam Systems 16:531–543 70. Rényi A (1957) Representations for real numbers and their ergodic properties. Acta Math Acad Sci Hungar 8:477–493 71. Riesz F (1938) Some mean ergodic theorems. J Lond Math Soc 13:274–278 72. Rokhlin VA (1948) A ‘general’ measure-preserving transformation is not mixing. Dokl Akad Nauk SSSR Ser Mat 60:349–351 73. Rokhlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 74. Rokhlin VA, Sinai Y (1961) Construction and properties of invariant measurable partitions. Dokl Akad Nauk SSSR 141:1038– 1041 75. Rudin W (1966) Real and Complex Analysis. McGraw Hill, New York 76. Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford, Oxford University Press 77. Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank. Funct Anal Appl 27:128–140 78. Shields P, Thouvenot J-P (1975) Entropy zero × Bernoulli pro¯ cesses are closed in the d-metric. Ann Probab 3:732–736 79. Simányi N (2003) Proof of the Boltzmann–Sinai ergodic hypothesis for typical hard disk systems. Invent Math 154:123– 178 80. Simányi N (2004) Proof of the ergodic hypothesis for typical hard ball systems. Ann Henri Poincaré 5:203–233 81. Simányi N, Szász D (1999) Hard ball systems are completely hyperbolic. Ann Math 149:35–96 82. Sinai YG (1959) On the notion of entropy of a dynamical system. Dokl Russ Acad Sci 124:768–771 83. Sinai YG (1964) On a weak isomorphism of transformations with invariant measure. Mat Sb (NS) 63:23–42 84. Sinai YG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat Nauk 25:141–192 85. Sinai Y (1976) Introduction to Ergodic Theory. Princeton, Princeton, translation of the 1973 Russian original
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Existence and Uniqueness of Solutions of Initial Value Problems
Existence and Uniqueness of Solutions of Initial Value Problems GIANNE DERKS Department of Mathematics, University of Surrey, Guildford, UK Article Outline Glossary Definition of the Subject Introduction Existence Uniqueness Continuous Dependence on Initial Conditions Extended Concept of Differential Equation Further Directions Bibliography Glossary Ordinary differential equation An ordinary differential equation is a relation between a (vector) function u : I ! Rm , (I an interval in R, m 2 N) and its derivatives. A function u, which satisfies this relation, is called a solution. Initial value problem An initial value problem is an ordinary differential equation with a prescribed value for the solution at one instance of its variable, often called the initial time. The initial value is the pair consisting of the initial time and the prescribed value of the solution. Vector field For an ordinary differential equation of the m ! Rm form du dt D f (t; u), the function f : R R is called the vector field. Thus, at a solution, the vector field gives the tangent to the solution. Flow map The flow map describes the solution of an ordinary differential equation for varying initial values. Hence, for an ordinary differential equation du dt D f (t; u), the flow map is a function ˚ : R R Rm ! Rm such that ˚(t; t0 ; u0 ) is a solution of the ordinary differential equation, starting at t D t0 in u D u0 . Functions: bounded, continuous, uniformly Lipschitz continuous Let D be a connected set in R k with k 2 N and let f : D ! Rm be a function on D: The function f is bounded if there is some M > 0 such that j f (x)j M for all x 2 D. The function f is continuous on D if lim x!x 0 f (x) D f (x0 ), for every x0 2 D. A continuous function on a bounded and closed set D is bounded. The function f is equicontinuous or
uniform continuous if the convergence of the limits in the definition of continuity is uniform for all x0 2 D, i. e., for every " > 0, there is a ı > 0, such that for all x0 2 D and all x 2 D with jx x0 j < ı it holds that j f (x) f (x0 )j < ". A continuous function on a compact interval is equicontinuous. Let D R Rm . The function f : D ! Rm is uniformly Lipschitz continuous on D with respect to its second variable, if f is continuous on D and there exists some constant L > 0 such that j f (t; u) f (t; u)j Lju vj ; for all (t; u); (t; v) 2 D : The constant L is called the Lipschitz constant. Pointwise and uniform convergence A sequence of functions fu n g with u n : I ! Rm is Pointwise convergent if lim n!1 u n (t) exists for every t 2 I. The sequence of functions fu n g is uniform convergent with limit function u : I ! Rm if lim n!1 supfju u n j j t 2 Ig D 0. A sequence of pointwise convergent, equicontinuous functions is uniform convergent and the limit function is equicontinuous. Notation u˙ Derivative of u, i. e., u(k)
du dt
The kth derivative of u, i. e.,
dk u dt k
I a (t0 ) The closed interval [t0 ; t0 C a] Bb (u0 ) The closed ball with radius b about u0 in Rm , i. e., Bb (u0 ) :D fu 2 Rm j ju u0 j bg kuk1 The supremum norm for a bounded function u : I ! Rm , i. e., kuk1 D supfju(t)j j t 2 Ig Definition of the Subject Many problems in physics, engineering, biology, economics, etc., can be modeled as relations between observables or states and their derivatives, hence as differential equations. When only derivatives with respect to one variable play a role, the differential equation is called an ordinary differential equation. The field of differential equations has a long history, starting with Newton and Leibniz in the seventeenth century. In the beginning of the study of differential equations, the focus is on finding explicit solutions as the emphasis is on solving the underlying physical problems. But soon one starts to wonder: If a starting point for a solution of a differential equation is given, does the solution always exist? And if such a solution exists, how long does it exist and is there only one such solution? These are the questions of existence and uniqueness of solutions of initial value problems. The first existence
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result is given in the middle of the nineteenth century by Cauchy. At the end of the nineteenth century and the beginning of the twentieth century, substantial progress is made on the existence and uniqueness of solutions of initial value problems and currently the heart of the topic is quite well understood. But there are many open questions as soon as one considers delay equations, functional differential equations, partial differential equations or stochastic differential equations. Another area of intensive current research, which uses the existence and uniqueness of differential equations, is the area of finite- and infinitedimensional dynamical systems. Introduction As indicated before, an initial value problem is the problem of finding a solution of an ordinary differential equation with a given initial condition. To be precise, let D be an open, connected set in R Rm . Given are a function f : D ! Rm and a point (t0 ; u0 ) 2 D. The initial value problem is the problem of finding an interval I R and a function u : I ! Rm such that t0 2 I, f(t; u(t)) j t 2 Ig D and du D f (t; u(t)) ; dt
t2I;
with u(t0 ) D u0 :
(1)
The function f is called the vector field and the point (t0 ; u0 ) the initial value. One often writes the deriva˙ tive as du dt D u. If f is continuous, then the initial value problem (1) is equivalent to finding a function u : I ! Rm such that Z
t
u(t) D u0 C
f (; u())d ;
for t 2 I :
(2)
t0
If f is continuous and a solution exists, then clearly this solution is continuously differentiable on I. It might seem restrictive to only consider first-order systems; however, most higher-order equations can be written as a system of first-order equations. Consider for example the nth order differential equation for v : I ! R v (n) (t) D F t; v; v˙; : : : ; v (n1) ; with initial conditions v (k) (t0 ) D v k ;
k D 0; : : : ; n 1 :
This can be written as a first-order system by using the vector function u : I ! Rm defined as u D (v; v˙; : : : ; v (n1) ).
The equivalent initial value problem for u is 0 B B u˙ D f (t; u) D B @
u2 :: : un F(t; u1 ; u2 ; : : : ; u n )
1 C C C; A 0
B B with u(t0 ) D B @
v0 v1 :: :
1 C C C: A
v n1 Obviously, this system is not a unique representation, there are many other ways of obtaining first-order systems from the nth-order problem. In the beginning of the study of differential equations, the focus is on finding explicit solutions. The first existence theorem is by Cauchy [3] in the middle of the nineteenth century and an initial value problem is also called a Cauchy problem. At the end of the nineteenth century substantial progress is made on the existence of solutions of an initial value problem when Peano [20] shows that if f is continuous, then a solution exists near the initial value (t0 ; u0 ), i. e., there is local existence of solutions. Global existence of solutions of initial value problems needs more than smoothness, as is illustrated by the following example. Example 1 Consider the initial value problem on D D R R given as u˙ D u2
and
u(0) D u0
for some u0 > 0. As can be verified easily, the solution is u(t) D u0 /(1 tu0 ), for t 2 (1; 1/u0 ). This solution cannot be extended for t 1/u0 , even though the vector field f (t; u) D u2 is infinitely differentiable on the full domain. As can be seen from later theorems, the lack of global existence is related to the fact that the vector field f is unbounded. Once existence of solutions of initial value problems is known, the next question is if such a solution is unique. As can be seen from the following example, continuity of f is not sufficient for uniqueness of solutions. Example 2 Consider the following initial value problem on D D R R: u˙ D juj˛
and u(0) D 0 :
If 0 < ˛ < 1, then there is an infinite number of solutions. Two obvious solutions for t 2 [0; 1) are u(t) D 0 and
Existence and Uniqueness of Solutions of Initial Value Problems
1
ˆ D ((1 ˛)t) 1˛ . But these solutions are members of u(t) a large family of solutions. For any c 0, the following functions are solutions for I D R: ( 0; t 0, define I a (t0 ) :D [t0 ; t0 C a] and Bb (u0 ) :D fu 2 Rm j ju u0 j bg and assume that the cylinder S :D I a (t0 ) Bb (u0 ) D. Let the vector field f be a continuous function on the cylinder S. This implies that f is bounded on S, say, M D maxfj f (t; u)j j (t; u) 2 Sg.
b . Then there exDefine the parameter D min a; M ists a solution u(t), with t0 t t0 C , which solves the initial value problem (1).
The theorem can also be phrased for an interval [t0 ; t0 ] or [t0 ; t0 C ]. The parameter gives a time interval such that that the solution (t; u(t)) is guaranteed to be within S. From the integral formulation of the initial value problem (2), it follows that Z t ju(t) u0 j j f (; u())jd M(t t0 ) : t0
To guarantee that u(t) is within S, the condition t t0 is sufficient as it gives M(t t0 ) b and t t0 a. In Fig. 1, this is illustrated in the case D R R.
Existence and Uniqueness of Solutions of Initial Value Problems, Figure 1 The parameter D min( Mb ; a) guarantees that the solution is within the shaded area S, on the left in the case a < if a > Mb
b M
and on the right
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Existence and Uniqueness of Solutions of Initial Value Problems
To prove the Cauchy–Peano existence theorem, a construction which goes back to Cauchy is used. This so-called Cauchy–Euler construction of approximate solutions uses joined line segments which are such that the tangent of each line segment is given by the vector field evaluated at the start of the line segment. To be specific, for any N 2 N, the interval I (t0 ) D [t0 ; t0 C ] is divided in N equal parts with intermediate points t k :D t0 C Nk , k D 1; : : : ; N. The approximate solution is the function u N : I (t0 ) ! R, with u N (t0 ) D u0 ; u N (t) D u N (t k1 ) C f (t k1 ; u N (t k1 )) (t t k1 ) ; (3) t k1 < t t k ;
k D 1; : : : ; N ;
see also Fig. 2. One sees immediately that function uN is continuous and piecewise differentiable and, using the bound on the vector field f , ju N (t) u N (s)j Mjt sj ;
for all t; s 2 I (t0 ) : (4)
This estimate implies that the sequence of functions fu N g is uniformly bounded and that the functions uN are equicontinuous. Indeed, the estimate above gives that for any t 2 I (t0 ) and any N 2 N, ju N (t)j ju N (t0 )j C Mjt t0 j ju0 j C M : And equicontinuity follows immediately from (4) as M does not depend on N. Arzela–Ascoli’s lemma states that a sequence of functions which is uniformly bounded and equicontinuous on a compact set has a subsequence which is uniformly convergent on this compact set [5]. As the sequence fu N g is uniformly bounded and equicontinuous on the com-
pact set I (t0 ), it follows immediately that it has a convergent subsequence. This convergent subsequence is denoted by u Nk and its limit by u. To prove the Cauchy– Peano theorem, we will show that this limit function u satisfies the integral equation (2) and hence is a solution of the initial value problem (1). Proof First it will be shown that if N is sufficiently large, then the functions uN are close to solutions of the differential equation in the following sense: for all " > 0, there is some N0 such that for all N > N0 ju˙ N (t) f (t; u N (t))j < " ; t 2 I (t0 ) ;
t ¤ tk ;
k D 0; : : : ; N :
(5)
Let " > 0. As f is a continuous function on the compact set S, it follows that f is uniform continuous. Thus, there is some ı" such that for all (t; u); (s; v) 2 S with jt sj C ju vj < ı" j f (t; u) f (s; v)j < " :
(6)
m l and let N > N0 . Then for Now define N0 D (MC1)
ın any t 2 I (t0 ), t ¤ t k , k D 0; : : : ; N, we have u˙ N (t) D f (t l ; u N (t l )), where l is such that t l < t < t l C1 . Hence, (6) gives ju˙ N (t) f (t; u N (t))j D j f (t l ; u N (t l )) f (t; u N (t))j < " as (4) shows that jt l tj C ju N (t l ) u N (t)j < (1 C M)jt l tj < (1CM)
(1CM)
ı" . N N0 Next it will be shown that (5) implies that the functions uN almost satisfy the integral equation (2) for N sufficiently large. From (5), it follows that f (t; u N (t)) " < u˙ N (t) < f (t; u N (t)) C " for all N > N0 and all t 2 I (t0 ), except at the special points tk , k D 0; : : : ; N. Hence, for any k D 1; : : : ; N and all t k1 < t t k , this gives Z
t
Z
t k1 t
u N (t) < u N (t k1 ) C u N (t k1 ) C
f (; u N ()) C " d
f (; u N ())d C t k1
" : N
Thus, R t k also for any k D" 1; : : : ; N, u N (t k ) u N (t k1 ) < t k1 f (; u N ())d C N and hence Existence and Uniqueness of Solutions of Initial Value Problems, Figure 2 The approximate solution uN , with N D 5, and the solution u(t). The dotted lines show the tangent given by the vector field f(t; u) at the points (ti ; u) for various values of u and i D 0; : : : ; 5
u N (t k ) u N (t0 ) D
k X u N (t j ) u N (t j1 ) jD1
Z
tk
N0 , the following holds: ˇ ˇ Z t ˇ ˇ ˇ < " ; ˇu N (t) u N (t0 ) f (; u ()d N ˇ ˇ t0
for all t 2 I (t0 ) :
(7)
Thus, the function uN satisfies the integral equation (2) up to an order " error if N is sufficiently large. As the subsequence u N k converges R t uniformly to u and f is continuous, this implies that t0 f (; u N k ()) ! Rt t 0 f (; u())d. Thus, from (7), it can be concluded that u satisfies the integral equation (2) exactly. Note that the full sequence fu N g does not necessarily converge. A counterexample can be found in exercise 12 in Chap. 1 of [5]. It is based on the initial value problem 1 u˙ D juj 4 sgn(u)Ct sin(/t) with u(0) D 0, and shows that on a small interval near t0 D 0, the even approximations are bounded below by a strictly positive constant, while the odd approximations are bounded above by a strictly negative constant. However, if it is known that the solution of the initial value problem is unique, then the full sequence of approximate solutions fu N g must converge to the solution. This can be seen by a contradiction argument. If there is a unique solution, but the sequence fu N g is not convergent, then there is a convergent subsequence and a remaining set of functions which does not converge to the unique solution. But Arzela–Ascoli’s lemma can be applied to the remaining set as well; thus, there exists a convergent subsequence in the remaining set, which must converge to a different solution. This is not possible as the solution is unique. In case of the initial value problem as presented in Example 2, the sequence of Cauchy–Euler approximate solutions fu N g converges to the zero function; hence, no subsequence is required to obtain a solution. As there are many solutions of this initial value problem, this implies that not all solutions of the initial value problem in Example 2 solutions can be approximated by the Cauchy–Euler construction of approximate solutions as defined in (3). As indicated in exercise 2.2 in Chap. 2 of [13], it is possible to get any solution by using appropriate Lipschitz continuous approximations of the vector field f , instead of the
original vector field itself, in the Cauchy–Euler construction. In the example p below this is illustrated for the vector field f (t; u) D juj. p Example 3 Let f : R R be defined as f (t; u) D juj. First we define the approximate functions fˆn : [0; 2] [0; 1) ! [0; 1) as (p u; u > n1 ˆf n (t; u) D p p1 1 n 1 u ; u 1 2 n n n and the definition of the Euler–Cauchy approximate functions is modified to u N (t) D u N (t k1 ) C fˆN (t k1 ; u N (t k1 )) (t t k1 ); t k1 < t t k ;
k D 1; : : : ; N :
This sequence converges to the solution u(t) D 4t 2 for t 2 [0; 2]. Next define the approximate functions f˜n : [0; 2] [0; 1) ! [0; 1) as f˜n (t; u) 8p u; ˆ ˆ ˆ1 1 ˆ ˆ u ˆ < 2n 2n 1 2 D Ca2 2n u ˆ 1 3 ˆ ˆ ˆ ˆ Ca3 n u ; ˆ : u;
t 1; u >
t 1;
1 2n
1 n
;
0 be such that the cylinder S :D I a (t0 ) Bb (u0 ) D. Let the vector field f be a uniformly Lipschitz continuous function on the cylinder S with respect to its second variable u and let M be the upper bound on f , i. e., M D maxf f (t; u) j (t; u) 2 Sg.
Existence and Uniqueness of Solutions of Initial Value Problems
b Define the parameter D min a; M . For t 2 [t0 ; t0 C ], there exists a unique solution u(t) of the initial value problem (1). Successive iterations play an important role in proving existence for more general differential equations. Thus, although the existence of solutions already follows from the Cauchy–Peano theorem, we will prove again existence by using the successive iterations instead of the Cauchy–Euler approximations as the ideas in the proof can be extended to more general differential equations. Proof By using the bound on the vector field f , we will show that if (t; u n (t)) 2 S for all t 2 [t0 ; t0 C 0 ], then (t; u nC1 (t)) 2 S for all t 2 [t0 ; t0 C 0 ]. Indeed, for any t 2 [t0 ; t0 C 0 ], Z t ju nC1 (t) u0 j j f (; u n ())jd
Z
t
ju nC2 (t) u nC1 (t)j
L ju nC1 () u n ()jd t0
L2 ku n u n1 k1
Z
t
( t0 )d t0
D
L2 ku n u n1 k1 (t t0 )2 : 2
Repeating this process, it follows for any k 2 N that Lk ku n u n1 k1 (t t0 ) k k! Lk k ku n u n1 k1 : k!
ju nCkC1 (t) u nCk (t)j
By using the triangle inequality repeatedly this gives for any n; k; 2 N and t 2 [t0 ; t0 C 0 ], ju nCkC1 (t) u n (t)j
t0 Z t
k X
ju nCiC1 (t) u nCi (t)j
iD0
Md M(t t0 ) b ;
(9)
t0
k X L nCi nCi ku1 u0 k1 (n C i)! iD0
thus, ju nC1 (t) u0 j M b and therefore u nC1 (t) 2 B b (u0 ). The boundedness of f also ensures that the functions fu n g are equicontinuous as for any n 2 N and any t0 t1 < t2 C t0 : ju n (t1 ) u n (t2 )j Z t2 j f (; u n ())jd Mjt2 t1 j : t1
Up to this point, we have only used the boundedness of f (which follows from the continuity of f ). But for the next part, in which it will be shown that for any t 2 [t0 ; t0 C
0 ] the sequence fu n (t)g is a Cauchy sequence in Rm , we will need the Lipschitz continuity. Let L be the Lipschitz constant of f on S, then for any n 2 N and t 2 [t0 ; t0 C 0 ], we have Z t ju nC1 (t) u n (t)j j f (; u n ()) f (; u n1 ())jd Z
t0 t
L ju n () u n1 ()jd
t0
Z
t
L ku n u n1 k1
d t0
D L ku n u n1 k1 (t t0 ) ; where ku n u n1 k1 D supfju n () u n1 ()j j t0 t0 C g. This implies that
k X Li i (L )n ku1 u0 k1 n! i! iD0
(L )n b eL : n! Thus, for any t 2 [t0 ; t0 C 0 ], the sequence fu n (t)g is a Cauchy sequence in Rm ; hence, the sequence has a limit, which will be denoted by u(t). In other words, the sequence of functions fu n g is pointwise convergent in I (t0 ). We have already seen that the functions un are equicontinuous; hence, the convergence is uniform and the limit function u(t) is equicontinuous [23]. To see that u satisfies the integral equation (2), observe that for any t 2 [t0 ; t0 C 0 ] ˇ ˇ Z t ˇ ˇ ˇu(t) u0 f (; u())d ˇˇ ˇ t0
ju(t) u nC1 (t)j ˇZ t ˇ ˇ ˇ ˇ Cˇ j f (; u n ()) f (; u())jd ˇˇ t0
ju(t) u nC1 (t)j C Lku u n k1 (t t0 ) (1 C L ) (ku u nC1 k1 C ku u n k1 ) for any n 2 N. As the sequence fu n g is uniformly convergent, this implies that u(t) satisfies the integral equation (2). Finally the uniqueness will be proved using techniques similar to those in “Existence.” Assume that there are two
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Existence and Uniqueness of Solutions of Initial Value Problems
solutions u and v of the integral equation (2). Hence, for any t0 t t0 C , Z t j f (; u()) f (; v())jd ju(t) v(t)j t0
Lku vk1 (t t0 ) : Thus, for k D 1, it holds that ju(t) v(t)j
L k ku vk1 (t t0 ) k ; k! for any t0 t t0 C :
(10)
For the induction step, assume that (10) holds for some k 2 N. Then for any t0 t t0 C , Z t k L ku vk1 ( t0 ) k d ju(t) v(t)j L k! t0 D
L kC1 ku vk1 (t t0 ) kC1 : (k C 1)!
By using the principle of induction, it follows that (10) holds for any k 2 N. This implies that for any t0kC1 satisfy ju(t) v(t)j (L ) t t0 C ,the solutions ju vk1 / (k C 1)! for any k 2 N. Since the expression on the right converges to 0 for k ! 1, this implies that u(t) D v(t) for all t 2 [t0 ; t0 C ] and hence the solution is unique. Implicitly, the proof shows that the integral equation gives rise to a contraction on the space of continuous functions with the supremum norm and this could also have been used to show existence and uniqueness; see [11,25], where Schauder’s fixed-point theorem is used to prove the Picard–Lindelöf theorem. With Schauder’s fixed-point theorem, existence and uniqueness can be shown for much more general differential equations than ordinary differential equations. The iteration process in (8) does not necessarily have to start with the initial condition. It can start with any continuously differentiable function u0 (t) and the interation process will still converge. In specific problems, there are often obvious choices for the initial function u0 which give a faster convergence or are more efficient, for instance, when proving unboundedness. Techniques similar to those in the proof can be used to derive a bound on the difference between the successive approximations and the solution, showing that ju n (t) u(t)j
ML n (t t0 )n ; (n C 1)!
t 2 [t0 ; t0 C ] ;
see [13]. This illustrates that the convergence is fast for t near t0 , but might be slow further away: see Fig. 3 for an example.
Existence and Uniqueness of Solutions of Initial Value Problems, Figure 3 The first four iterations for initial value problem with the vector field f(t; u) D t cos u C sin t and the initial value u(0) D 0. Note that the convergence in the interval [0; 1] is very fast (two steps seem sufficient for graphical purposes), but near t D 2, the convergence has not yet been reached in three steps
Continuous Dependence on Initial Conditions In this section, the vector field f is uniformly Lipschitz continuous on D with respect to its second variable u and its Lipschitz constant is denoted by L. By combining the Picard–Lindelöf theorem (Theorem 4) and the extension theorem (Theorem 3), it follows that for every (t0 ; u0 ) 2 D, there exists a unique solution of (1) passing through (t0 ; u0 ) with a maximal interval of existence (t (t0 ; u0 ); tC (t0 ; u0 )). The trajectory through (t0 ; u0 ) is the set of points (t; u(t)), where u(t) solves (1) and t (t0 ; u0 ) < t < tC (t0 ; u0 ). Now define the set E RnC2 as E :D f (t; t0 ; u0 ) j t (t0 ; u0 ) < t < tC (t0 ; u0 ); (t0 ; u0 ) 2 D g : The flow map ˚ : E ! Rm is a mapping which describes the solution of the initial value problem with the initial condition varying through D, i. e., d ˚(t; t0 ; u0 ) D f (t; ˚ (t; t0 ; u0 )) and dt
Existence and Uniqueness of Solutions of Initial Value Problems
˚(t0 ; t0 ; u0 ) D u0 ;
for (t; t0 ; u0 ) 2 E :
From the integral equation (2), it follows that the flow map satisfies Z
Z
t
(t) K C
()()d ;
for all a t b :
a
Then
t
˚(t; t0 ; u0 ) D u0 C
such that
f (; ˚ (; t0 ; u0 ))d ; t0
Z (t) K exp
()d
Lemma 5 (Gronwall’s Lemma) Let : [a; b] ! R and : [a; b] ! R be nonnegative continuous functions on an interval [a; b]. Let K 0 be some nonnegative constant
;
for all a t b :
a
for (t; t0 ; u0 ) 2 E : Furthermore, the uniqueness implies for the flow map that starting at the point (t0 ; u0 ), flowing to (t; ˚ (t; t0 ; u0 )) is the same as starting at the point (t0 ; u0 ) and flowing to the intermediate point (s; ˚ (s; t0 ; u0 )) and continuing to the point (t; ˚ (t; t0 ; u0 )). This implies that the flow map has a group structure: for any (t0 ; u0 ) 2 D and any s; t 2 (t (t0 ; u0 ); tC (t0 ; u0 )), the flow map satisfies ˚(t; s; ˚ (s; t0 ; u0 )) D ˚(t; t0 ; u0 ); the identity map is given by ˚(t0 ; t0 ; u0 ) D u0 ; and by combining the last two results, it follows that ˚ (t0 ; t; ˚ (t; t0 ; u0 )) D ˚ (t0 ; t0 ; u0 ) D u0 , and hence there is an inverse. See also Fig. 4 for a sketch of the flow map for D R R. To show that the flow map is continuous in all its variables, Gronwall’s lemma [9] will be very useful. There are many versions of Gronwall’s lemma; the one presented here follows [1].
t
Rt Proof Define F(t) D K C a ()()d, for a t b. Then the assumption in the lemma gives (t) F(t) for a t b. Furthermore, F is differentiable, with F˙ D ; hence, for a t b
Z t d ()d F(t) exp dt a
Z t D (t)(t) exp ()d a
Z t F(t) (t) exp ()d 0 a
as (t) F(t). Integrating the left-hand side gives
Z t ()d F(a) D K ; F(t) exp a
which implies for that (t) F(t) K exp
hR
t a
i ()d .
Gronwall’s lemma will be used to show that small variations in the initial conditions give locally small variations in the solution, or Theorem 6 The flow map ˚ : E ! RnC2 is continuous in E.
Existence and Uniqueness of Solutions of Initial Value Problems, Figure 4 The flow map ˚(t; t0 ; u0 ). Note the group property ˚(t; s; ˚(s; t0 ; u0 )) D ˚(t; t0 ; u0 ). The lightly shaded area is the tube Uı1 and the darker shaded area is the tube Uı as used in the proof of Theorem 6. If a solution starts from within Uı , then it will stay within the tube Uı1 for t1 t t2 , as the solution ˚(t; s0 ; v0 ) demonstrates
Proof Let (t; t0 ; u0 ) 2 E. As t; t0 2 (t (t0 ; u0 ); tC (t0 ; u0 )), there is some t1 < t2 such that t; t0 2 (t1 ; t2 ), [t1 ; t2 ]
(t (t0 ; u0 ); tC (t0 ; u0 )) and the solution ˚(s; t0 ; u0 ) exists for any s 2 [t1 ; t2 ]. First we will show that there is some tube around the solution curve f(s; ˚ (s; t0 ; u0 )) j t1 s t2 g such that for any (s0 ; v0 ) in this tube, the solution ˚(s; s0 ; v0 ) exists for t1 s t2 . As D is open, there is some ı1 > 0 such that the closed tube Uı1 :D f(s; v) j j˚(s; t0 ; u0 )vj ı1 ; t1 s t2 g D. As f is Lipschitz continuous, hence continuous, there is some M > 0 such that j f (s; v)j M for all (s; v) 2 Uı1 . Recall that the Lipschitz constant of f on D is denoted by L. Now define ı :D ı1 eL(t2 t1 ) < ı1 and the open tube Uı D f(s; v) j j˚(s; t0 ; u0 ) vj < ı; t1 < s < t2 g ; thus, Uı Uı1 D. Thus, for every (s0 ; v0 ) 2 Uı , the solution ˚(s; s0 ; v0 ) exists for s in some closed interval in
391
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Existence and Uniqueness of Solutions of Initial Value Problems
[t1 ; t2 ] around s0 and the solution in this interval satisfies Z s ˚(s; s0 ; v0 ) D v0 C f (; ˚ (; s0 ; v0 ))d ; s0
see also Fig. 4. Furthermore, for any s 2 [t1 ; t2 ], we have ˚ (s; t0 ; u0 ) D ˚(s; s0 ; ˚ (s0 ; t0 ; u0 )) Z s f (; ˚ (; s0 ; ˚ (s0 ; t0 ; u0 )))d D ˚(s0 ; t0 ; u0 ) C s Z 0s f (; ˚ (; t0 ; u0 ))d : D ˚ (s0 ; t0 ; u0 ) C s0
Subtracting these expressions gives j˚(s; s0 ; v0 ) ˚(s; t0 ; u0 )j jv0 ˚(s0 ; t0 ; u0 )j Z s C j f (; ˚ (; s0 ; v0 )) f (; ˚ (; t0 ; u0 ))j d s0 Z s L j˚(; s0 ; v0 ) ˚(; t0 ; u0 )jd ; ıC s0
so Gronwall’s lemma implies that j˚(s; s0 ; v0 ) ˚ (s; t0 ; u0 )j ıe L(ss 0 ) ı1 . Hence, for any s, ˚ (s; s0 ; v0 ) 2 Uı1 D, and hence this solution can be extended to its maximal interval of existence, which contains the interval [t1 ; t2 ]. So it can be concluded that for any (s0 ; v0 ) 2 Uı , the solution ˚(s; s0 ; v0 ) exists for any s 2 [t1 ; t2 ]. Next we will show continuity of the flow map in its last two variables, i. e., in the initial value. For any (s0 ; v0 ) 2 Uı and t1 t t2 , we have j˚(t; s0 ; v0 ) ˚(t; t0 ; u0 )j jv0 u0 j ˇZ t ˇ Z t ˇ ˇ ˇ C ˇ f (; ˚ (; s0 ; v0 ))d f (; ˚ (; t0 ; u0 ))dˇˇ s0 t0 ˇZ t 0 ˇ ˇ ˇ ˇ j f (; ˚ (; s0 ; v0 ))jd ˇˇ jv0 u0 j C ˇ s0 ˇ ˇZ t ˇ ˇ ˇ j f (; ˚ (; t0 ; u0 )) f (; ˚ (; s0 ; v0 ))jd ˇˇ Cˇ t0
jv0 u0 j C Mjt0 s0 j ˇ ˇZ t ˇ ˇ ˇ Lj˚(; t0 ; u0 ) ˚(; s0 ; v0 )jd ˇˇ : Cˇ t0
Thus, Gronwall’s lemma implies that j˚(t; s0 ; v0 ) ˚(t; t0 ; u0 )j (jv0 u0 j C Mjt0 s0 j) e Ljtt0 j (jv0 u0 j C Mjt0 s0 j) e Ljt2 t1 j ;
and it follows that ˚ is continuous in its last two arguments. The continuity of ˚ in its first argument follows immediately from the fact that the solution of the initial value problem is continuous. If the vector field f is smooth, then the flow map ˚ is smooth as well. Theorem 7 Let f 2 C 1 (D; R m ). Then the flow map ˚ 2 C 1 (E; R m ) and det(Du 0 ˚(t; t0 ; u0 )) Z t D exp tr Du f (; ˚ (; t0 ; u0 )) d t0
for any (t; t0 ; u0 ) 2 E. In this theorem, tr(Du f (; ˚ (; t0 ; u0 ))) stands for the trace of the matrix Du f (; ˚ (; t0 ; u0 )). For second-order linear systems, this identity is known as Abel’s identity and det(Du 0 ˚(t; t0 ; u0 )) is the Wronskian. The proof of Theorem 7 uses the fact that Du 0 ˚(t; t0 ; u0 ) satisfies the linear differential equation d Du ˚(t; t0 ; u0 ) D Du f (t; ˚ (t; t0 ; u0 )) Du 0 ˚(t; t0 ; u0 ); dt 0 with the initial condition Du 0 ˚(t0 ; t0 ; u0 )) D I. This last fact follows immediately from ˚ (t0 ; t0 ; u0 ) D u0 . And the linear differential equation follows by differentiating the differential equation for ˚(t; t0 ; u0 ) with respect to u0 . The full details of the proof can be found, for example, in [5,11]. Extended Concept of Differential Equation Until now, we have looked for continuously differentiable functions u(t), which satisfy the initial value problem (1). But the initial value problem can be phrased for less smooth functions as well. For example, one can define a solution as an absolute continuous function u(t) which satisfies (1). A function u : I ! Rm is absolutely continuous on I if for every positive number ", no matter how small, there is a positive number ı small enough so that whenever a sequence of pairwise disjoint subintervals P [s k ; t k ] of I, k D 1; 2; : : : ; N satisfies N kD1 jt k s k j < ı then n X
d (u(t k ); u(s k )) < " :
kD1
An absolute continuous function has a derivative almost everywhere and is uniformly continuous, thus continuous [24]. Thus, the initial value problem for an absolute continuous function can be stated almost everywhere.
Existence and Uniqueness of Solutions of Initial Value Problems
For the existence of absolute continuous solutions, the continuity condition for the vector field in the Cauchy– Peano theorem (Theorem 1) is replaced by the so-called Carathéodory conditions on the vector field. A function f : D ! Rm satisfies the Carathéodory conditions if [2] f (t; u) is Lebesgue measurable in t for each fixed u; f (t; u) is continuous in u for each fixed t; for each compact set A D, there is a measurable function m A (t) such that j f (t; u)j m A (t) for any (t; u) 2 A. Similar theorems to the ones in the previous sections about existence and uniqueness can be stated in this case. Theorem 8 (Carathéodory) If D is an open set in RnC1 and f : D ! Rm satisfies the Carathéodory conditions on D, then for every (t0 ; u0 ) 2 D there is a solution of the initial value problem (1) through (t0 ; u0 ). Theorem 9 If D is an open set in RnC1 , f : D ! Rm satisfies the Carathéodory conditions on D and u(t) satisfies the initial value problem (1) on some interval, then there exists a maximal open interval of existence. Furthermore, if (t ; tC ) denotes the maximal interval of existence, then the solution u(t) tends to the boundary of D if t # t or t " tC . Theorem 10 If D is an open set in RnC1 , f : D ! Rm satisfies the Carathéodory conditions on D and for every compact set U D there is an integrable function LU (t) such that j f (t; u) f (t; v)j LU juvj;
for all (t; u); (t; v) 2 U ;
then for every (t0 ; u0 ) 2 D, there is a unique solution ˚ (t; t0 ; u0 ) of the initial value problem (1). The domain E of the flow map ˚ is open and ˚ is continuous in E. Proofs of those theorems can be found, for example, in Sect. I.5 in [11]. Many other generalizations are possible, for example, to vector fields which are discontinuous in u. Such vector fields play an important role in control theory. More details can be found in [6,7,17].
similar to the ones for ordinary differential equations as presented in the earlier sections and can be proved by using Schauder’s fixed-point theorem. But not all solutions can be extended as in the extension theorem (Theorem 3); see Chap. 2 in [12]. If one considers differential equations with more variables, i. e., partial differential equations, then there are no straightforward general existence and uniqueness theorems. For partial differential equations, the existence and uniqueness depends very much on the details of the partial differential equation. One possible theorem is the Cauchy– Kowaleskaya theorem, which applies to partial differential 2 equations of the form @m t u D F(t; u; @x u; @ t u; @ tx u; : : : ), where the total number of derivatives of u in F should be less than or equal to m [22]. But there are still many open questions as well. A famous open question is the existence and uniqueness of solutions of the three-dimensional Navier–Stokes equations, a system of equations which describes fluid motion. Existence and uniqueness is known for a fluid in a two-dimensional domain, but not in a threedimensional domain. The question is one of the seven “Millennium Problems,” stated as prize problems at the beginning of the third millennium by the Clay Mathematics Institute [26]. Existence and uniqueness results are also used in dynamical systems. An area of dynamical systems is bifurcation theory and for bifurcation theory the smooth dependence on parameters is crucial. The following theorem gives sufficient conditions for the smooth dependence parameters; see Theorem 3.3 in Chap. 1 of [11]. Theorem 11 If the vector field depends on parameters 2 R k , i. e., f : D R k ! Rm and f 2 C 1 (D R k ; Rm ), then the flow map is continuously differentiable with respect to its parameters . Furthermore, D ˚ satisfies an inhomogeneous linear equation d D ˚(t; t0 ; u0 ; ) dt D Du f (t; ˚ (t; t0 ; u0 ; ); ) D ˚(t; t0 ; u0 ; ) C D f (t; ˚ (t; t0 ; u0 ; ); ) with initial condition D ˚(t0 ; t0 ; u0 ; ) D 0.
Further Directions As follows from the previous sections, the theory of initial value problems for ordinary differential equations is quite well understood, at least for continuous vector fields. For more general differential equations, the situation is quite different. Consider for example a retarded differential equation, hence a differential equation with delay effects [12]. For retarded differential equations, there are local existence and uniqueness theorems which are quite
Results and overviews of bifurcation theory can be found, for example, in [4,8,10,14,16]. Another area of dynamical systems is stability theory. Roughly speaking, a solution is called stable if other solutions which start near this solution stay near it for all time. Note that the local continuity result in Theorem 6 is not a stability result as it only states that nearby solutions will stay nearby for a short time. For stability results, long-time existence of nearby solutions is a crucial property [10,14].
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Bibliography Primary Literature 1. Bellman R (1943) The stability of solutions of linear differential equations. Duke Math J 10:643–647 2. Carathéodory C (1918) Vorlesungen über Reelle Funktionen. Teubner, Leipzig (reprinted: (1948) Chelsea Publishing Company, New York) 3. Cauchy AL (1888) Oevres complètes (1) 6. Gauthiers-Villars, Paris 4. Chow S-N, Hale JK (1982) Methods of bifurcation theory. Springer, New York 5. Coddington EA, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York 6. Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, Dordrecht 7. Flugge-Lotz I (1953) Discontinuous automatic control. Princeton University Press, Princeton 8. Golubitsky M, Stewart I, Schaeffer DG (1985–1988) Singularities and groups in bifurcation theory, vol 1 and 2. Springer, New York 9. Gronwall TH (1919) Note on the derivative with respect to a parameter of the solutions of a system of differential equations. Ann Math 20:292–296 10. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York 11. Hale JK (1969) Ordinary Differential Equations. Wiley, New York 12. Hale JK, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations. Springer, New York 13. Hartman P (1973) Ordinary Differential Equations. Wiley, Baltimore 14. Iooss G, Joseph DD (1980) Elementary stability and bifurcation theory. Springer, New York 15. Kneser H (1923) Ueber die Lösungen eines Systems gewöhnlicher Differentialgleichungen das der Lipschitzschen Bedingung nicht genügt. S-B Preuss Akad Wiss Phys-Math Kl 171– 174 16. Kuznetsov YA (1995) Elements of applied bifurcation analysis. Springer, New York 17. Lee B, Markus L (1967) Optimal Control Theory. Wiley, New York
18. Lindelöf ME (1894) Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre. Comptes rendus hebdomadaires des séances de l’Académie des sciences 114:454–457 19. Müller M (1928) Beweis eines Satzes des Herrn H. Kneser über die Gesamtheit der Lösungen, die ein System gewöhnlicher Differentialgleichungen durch einen Punkt schickt. Math Zeit 28:349–355 20. Peano G (1890) Démonstration de l’integrabilité des équations differentielles ordinaires. Math Ann 37:182–228 21. Picard É (1890) Mémoire sur la théorie de équations aux dérivées partielles et la méthode des approximations successives. J Math, ser 4, 6:145–210 22. Rauch J (1991) Partial Differential Equations. Springer, New York 23. Rudin W (1976) Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York 24. Rudin W (1987) Real and Complex Analysis, 3rd edn. McGrawHill, New York 25. Zeidler E (1995) Applied functional analysis. Vol 1 Applications to mathematical physics. Springer, New York 26. Clay Mathematics Institute (2000) The Millenium Problems: Navier Stokes equation. http://www.claymath.org/ millennium/Navier-Stokes_Equations/
Books and Reviews Arnold VI (1992) Ordinary Differential Equations. Springer, Berlin Arrowsmith DK, Place CM (1990) An Introduction to Dynamical Systems. Cambridge University Press, Cambridge Braun M (1993) Differential Equations and their Applications. Springer, Berlin Brock WA, Malliaris AG (1989) Differential Equations, stability and chaos in Dynamic Economics. Elsevier, Amsterdam Grimshaw R (1993) Nonlinear Ordinary Differential Equations. CRC Press, Boca Raton Ince EL (1927) Ordinary differential equations. Longman, Green, New York Jordan DW, Smith P (1987) Nonlinear Differential Equations. Oxford University Press, Oxford Werner H, Arndt H (1986) Gewöhnliche Differentialgleichungen: eine Einführung in Theorie und Praxis. Springer, Berlin
Finite Dimensional Controllability
Finite Dimensional Controllability LIONEL ROSIER Institut Elie Cartan, Vandoeuvre-lès-Nancy, France Article Outline Glossary Definition of the Subject Introduction Control Systems Linear Systems Linearization Principle High Order Tests Controllability and Observability Controllability and Stabilizability Flatness Future Directions Bibliography Glossary Control system A control system is a dynamical system incorporating a control input designed to achieve a control objective. It is finite dimensional if the phase space (e. g. a vector space or a manifold) is of finite dimension. A continuous-time control system takes the form dx/dt D f (x; u), x 2 X, u 2 U and t 2 R denoting respectively the state, the input, and the continuous time. A discrete-time system assumes the form x kC1 D f (x k ; u k ), where k 2 Z is the discrete time. Open/closed loop A control system is said to be in open loop form when the input u is any function of time, and in closed loop form when the input u is a function of the state only, i. e., it takes the more restrictive form u D h(x(t)), where h : X ! U is a given function called a feedback law. Controllability A control system is controllable if any pair of states may be connected by a trajectory of the system corresponding to an appropriate choice of the control input. Stabilizability A control system is asymptotically stabilizable around an equilibrium point if there exists a feedback law such that the corresponding closed loop system is asymptotically stable at the equilibrium point. Output function An output function is any function of the state. Observability A control system given together with an output function is said to be observable if two different
states give rise to two different outputs for a convenient choice of the input function. Flatness An output function is said to be flat if the state and the input can be expressed as functions of the output and of a finite number of its derivatives. Definition of the Subject A control system is controllable if any state can be steered to another one in the phase space by an appropriate choice of the control input. While the stabilization issue has been addressed since the XIXth century (Watt’s steam engine governor providing a famous instance of a stabilization mechanism at the beginning of the English Industrial Revolution), the controllability issue has been addressed for the first time by Kalman in the 1960s [20]. The controllability is a basic mathematical property which characterizes the degrees of freedom available when we try to control a system. It is strongly connected to other control concepts: optimal control, observability, and stabilizability. While the controllability of linear finite-dimensional systems is well understood since Kalman’s seminal papers, the situation is more tricky for nonlinear systems. For the later, concepts borrowed from differential geometry (e. g. Lie brackets, holonomy, . . . ) come into play, and the study of their controllability is still a field of active research. The controllability of finite dimensional systems is a basic concept in control theory, as well as a notion involved in many applications, such as spatial dynamics (with e. g. spatial rendezvous), airplane autopilot, industrial robots, quantic chemistry. Introduction A very familiar example of a controllable finite dimensional system is given by a car that one attempts to park at some place in a parking. The phase space is roughly the three dimensional Euclidean space R3 , a state being composed of the two coordinates of the center of mass together with the angle formed by some axis linked to the car with the (fixed) abscissa axis. The driver may act on the angle of the wheels and on their velocity, which may thus be taken as control inputs. In general, the presence of obstacles (e. g. other cars) impose to change the phase space R3 to a subset of it. The controllability issue is roughly how to combine changes of direction and of velocity to drive the car from a position to another one. Note that the system is controllable, even if the number of control inputs (2) is less than the number of independent coordinates (3). This is an important property resting upon the many connections between the coordinates of the state. While in nature each motion is generally controlled by an input (think of
395
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Finite Dimensional Controllability
the muscles in an arm), the control theory focuses on the study of systems in which an input living in a space of low dimension (typically, one) is sufficient to control the coordinates of a state living in a space of high dimension. The article is outlined as follows. In Sect. “Control Systems”, we introduce the basic concepts (controllability, stabilizability) used thereafter. In the next section, we review the linear theory, recalling the Kalman and Hautus tests for the controllability of a time invariant system, and the Gramian test for a time dependent system. Section “Linearization Principle” is devoted to the linearization principle, which allows to deduce the controllability of a nonlinear system from the controllability of its linearization along a trajectory. The focus in Sect. “High Order Tests” is on nonlinear systems for which the linear test fails, i. e., the linearized system fails to be controllable. High order conditions based upon Lie brackets ensuring controllability will be given, first for systems without drift, and next for systems with a drift. Section “Controllability and Observability” explores the connections between controllability and observability, while Sect. “Controllability and Stabilizability” shows how to derive stabilization results from the controllability property. A final section on the flatness, a new theory used in many applications to design explicit control inputs, is followed by some thoughts on future directions. Control Systems A finite dimensional (continuous-time) control system is a differential equation of the form x˙ D f (x; u)
(1)
where x 2 X is the state, u 2 U is the input, f : X U ! U is a smooth (typically real analytic) nonlinear function, x˙ D dx/dt, and X and U denote finite dimensional manifolds. For the sake of simplicity, we shall assume here that X Rn and U Rm are open sets. Sometimes, we impose U to be bounded (or to be a compact set) to force the control input to be bounded. Given some control input u 2 L1 (I; U), i. e. a measurable essentially bounded function u : I ! U, a solution of (1) is a locally Lipschitz continuous function x() : J ! X, where J I, such that x˙ (t) D f (x(t); u(t))
for almost every t 2 J :
(2)
Note that J I only, that is, x needs not exist on all I, as it may escape to infinity in finite time. In general, u is piecewise smooth, so that (2) holds actually for all t except for finitely many values. The basic problem of the controllability is the issue whether, given an initial state x0 2 X,
a terminal state x T 2 X, and a control time T > 0, one may design a control input u 2 L1 ([0; T]; U) such that the solution of the system (
x˙ (t) D f (x(t) ; u(t)) x(0) Dx0
(3)
satisfies x(T) D x T . An equilibrium position for (1) is a point x 2 X such that there exists a value u 2 U (typically, 0) such that f (x; u) D 0. An asymptotically stabilizing feedback law is a function k : X ! U with k(x) D u, such that the closed loop system x˙ D f (x; k(x))
(4)
obtained by plugging the control input u(t) :D k(x(t))
(5)
into (1), is locally asymptotically stable at x. Recall that it means that (i) the equilibrium point is stable: for any " > 0, there exists some ı > 0 such that any solution of (4) starting from a point x0 such that jx0 xj < ı at t D 0, is defined on RC and satisfies jx(t)j " for all t 0; (ii) the equilibrium point is attractive. For some ı > 0 as in (i), we have also that x(t) ! 0 as t ! 1 whenever jx0 j < ı. The feedback laws considered here will be continuous, and we shall mean by a solution of (4) any function x(t) satisfying (4) for all t. Notice that the solutions of the Cauchy problems exist (locally in time) by virtue of Peano theorem. A control system is asymptotically stabilizable around an equilibrium position if an asymptotically stabilizing feedback law as above does exist. In the following, we shall also consider time-varying systems, i. e. systems of the form x˙ D f (x; t; u)
(6)
where f : X I U ! X is smooth and I R denotes some interval. The controllability and stabilizability concepts extend in a natural way to that setting. Time-varying feedback laws u(t) D k(x(t); t)
(7)
where k : X R ! U is a smooth (generally time periodic) function, prove to be useful in situations where the classical static stabilization defined above fails.
Finite Dimensional Controllability
and let r D rank R(A; B) < n. It may be seen that the reachable space from the origin, that is the set ( R D x T 2 Rn ; 9T > 0; 9u 2 L2 [0; T] ; Rn ;
Linear Systems Time Invariant Linear Systems A time invariant linear system is a system of the form x˙ D Ax C Bu
(8)
B 2 denote some time invariwhere A 2 ant matrices. This corresponds to the situation where the function f in (1) is linear in both the state and the input. Here, x 2 Rn and u 2 Rm , and we consider square integrable inputs u 2 L2 ([0; T]; Rm ). We say that (8) is controllable in time T if for any pair of states x0 ; x T 2 Rn , one may construct an input u 2 L2 ([0; T]; Rm ) that steers (8) from x0 to xT . Recall that the solution of (8) emanating from x0 at t D 0 is given by Duhamel formula x(t) D e tA x0 C
Z
t
e(ts)A Bu(s) ds :
0
Let us introduce the nnm matrix R(A; B) D (BjABjA2 Bj jAn1 B) obtained by gathering together the matrices B; AB; A2 B; : : : ; An1 B. Then we have the following rank condition due to Kalman [20] for the controllability of a linear system. Theorem 1 (Kalman) The linear system (8) is controllable in time T if and only if rank R(A; B) D n. We notice that the controllability of a linear system does not depend of the control time T, and that the control input may actually be chosen very smooth (e. g. in C 1 ([0; T]; Rn )). Example 1 Consider a pendulum to which is applied a torque as a control input. A simplified model is then given by the following linear system x˙1 D x2 x˙2 D x1 C u
Z
(9) (10)
where x1 ; x2 and u stand respectively for the angle with the vertical, the angular velocity, and the torque. Here, n D 2, m D 1, and 0 1 0 AD and B D : (11) 1 0 1 As rank(B; AB) D 2, we infer from Kalman rank test that (9)–(10) is controllable. When (8) fails to be controllable, it may be important (e. g. when studying the stabilizability) to identify the uncontrollable part of (8). Assume that (8) is not controllable,
)
T
(Ts)A
e
Rnm
Rnn ,
Bu(s) ds D x T
(12) ;
0
coincides with the space spanned by the columns of the matrix R(A; B): R D fR(A; B)V ; V 2 Rnm g
(13)
In particular, dim R D rank R(A; B) D r < n. Let e D fe1 ; : : : ; e n g be the canonical basis of Rn , and let f f1 ; : : : ; f r g be a basis of R, that we complete in a basis f D ˜ denotes the vector of the f f1 ; : : : ; f n g of Rn . If x (resp. x) coordinates of a point in the basis e (resp. f), then x D T x˜ where T D ( f1 ; f2 ; : : : ; f n ) 2 Rnn . In the new coordi˜ (8) may be written nates x, ˜x C Bu ˜ x˙˜ D A˜
(14)
where A˜ :D T 1 AT and B˜ :D T 1 B read A˜1 A˜ 2 B˜ 1 ˜D A˜ D B 0 A˜ 3 0
(15)
with A˜ 1 2 Rrr , B˜ 1 2 Rrm . Writing x˜ D (x˜1 ; x˜2 ) 2 Rr Rnr , we have x˙˜ 1 D A˜ 1 x˜1 C A˜ 2 x˜2 C B˜ 1 u
(16)
x˙˜ 2 D A˜ 3 x˜2
(17)
and it may be proved that rank R(A˜1 ; B˜ 1 ) D r :
(18)
This is the Kalman controllability decomposition. By (18), the dynamics of x˜1 is well controlled. Actually, a solution of (18) evaluated at t D T assumes the form Z T ˜ ˜ x˜1 (T) D e T A 1 x˜1 (0) C e(Ts)A1 A˜ 2 x˜2 (s) ds 0
Z
T
C ˜
˜
e(Ts)A1 B˜ 1 u(s)ds
0
D eT A1 x 1 C
Z
T
˜
e(Ts)A1 B˜ 1 u(s)ds
0
RT ˜ if we set x 1 D x˜1 (0) C 0 es A 1 A˜ 2 x˜2 (s)ds. Hence x˜1 (T) r may be given any value in R . On the other hand, no control input is present in (17). Thus x˜2 stands for the uncontrolled part of the dynamics of (8).
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Finite Dimensional Controllability
Another test based upon a spectral analysis has been furnished by Hautus in [16]. Theorem 2 (Hautus) The control system (8) is controllable in time T if and only if rank(I A; B) D n for all 2 C. Notice that in Hautus test we may restrict ourselves to the complex numbers which are eigenvalues of A, for otherwise rank(I A) D n.
Note that the Gramian test provides a third criterion to test whether a time invariant linear system (8) is controllable or not. Corollary 4 (8) is controllable in time T if and only if the Gramian Z T GD e(Tt)A BB e(Tt)A dt 0
is invertible. Time-Varying Linear Systems Let us now turn to the controllability issue for a time-varying linear system x˙ (t) D A(t)x(t) C B(t)u(t)
(19)
where A 2 L1 ([0; T] ; Rnn ), B 2 L1 ([0; T] ; Rnm ) denote time-varying matrices. Such a system arises in a natural way when linearizing a control system (1) along a trajectory. The input u(t) is any function in L1 ([0; T] ; Rm ) and a solution of (19) is any locally Lipschitz continuous function satisfying (19) almost everywhere. We define the fundamental solution associated with A as follows. Pick any s 2 [0; T], and let M : [0; T] ! Rnn denote the solution of the system ˙ M(t) D A(t)M(t)
(20)
M(s) D I
Z
t
x(t) D (t; t0 )x0 C
(t; s)B(s)u(s)ds : t0
The controllability Gramian of (20) is the matrix T
(T; t)B(t)B (t) (T; t)dt
GD
u(t) D B (t) (T; t)G 1 (x T (T; 0)x0 ) :
(21)
A remarkable property of the control input u is that u minimizes the control cost Z T ju(t)j2 dt E(u) D 0
among all the control inputs u 2 L1 ([0; T] ; Rm ) (or u 2 L2 ([0; T]; Rm )) steering (19) from x0 to xT . Actually a little more can be said. Proposition 5 If u 2 L2 ([0; T] ; Rm ) is such that the solution x of (19) emanating from x0 at t D 0 reaches xT at time T, and if u ¤ u, then E(u) < E(u):
Then (t; s) :D M(t). Notice that (t; s) D e(ts)A when A is constant. The solution x of (20) starting from x0 at time t0 reads then
Z
As the value of the control time T plays no role according to Kalman test, it follows that the Gramian G is invertible for all T > 0 whenever it is invertible for one T > 0. If (19) is controllable, then an explicit control input steering (19) from x0 to xT is given by
0
where denotes transpose. Note that G 2 Rnn , and that G is a nonnegative symmetric matrix. Then the following result holds. Theorem 3 (Gramian test) The system (19) is controllable on [0; T] if and only if the Gramian G is invertible.
The main drawback of the Gramian test is that the knowledge of the fundamental solution (t; s) and the computation of an integral term are both required. In the situation where A(t) and B(t) are smooth functions of time, a criterion based only upon derivatives in time is also available. Assume that A 2 C 1 ([0; T] ; Rnn ) and that B 2 C 1 ([0; T] ; Rnm ), and define a sequence of functions B i 2 C 1 ([0; T] ; Rnm ) by induction on i by B0 D B B i D AB i1
dB i1 : dt
(22)
Then the following result holds (see, e. g., [11]). Theorem 6 Assume that there exists a time t 2 [0; T] such that span(B i (t)v ; v 2 Rm ; i 0) D Rn : Then (19) is controllable.
(23)
Finite Dimensional Controllability
The converse of Theorem 6 is true when A and B are real analytic functions of time. More precisely, we have the following result.
Hautus test admits the following extension due to Liu [25]: (24) is (exactly) controllable in time T if and only if there exists some constant ı > 0 such that
Theorem 7 If A and B are real analytic on [0; T], then (20) is controllable if and only if for all t 2 [0; T] span(B i (t)v ; v 2 R ; i 0) D R : m
jj(A I)zjj2 C jjB zjj2 ıjjzjj2 8z 2 D(A ) ; 8 2 C :
(25)
n
Clearly, Theorem 7 is not valid when A and B are merely of class C 1 . (Take n D m D 1, A(t) D 0, B(t) D exp(t 1 ) and t D 0.) Linear Control Systems in Infinite Dimension Let us end this section with some comments concerning the extensions of the above controllability tests to control systems in infinite dimension (see [30] for more details). Let us consider a control system of the form x˙ D Ax C Bu
(24)
where A: D(A) X ! X is an (unbounded) operator generating a strongly continuous semigroup (S(t)) t0 on a (complex) Hilbert space X, and B : U ! X is a bounded operator, U denoting another Hilbert space. Definition 8 We shall say that (24) is Exactly controllable in time T if for any x0 ; x T 2 X there exists u 2 L2 ([0; T]; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies x(T) D x T ; Null controllable in time T if for any x0 2 X there exists u 2 L2 ([0; T] ; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies x(T) D 0; Approximatively controllable in time T if for any x0 ; x T 2 X and any " > 0 there exists u 2 L2 ([0; T]; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies jjx(T) x T jj < " (jj jj denoting the norm in X). This setting is convenient for a partial differential equation with an internal control Bu :D gu(t), where g D g(x) is such that gU X. Let us review the above controllability tests. Kalman rank test, which is based on a computation of the dimension of the reachable space, possesses some extension giving the approximate (not exact!) controllability of (24) (See [13], Theorem 3.16). More interestingly, a Kalman-type condition has been introduced in [23] to investigate the null controllability of a system of coupled parabolic equations.
In (25), A (resp. B ) denotes the adjoint of the operator A (resp. B), and jj jj denotes the norm in X. The Gramian test admits the following extension, due to Dolecky–Russell [14] and J.L. Lions [24]: (24) is exactly controllable in time T if and only if there exists some constant ı > 0 such that Z T jjB S (t)x0 jj2 dt ıjjx0jj2 8x0 2 X : 0
Linearization Principle Assume given a smooth nonlinear control system x˙ D f (x; u)
(26)
where f : X U ! Rn is a smooth map (i. e. of class C 1 ) and X Rn , U Rm denote some open sets. Assume also given a reference trajectory (x; u) : [0; T] ! X U where u 2 L1 ([0; T] ; U) is the control input and x solves (26) for u u. We introduce the linearized system along the reference trajectory defined as y˙ D A(t)y C B(t)v
(27)
where @f (x(t); u(t)) 2 Rnn ; @x @f (x(t); u(t)) 2 Rnm B(t) D @u
A(t) D
(28)
and y 2 Rn , v 2 Rm . (27) is formally derived from (26) by letting x D x C y, u D u C v and observing that y˙ D x˙ x˙ D f (x C y; u C v) f (x; u)
@f @f (x; u)y C (x; u)v : @x @u
Notice that if (x 0 ; u0 ) is an equilibrium point of f (i. e. f (x 0 ; u 0 ) D 0) and x(t) x0 , u(t) u0 , then A(t) D @f @f A D @x (x 0 ; u 0 ) and B(t) D B D @u (x 0 ; u 0 ) take constant values. Equation (27) is in that case the time invariant linear system y˙ D Ay C Bv :
(29)
399
400
Finite Dimensional Controllability
Let x0 ; x T 2 X. We seek for a trajectory x of (26) connecting x0 to xT when x0 (resp. xT ) is close to x(0) (resp. x(T)). In addition, we will impose that the trajectory (x, u) be uniformly close to the reference trajectory (x; u). We are led to the following Definition 9 The system (26) is said to be controllable along (x; u) if for each " > 0, there exists some ı > 0 such that for each x0 ; x T 2 X with jjx0 x(0)jj < ı, jjx T x(T)jj < ı, there exists a control input u 2 L1 ([0; T] ; U) such that the solution of (26) starting from x0 at t D 0 satisfies x(T) D x T and sup jx(t) x(t)j C ju(t) u(t)j " :
tory such that the linearization of (30)–(32) along it is controllable. Pick any time T > 0 and let u 1 (t) D cos( t/T), u 2 (t) D 0 for t 2 [0; T] and x0 D 0. Let x denote the corresponding solution of (30)–(32). Notice that x 1 (t) D (T/) sin( t/T) (hence x 1 (T) D 0), and x 2 D x 3 0. The linearization of (30)–(32) along (x; u) is (27) with 0 1 0 0 0 A(t) D @ 0 0 0 A; 0 cos( t/T) 0 (33) 0 1 1 0 A: B(t) D @ 0 1 0 (T/) sin( t/T)
t2[0;T]
We are in a position to state the linearization principle. Theorem 10 Let (x; u) be a trajectory of (26). If the linearized system (27) along (x; u) is controllable, then the system (26) is controllable along the trajectory (x; u). When the reference trajectory is stationary, we obtain the following result. Corollary 11 Let (x0 ; u0 ) be such that f (x0 ; u0 ) D 0. If the linearized system (29) is controllable, then the system (26) is controllable along the stationary trajectory (x; u) D (x0 ; u0 ). Notice that the converse of Corollary 11 is not true. (Consider the system x˙ D u3 with n D m D 1, x0 D u0 D 0, which is controllable at the origin as it may be seen by performing the change of inputs v D u3 .) Often, the nonlinear part of f plays a crucial role in the controllability of (26). This will be explained in the next section using Lie algebraic techniques. Another way to use the nonlinear contribution in f is to consider a linearization of (26) along a convenient (not stationary) trajectory. We consider a system introduced by Brockett in [6] to exhibit an obstruction to stabilizability. Example 2 The Brockett’s system reads
Notice that A and B are real analytic, so that we may apply Theorem 8 to check whether (27) is controllable or not on [0; T]. Simple computations give 0 1 0 0 A: B1 (t) D @ 0 0 0 2 cos( t/T) Clearly, span B0 (0)
1 0
; B0 (0)
0 1
; B1 (0)
0 1
!
D R3
hence (27) is controllable. We infer from Theorem 10 that (30)–(32) is controllable along (x; u). Notice that if we want to prove the controllability around an equilibrium point as above for Brockett’s system, we have to design a reference control input u so that x0 D x(0) D x(T) :
(34)
When f is odd with respect to the control, i. e. f (x; u) D f (x; u) ;
(35)
then (34) is automatically satisfied whenever u fulfills
x˙1 D u1
(30)
x˙2 D u2
(31)
x˙3 D x1 u2 x2 u1 :
(32)
u(t) D u(T t) 8t 2 [0; T] :
(36)
Indeed, it follows from (35)–(36) that the solution x to (26) starting from x0 at t D 0 satisfies
Its linearization along (x0 ; u0 ) D (0; 0), which reads y˙1 D v1 y˙2 D v2 y˙3 D 0 ; is not controllable, by virtue of the Kalman rank condition. We may however construct a smooth closed trajec-
x(t) D x(T t) 8t 2 [0; T] :
(37)
In particular, x(T) D x(0) D x0 . Of course, the control inputs of interest are those for which the linearized system (27) is controllable, and the latter property is “generically” satisfied for a controllable system. The above construction of the reference trajectory is due to J.-M. Coron,
Finite Dimensional Controllability
and is referred to as the return method. A precursor to that method is [34]. Beside giving interesting results for the stabilization of finite dimensional systems (see below Sect. “Controllability and Stabilizability”), the return method has also been successfully applied for the control of some important partial differential equations arising in Fluid Mechanics (see [11]). High Order Tests In this section, we shall derive new controllability tests for systems for which the linearization principle is inconclusive; that is, the linearization at an equilibrium point fails to be controllable. To simplify the exposition, we shall limit ourselves to systems affine in the control, i. e. systems of the form x˙ D f0 (x) C u1 f1 (x) C C u m f m (x) ; x 2 Rn ; juj1 ı
(38)
where juj1 :D sup1im ju i j and ı > 0 denotes a fixed number. We assume that f0 (0) D 0 and that f i 2 C 1 (Rn ; Rn ) for each i 2 f0; : : : ; mg. To state the results we need to introduce a few notations. Let v D (v1 ; : : : ; v n ) and w D (w1 ; : : : ; w n ) be two vector fields of class C 1 on Rn . The Lie bracket of v and w, denoted by [v; w], is the vector field [v; w] D
@v @w v w @x @x
where @w/@x is the Jacobian matrix (@w i /@x j ) i; jD1;:::;n , and the vector v(x) !D (v1 (x); : : : ; v n (x)) is identified v 1 (x)
to the column
: : : v n (x)
. As [v; w] is still a smooth vector
field, we may bracket it with v, or w, etc. Vector fields like [v; [v; w]], [[v; w]; [v; [v; w]]], etc. are termed iterated Lie brackets of v, w. The Lie bracketing of vector fields is an operation satisfying the two following properties (easy to check) 1. Anticommutativity: [w; v] D [v; w]; 2. Jacobi identity: [ f ; [g; h]]C[g; [h; f ]]C[h; [ f ; g]] D 0. The Lie algebra generated by f 1 , . . . ,f m , denoted Lie( f1 ; : : : ; f m ), is the smallest vector subspace v of C 1 (Rn ; Rn ) which contains f 1 , . . . ,f m and which is closed under Lie bracketing (i. e. v; w 2 V ) [v; w] 2 V ). It is easily to see that Lie( f1 ; : : : ; f m ) is the vector space spanned by all the iterated Lie brackets of f 1 ,. . . ,f m . (Actually, using the anticommutativity and Jacobi identity, we may restrict ourselves to iterated Lie brackets of the form
[ f i 1 ; [ f i 2 ; [: : : [ f i p1 ; f i p ] : : : ]]] to span Lie( f1 ; : : : ; f m ).) For any x 2 Rn , we set Lie( f1 ; : : : ; f m )(x) D fg(x); g 2 Lie( f1 ; : : : ; f m )g Rn : Example 3 Let us consider the system 0
1 0 1 1 0 x˙ D u1 f1 (x) C u2 f2 (x) D u1 @ 0 A C u2 @ 1 A (39) 0 x1 where x D (x1 ; x2 ; x3 ) 2 R3 . Clearly, the linearized system at (x0 ; u0 ) D (0; 0), which reduces to x˙ D u1 f1 (0) C u2 f2 (0), is not controllable. However, we shall see later that the system (39) is controllable. First, each point (˙"; 0; 0) (resp. (0; ˙"; 0)) may be reached from the origin by letting u(t) D (˙1; 0) (resp. u(t) D (0; ˙1)) on the time interval [0; "]. More interestingly, any point (0; 0; "2 ) may also be reached from the origin in a time T D O("), even though (0; 0; 1) 62 span( f1 (0); f2 (0)). To prove the last claim, let us introduce for i D 1; 2 the flow map it defined by it (x0 ) D x(t), where x() solves x˙ D f i (x) ;
x(0) D x0 :
Then, it is easy to see that 2" 1" 2" 1" (0) D (0; 0; "2 ) : It means that the control 8 (1; 0) if ˆ ˆ < (0; 1) if u(t) D (1; 0) if ˆ ˆ : (0; 1) if
0t 0 and any x0 ; x T 2 Rn , there exists a control u 2 L1 ([0; T] ; Rm ) such that the solution x(t) of (41) emanating from x0 at t D 0 reaches xT at time t D T. Note that the controllability may be supplemented with the words: local, with small controls, in small time, in large time, etc. An important instance, the small-time local controllability, is introduced below in Definition 14. The following result has been obtained by Rashevski [28] and Chow [8]. Theorem 13 (Rashevski–Chow) Assume that f i 2 C 1 (Rn ; Rn ) for all i 2 f1; : : : ; mg. If Lie( f1 ; : : : ; f m )(x) D Rn
8x 2 Rn ;
(43)
then (41) is controllable. Example 3 continued (39) is controllable, since [ f1 ; f2 ] D (0; 0; 1) gives span( f1 (x); f2 (x); [ f1 ; f2 ](x)) D R3
(45)
8x 2 R3 : (44)
˚
A (T) D x(T; u); u(t) 2 U 8t 2 [0; T] :
Definition 14 We shall say that the control system (45) is accessible from the origin if the interior of A (T) is nonempty for any T > 0; small time locally controllable (STLC) at the origin if for any T > 0 there exists a number ı > 0 such that for any pair of states (x0 ; x T ) with jx0 j < ı; jx T j < ı, there exists a control input u 2 L1 ([0; T] ; U ) such that the solution x of x˙ D f0 (x)Cu1 f1 (x)C Cu m f m (x);
x(0) D x0 (46)
satisfies x(T) D x T . (Notice that in the definition of the small time local controllability, certain authors assume x0 D 0 in above definition, or require the existence of ı > 0 for any T > 0 and any > 0.) The accessibility property turns out to be easy to characterize. As a STLC system is clearly accessible, the accessibility property is often considered as the first property to test before investigating the controllability of a given affine system. The following result provides an accessibility test based upon the rank at the origin of the Lie algebra generated by all the vectors fields involved in the control system.
Finite Dimensional Controllability
Theorem 15 (Hermann–Nagano) If dim Lie( f0 ; f1 ; : : : ; f m )(0) D n
(47)
then the system (45) is accessible from the origin. Conversely, if (45) is accessible from the origin and the vector fields f 0 , f 1 , . . . , f m are real analytic, then (47) has to be satisfied. Example 4 Pick any k 2 N and consider as in [22] the system
x˙1 D u; juj 1 ; (48) x˙2 D (x1 ) k so that n D 2, m D 1, and f0 (x) D (0; (x1 ) k ), f1 (x) D (1; 0). Setting
h D [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f3 ]] : Finally, we set X (h) D h : 2S m
With h ˚ as in (49), and S3 D idf1;2;3g ; (1 2 3); (1 3 2); (1 2); (1 3); (2 3) , we obtain (h) D [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f2 ]] C [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f3 ]]
(ad f1 ; f0 ) D [ f1 ; f0 ] (ad iC1 f1 ; f0 ) D [ f1 ; (ad i f1 ; f0 )]
Let Sm denote the usual symmetric group, i. e. Sm is the group of all the permutations of the set f1; : : : ; mg. If 2 S m and h is an iterated Lie bracket of f0 ; f1 ; : : : ; f m , we denote by h the iterated Lie bracket obtained by replacing, in the definition of h, f i by f (i) for each i 2 f1; : : : ; mg. For instance, if D (1 2 3) and h is as in (49), then
C [[ f3 ; [ f3 ; f0 ]]; [ f3 ; f1 ]] C [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f1 ]]
8i 1
C [[ f3 ; [ f3 ; f0 ]]; [ f3 ; f2 ]] C [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f3 ]] :
we obtain at once that
We need the following
k
(ad f1 ; f0 )(x) D (0; k !)
Definition 16 Let 2 [0; 1]. We say that the system (45) satisfies the condition S( ) if, for any iterated Lie bracket h with ı0 (h) odd and ı i (h) even for all i 2 f1; : : : ; mg, the vector (h)(0) belongs to the vector space spanned by the vectors g(0) where g is an iterated Lie bracket satisfying
hence Lie( f0 ; f1 )(0) D R2 : It follows from Theorem 15 that (48) is accessible from the origin. On the other hand, it is clear that (48) is not STLC at the origin when k is even, since x˙2 D (x1 ) k 0, hence x2 is nondecreasing. Using a controllability test due to Hermes [17], it may be shown that (48) is STLC at the origin if and only if k is odd. Note that the linearized system at the origin fails to be controllable whenever k 2. Let us consider some affine system (45) which is accessible from the origin. We exclude the trivial situation when the linearization principle may be applied, and seek for a Lie algebraic condition ensuring that (45) is STLC. If h is a given iterated Lie bracket of f0 ; f1 ; : : : ; f m , we let ı i (h) denote the number of occurrences of f i in the definition of h. For instance, if m D 3 and h D [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f2 ]]
(49)
then ı0 (h) D 1 ; ı1 (h) D 3 ; ı2 (h) D 1 ; ı3 (h) D 0 : Notice that the fields f0 ; f1 ; : : : ; f m are considered as indeterminates when computing the ı i (h)’s, their effective values as vector fields being ignored.
ı0 (g) C
m X
ı i (g) < ı0 (h) C
iD1
m X
ı i (h) :
iD1
Then we have the following result proved in [35]. Theorem 17 (Sussmann) If the condition S( ) is satisfied for some 2 [0; 1], then the system (45) is small time locally controllable at the origin. When D 0, Sussmann’s theorem is nothing else than Hermes’ theorem. Sussmann’s theorem, which is in itself very useful in Engineering (see, e. g., [5]), has been extended in [1,4,18]. Controllability and Observability Assume given a control system x˙ D f (x; u)
(50)
together with an output function y D h(x)
(51)
where f : X U ! Rn and h : X ! Y are smooth functions, and X Rn , U Rm and Y R p denote some
403
404
Finite Dimensional Controllability
open sets. y typically stands for a (partial) measurement of the state, e. g. the p first coordinates, where p < n. Often, only y is available, and for the stabilization of (50) we should consider an output feedback law of the form u D k(y). We shall say that (50)–(51) is observable on the interval [0; T] if, for any pair x0 ; x˜0 of points in X, one may find a control input u 2 L1 ([0; T] ; U) such that if x (resp. x˜ ) denotes the solution of (50) emanating from x0 (resp. x˜0 ) at time t D 0, and y (resp. y˜) denotes the corresponding output function, we have y(t) ¤ y˜(t) for some t 2 [0; T] :
(52)
For a time invariant linear control system and a linear output function x˙ D Ax C Bu
(53)
y D Cx
(54)
with A 2 Rnn , B 2 Rnm , C 2 R pn , the output is found to be Z t y(t) D Cx(t) D Ce tA x0 C C e(ts)A Bu(s)ds : (55) 0
In particular, y(t) y˜(t) D Ce tA (x0 x˜0 )
Notice that the observability of the adjoint system is easily shown to be equivalent to the existence of a constant ı > 0 such that Z T jjB e tA x0 jj2 dt ıjjx0 jj2 8x0 2 Rn : (59) 0
As it has been pointed out in Sect. “Linear Systems”, the equivalence between the controllability of a system and the observability of its adjoint, expressed in the form (59), is still true in infinite dimension, and provides a very useful way to investigate the controllability of a partial differential equation. Finally, using the duality principle, we see that any controllability test gives rise to an observability test, and vice-versa. Controllability and Stabilizability In this section we shall explore the connections between the controllability and the stabilizability of a system. Let us begin with a linear control system x˙ D Ax C Bu :
(56)
and B does not play any role. Therefore, (53)–(54) is observable in time T if and only if the only state x 2 Rn such that Ce tA x D 0 for any t 2 [0; T] is x D 0. Differentiating in time and applying Cayley–Hamilton theorem, we obtain the following result. Theorem 18 System (53)–(54) is observable in time T if and only if rank O(A; C) D n, where the observability matrix O(A; C) 2 Rn pn is defined by 1 0 C B CA C C B O(A; C) D B C :: A @ : n1 CA Noticing that R(A; B) D O(A ; B )
and introducing the adjoint system (57)
y˜ D B x˜
(58)
(60)
Performing a linear change of coordinates if needed, we may assume that (60) has the block structure given by the Kalman controllability decomposition x˙1 A1 A2 x1 B1 u (61) D C x˙2 0 A3 x2 0 where A1 2 Rrr , B1 2 Rrm , and where rank R (A1 ; B1 ) D r. For any square matrix M 2 C nn we denote by (M) its spectrum, i. e. (M) D f 2 C ; det(M I) D 0g : Let us note C :D f 2 C ; Re < 0g. Then the asymptotic stabilizability of (60) may be characterized as follows. Theorem 20 There exists a (continuous) asymptotically stabilizing feedback law u D k(x) with k(0) D 0 for (60) if and only if (A3 ) C . If it is the case, then for any family S D ( i )1ir of elements of C invariant by conjugation, there exists a linear asymptotically stabilizing feedback law u(x) D Kx, with K 2 Rmn , such that (A C BK) D (A3 ) [ S :
x˙˜ D A x˜
we arrive to the following duality principle.
Theorem 19 A time invariant linear system is controllable if and only if its adjoint system is observable.
(62)
The property (62), which shows to what extend the spectrum of the closed loop system can be assigned, is referred to as the pole shifting theorem.
Finite Dimensional Controllability
As a direct consequence of Theorem 20, we obtain the following result. Corollary 21 A time invariant linear system which is controllable is also asymptotically stabilizable. It is natural to ask whether Corollary 21 is still true for a nonlinear system, i. e. if a controllable system is necessarily asymptotically stabilizable. A general result cannot be obtained, as is shown by Brockett’s system. Example 2 continued Brockett’s system (30)–(32) may be written x˙ D u1 f1 (x) C u2 f2 (x) D f (x; u), and it follows from Theorem 13 that (30)–(32) is controllable around (x0 ; u0 ) D (0; 0). Let us now recall a necessary condition for stabilizability due to R. Brockett [6]. Brockett third condition for stabilizability. Let f 2 C(Rn Rm ; Rn ) with f (x0 ; u0 ) D 0. If the control system x˙ D f (x; u) can be locally asymptotically stabilized at x0 by means of a continuous feedback law u satisfying u(x0 ) D u0 , then the image by f of any open neighborhood of (x0 ; u0 ) 2 Rn Rm contains an open neighborhood of 0 2 Rn . Here, for any neighborhood V of (x0 ; u0 ) in R5 , f (V) does not cover an open neighborhood of 0 in R3 , since (0; 0; ") 62 f (V) for any " 2 R. According to Brockett’s condition, the system (30)–(32) is not asymptotically stabilizable at the origin. Thus a controllable system may fail to be asymptotically stabilizable by a continuous feedback law u D k(x) due to topological obstructions. It turns out that this phenomenon does not occur when the phase space is the plane, as it has been demonstrated by Kawski in [21]. Theorem 22 (Kawski) Let f0 ; f1 be real analytic vector fields on R2 with f0 (0) D 0 and f1 (0) ¤ 0. Assume that the system x˙ D f0 (x) C u f1 (x) ;
u2R
(63)
is small time locally controllable at the origin. Then it is also asymptotically stabilizable at the origin by a Hölder continuous feedback law u D k(x). In larger dimension (n 3), a way to go round the topological obstruction is to consider a time-varying feedback law u D k(x; t). It has been first observed by Sontag and Sussmann in [33] that for one-dimensional state and input (n D m D 1), the controllability implies the asymptotic stabilizability by means of a time-varying feedback law. This kind of stabilizability was later established by Samson in [31] for Brockett’s system (n D 3 and m D 2). Finally, using the return method, Coron proved that the implication
Controllability ) Asymptotic Stabilizability by TimeVarying Feedback was a principle verified in most cases. To state precise results, we consider affine systems x˙ D f0 (x) C
m X
u i f i (x) ;
x 2 Rn
iD1
and distinguish again two cases: (1) f0 0 (no drift); (2) f0 6 0 (a drift). (1) System without drift Theorem 23 (Coron [9]) Assume that (43) holds for the system (41). Pick any number T > 0. Then there exists a feedback law u D (u1 ; : : : ; u m ) 2 C 1 (Rn R; Rm ) such that u(0; t) D 0
8t 2 R
u(x; t C T) D u(x; t)
(64) 8(x; t) 2 Rn R
(65)
and such that 0 is globally asymptotically stable for the system x˙ D
m X
u i (x; t) f i (x) :
iD1
(2) System with a drift Theorem 24 (Coron [10]) Assume that the system (45) satisfies the condition S( ) for some 2 [0; 1]. Assume also that n 62 f2; 3g and that Lie( f0 ; f1 ; : : : ; f m )(0) D Rn : Pick any T > 0. Then there exists a feedback law u D (u1 ; : : : ; u m ) 2 C 0 (Rn R; Rm ) with u 2 C 1 ((Rn nf0g) R; Rm ) such that (64)–(65) hold and such that 0 is locally asymptotically stable for the system x˙ D f0 (x) C
m X
u i (x; t) f i (x) :
iD1
Flatness While powerful criterions enable us to decide whether a control system is controllable or not, most of them do not provide any indication on how to design an explicit control input steering the system from a point to another one. Fortunately, there exists a large class of systems, the so-called flat systems, for which explicit control inputs may easily be found. The flatness theory has been introduced by Fliess, Levine, Martin, and Rouchon in [15], and since then it has attracted the interest of many researchers
405
406
Finite Dimensional Controllability
thanks to its numerous applications in Engineering. Here, we only sketch the main ideas, referring the interested reader to [27] for a comprehensive introduction to the subject. Let us consider a smooth control system x 2 Rn ; u 2 Rm
x˙ D f (x; u) ;
given together with an output y 2 Rm depending on x, u, and a finite number of derivatives of u ˙ : : : ; u(r) ) : y D h(x; u; u;
Example 6 Consider now the nonlinear system x˙1 D u1
(72)
x˙2 D u2
(73)
x˙3 D x2 u1 :
(74)
Eliminating the input u1 in (72)–(74) yields x˙3 D x2 x˙1 , so that x2 may be expressed as a function of x1 ; x3 and their derivatives. The same is true for u2 , thanks to (73). Let us pick y D (y1 ; y2 ) D (x1 ; x3 ) :
Following [15], we shall say that y is a flat output if the components of y are differentially independent, and both x and u may be expressed as functions of y and of a finite number of its derivatives (66) x D k y; y˙; : : : ; y (p) u D l y; y˙; : : : ; y (q) : (67) In (66)–(67), p and q denote some nonnegative integers, and k and l denote some smooth functions. Since the state x and the input u are parameterized by the flat output y, to solve the controllability problem
We claim that y is a flat output. Indeed, y˙2 (x1 ; x2 ; x3 ) D y1 ; ; y2 ; y˙1 y¨2 y˙1 y˙2 y¨1 : (u1 ; u2 ) D y˙1 ; ( y˙1 )2
(75) (76)
Pick x0 D (0; 0; 0) and x T D (0; 0; 1). Notice that, by the mean value theorem, y˙1 has to vanish somewhere, say at t D ¯t . We shall construct y1 in such a way that y˙1 vanishes only at t D ¯t . For x2 not to be singular at ¯t , we have to impose the condition y˙2 ( ¯t ) D 0 :
x˙ D f (x; u) x(0) D x0 ;
(68) x(T) D x T
(69)
it is sufficient to pick any function y 2 C max(p;q) ([0; T]; Rm ) such that k(y; y˙; : : : ; y (p) )(0) D x(0) D x0 k(y; y˙; : : : ; y
(p)
)(T) D x(T) D x T :
(70)
If y1 and y2 are both analytic near ¯t , we notice that x2 D y˙2 / y˙1 is analytic near ¯t , hence u2 D x˙2 is well-defined (and analytic) near t¯. To steer the solution of (72)–(74) from x0 D (0; 0; 0) to x T D (0; 0; 1), it is then sufficient to pick a function y D (y1 ; y2 ) 2 C ! ([0; T]; R2 ) such that y1 (0) D y2 (0) D 0 ; y1 (T) D 0 ;
The constraints (70)–(71) are generally very easy to satisfy. The desired control input u is then given by (67). Let us show how this program may be carried out on two simple examples.
y˙1
y2 (T) D 1 ;
T D0; 2
Example 5 For the simple integrator x˙1 Dx2
and y˙1 (0) ¤ 0 ;
y˙2 (T) D 0 ;
and y˙1 (T) ¤ 0 : T T y˙2 D 0 ; y¨1 ¤0; 2 2 T and y˙1 (t) ¤ 0 for t ¤ : 2
Clearly,
x˙2 Du the output y D x1 is flat, for x2 DR y˙ and u D y¨. The output z D x2 is not flat, as x1 D z(t) dt. To steer the system from x0 D (0; 0) to x T D (1; 0) in time T, we only have to pick a function y 2 C 2 ([0; T]; R) such that y(0) D 0 ;
y˙2 (0) D 0 ;
(71)
y˙(0) D 0 ;
y(T) D 1 ; and y˙(T) D 0 :
Clearly, y(t) D t 2 (2T t)2 /T 4 is convenient.
t 4 Tt 3 T 2 t2 C (y1 (t); y2 (t)) D t(T t); 4 2 4
is convenient. Future Directions Lie algebraic techniques have been used to provide powerful controllability tests for affine systems. However, there is still an important gap between the known necessary con-
Finite Dimensional Controllability
ditions (e. g. the Legendre Clebsh condition or its extensions) and the sufficient conditions (e. g. the S( ) condition) for the small time local controllability. On the other hand, it has been noticed by Kawski in ([22], Example 6.1) that certain systems can be controlled on small time intervals [0; T] only by using faster switching control variations, the number of switchings tending to infinity as T tends to zero. As for switched systems [2], it is not clear whether purely algebraic conditions be sufficient to characterize the controllability of systems with drift. Of great interest for applications is the development of methods providing explicit control inputs, or control inputs computed in real time with the aid of some numerical schemes, both for the motion planning and the stabilization issue. The flatness theory seems to be a very promising method, and it has been successfully applied in Engineering. An active research is devoted to filling the gap between necessary and sufficient conditions for the existence of flat outputs, and to extending the theory to partial differential equations. The control of nonlinear partial differential equations may sometimes be reduced to the control of a family of finite-dimensional systems by means of the Galerkin method [3]. On the other hand, the spatial discretization of partial differential equations by means of finite differences, finite elements, or spectral methods leads in a natural way to the control of finite dimensional systems (see, e. g., [38]). Of great importance is the uniform boundedness with respect to the discretization parameter of the L2 (0; T)-norms of the control inputs associated with the finite dimensional approximations. Geometric ideas borrowed from the control of finite dimensional systems (e. g. the return method, the power series expansion, the quasistatic deformation) have been applied to prove the controllability of certain nonlinear partial differential equations whose linearization fails to be controllable. (See [11] for a survey of these techniques.) The Korteweg–de Vries equation provides an interesting example of a partial differential equation whose linearization fails to be controllable for certain lengths of the space domain [29]. However, for these critical lengths the reachable space proves to be of finite codimension, and it may be proved that the full equation is controllable by using the nonlinear term in order to reach the missing directions [7,12]. Bibliography Primary Literature 1. Agrachev AA, Gamkrelidze RV (1993) Local controllability and semigroups of diffeomorphisms. Acta Appl Math 32:1–57
2. Agrachev A, Liberzon D (2001) Lie-algebraic stability criteria for switched systems. SIAM J Control Optim 40(1):253–269 3. Agrachev A, Sarychev A (2005) Navier–Stokes equations: controllability by means of low modes forcing. J Math Fluid Mech 7(1):108–152 4. Bianchini RM, Stefani G (1986) Sufficient conditions of local controllability. In: Proc. 25th IEEE–Conf. Decision & Control. Athens 5. Bonnard B (1992) Contrôle de l’altitude d’un satellite rigide. RAIRO Autom Syst Anal Control 16:85–93 6. Brockett RW (1983) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control Theory. Progr. Math., vol 27. Birkhäuser, Basel, pp 181–191 7. Cerpa E, Crépeau E (2007) Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain 8. Chow WL (1940) Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math Ann 117:98–105 9. Coron JM (1992) Global asymptotic stabilization for controllable systems without drift. Math Control Signals Syst 5: 295–312 10. Coron JM (1995) Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws. SIAM J Control Optim 33:804–833 11. Coron JM (2007) Control and nonlinearity, mathematical surveys and monographs. American Mathematical Society, Providence 12. Coron JM, Crépeau E (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths. J Eur Math Soc 6(3):367–398 13. Curtain RF, Pritchard AJ (1978) Infinite Dimensional Linear Systems Theory. Springer, New York 14. Dolecky S, Russell DL (1977) A general theory of observation and control. SIAM J Control Optim 15:185–220 15. Fliess M, Lévine J, Martin PH, Rouchon P (1995) Flatness and defect of nonlinear systems: introductory theory and examples. Intern J Control 31:1327–1361 16. Hautus MLJ (1969) Controllability and observability conditions for linear autonomous systems. Ned Akad Wetenschappen Proc Ser A 72:443–448 17. Hermes H (1982) Control systems which generate decomposable Lie algebras. J Differ Equ 44:166–187 18. Hermes H, Kawski M (1986) Local controllability of a singleinput, affine system. In: Proc. 7th Int. Conf. Nonlinear Analysis. Dallas 19. Hirschorn R, Lewis AD (2004) High-order variations for families of vector fields. SIAM J Control Optim 43:301–324 20. Kalman RE (1960) Contribution to the theory of optimal control. Bol Soc Mat Mex 5:102–119 21. Kawski M (1989) Stabilization of nonlinear systems in the plane. Syst Control Lett 12:169–175 22. Kawski M (1990) High-order small time local controllability. In: Sussmann HJ (ed) Nonlinear controllability and optimal control. Textbooks Pure Appl. Math., vol 113. Dekker, New York, pp 431–467 23. Khodja FA, Benabdallah A, Dupaix C, Kostin I (2005) Nullcontrollability of some systems of parabolic type by one control force. ESAIM Control Optim Calc Var 11(3):426–448 24. Lions JL (1988) Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. In: Recherches en Mathématiques Appliquées, Tome 1, vol 8. Masson, Paris
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25. Liu K (1997) Locally distributed control and damping for the conservative systems. SIAM J Control Optim 35:1574– 1590 26. Lobry C (1970) Contrôlabilité des systèmes non linéaires. SIAM J Control 8:573–605 27. Martin PH, Murray RM, Rouchon P (2002) Flat systems. Mathematical Control Theory, Part 1,2 (Trieste 2001). In: ICTP Lect. Notes, vol VIII. Abdus Salam Int Cent Theoret Phys, Trieste 28. Rashevski PK (1938) About connecting two points of complete nonholonomic space by admissible curve. Uch Zapiski ped inst Libknexta 2:83–94 29. Rosier L (1997) Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM: Control Optim Calc Var 2:33–55 30. Rosier L (2007) A Survey of controllability and stabilization results for partial differential equations. J Eur Syst Autom (JESA) 41(3–4):365–412 31. Samson C (1991) Velocity and torque feedback control of a nonholonomic cart. In: de Witt C (ed) Proceedings of the International workshop on nonlinear and adaptative control: Issues in robotics, Grenoble, France nov. 1990. Lecture Notes in Control and Information Sciences, vol 162. Springer, Berlin, pp 125–151 32. Silverman LM, Meadows HE (1967) Controllability and observability in time-variable linear systems. SIAM J Control 5: 64–73 33. Sontag ED, Sussmannn H (1980) Remarks on continuous feedbacks. In: Proc. IEEE Conf. Decision and Control, (Albuquerque 1980). pp 916–921
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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks SIDNEY REDNER Center for Polymer Studies and Department of Physics, Boston University, Boston, USA Article Outline Glossary Definition of the Subject Introduction Solving Resistor Networks Conduction Near the Percolation Threshold Voltage Distribution in Random Networks Random Walks and Resistor Networks Future Directions Bibliography Glossary Conductance (G) The relation between the current I in an electrical network and the applied voltage V: I D GV . Conductance exponent (t) The relation between the conductance G and the resistor (or conductor) concentration p near the percolation threshold: G (p p c ) t . Effective medium theory (EMT) A theory to calculate the conductance of a heterogeneous system that is based on a homogenization procedure. Fractal A geometrical object that is invariant at any scale of magnification or reduction. Multifractal A generalization of a fractal in which different subsets of an object have different scaling behaviors. Percolation Connectivity of a random porous network. Percolation threshold pc The transition between a connected and disconnected network as the density of links is varied. Random resistor network A percolation network in which the connections consist of electrical resistors that are present with probability p and absent with probability 1 p. Definition of the Subject Consider an arbitrary network of nodes connected by links, each of which is a resistor with a specified electrical resistance. Suppose that this network is connected to the leads of a battery. Two natural scenarios are: (a) the “busbar geometry” (Fig. 1), in which the network is connected
to two parallel lines (in two dimensions), plates (in three dimensions), etc., and the battery is connected across the two plates, and (b) the “two-point geometry”, in which a battery is connected to two distinct nodes, so that a current I injected at a one node and the same current withdrawn from the other node. In both cases, a basic question is: what is the nature of the current flow through the network? There are many reasons why current flows in resistor networks have been the focus of more than a century of research. First, understanding currents in networks is one of the earliest subjects in electrical engineering. Second, the development of this topic has been characterized by beautiful mathematical advancements, such as Kirchhoff’s formal solution for current flows in networks in terms of tree matrices [52], symmetry arguments to determine the electrical conductance of continuous two-component media [10,29,48,69,74], clever geometrical methods to simplify networks [33,35,61,62], and the use of integral transform methods to solve node voltages on regular networks [6,16,94,95]. Third, the nodes voltages of a network through which a steady electrical current flows are harmonic [26]; that is, the voltage at a given node is a suitably-weighted average of the voltages at neighboring nodes. This same harmonicity also occurs in the probability distribution of random walks. Consequently, there are deep connections between the probability distribution of random walks on a given network and the node voltages on the same network [26]. Another important theme in the subject of resistor networks is the essential role played by randomness on current-carrying properties. When the randomness is weak, effective medium theory [10,53,54,55,57,74] is appropriate to characterize how the randomness affects the conductance. When the randomness is strong, as embodied by a network consisting of a random mixture of resistors and insulators, this random resistor network undergoes a transition between a conducting phase and an insulating phase when the resistor concentration passes through a percolation threshold [54]. The feature underlying this phase change is that for a small density of resistors, the network consists of disconnected clusters. However, when the resistor density passes through the percolation threshold, a macroscopic cluster of resistors spans the system through which current can flow. Percolation phenomenology has motivated theoretical developments, such as scaling, critical point exponents, and multifractals that have advanced our understanding of electrical conduction in random resistor networks. This article begins with an introduction to electrical current flows in networks. Next, we briefly discuss ana-
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 1 Resistor networks in a the bus-bar geometry, and b the two-point geometry
lytical methods to solve the conductance of an arbitrary resistor network. We then turn to basic results related to percolation: namely, the conduction properties of a large random resistor network as the fraction of resistors is varied. We will focus on how the conductance of such a network vanishes as the percolation threshold is approached from above. Next, we investigate the more microscopic current distribution within each resistor of a large network. At the percolation threshold, this distribution is multifractal in that all moments of this distribution have independent scaling properties. We will discuss the meaning of multifractal scaling and its implications for current flows in networks, especially the largest current in the network. Finally, we discuss the relation between resistor networks and random walks and show how the classic phenomena of recurrence and transience of random walks are simply related to the conductance of a corresponding electrical network. The subject of current flows on resistor networks is a vast subject, with extensive literature in physics, mathematics, and engineering journals. This review has the modest goal of providing an overview, from my own myopic perspective, on some of the basic properties of random resistor networks near the percolation threshold. Thus many important topics are simply not mentioned and the reference list is incomplete because of space limitations. The reader is encouraged to consult the review articles listed in the reference list to obtain a more complete perspective. Introduction In an elementary electromagnetism course, the following classic problem has been assigned to many generations
of physics and engineering students: consider an infinite square lattice in which each bond is a 1 ohm resistor; equivalently, the conductance of each resistor (the inverse resistance) also equals 1. There are perfect electrical connections at all vertices where four resistors meet. A current I is injected at one point and the same current I is extracted at a nearest-neighbor lattice point. What is the electrical resistance between the input and output? A more challenging question is: what is the resistance between two diagonal points, or between two arbitrary points? As we shall discuss, the latter questions can be solved elegantly using Fourier transform methods. For the resistance between neighboring points, superposition provides a simple solution. Decompose the current source and sink into its two constituents. For a current source I, symmetry tells us that a current I/4 flows from the source along each resistor joined to this input. Similarly, for a current sink I, a current I/4 flows into the sink along each adjoining resistor. For the source/sink combination, superposition tells us that a current I/2 flows along the resistor directly between the source and sink. Since the total current is I, a current of I/2 flows indirectly from source to sink via the rest of the lattice. Because the direct and indirect currents between the input and output points are the same, the resistance of the direct resistor and the resistance of rest of the lattice are the same, and thus both equal to 1. Finally, since these two elements are connected in parallel, the resistance of the infinite lattice between the source and the sink equals 1/2 (conductance 2). As we shall see in Sect. “Effective Medium Theory”, this argument is the basis for constructing an effective medium theory for the conductance of a random network. More generally, suppose that currents I i are injected at each node of a lattice network (normally many of these
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currents are zero and there would be both positive and negative currents in the steady state). Let V i denote the voltage at node i. Then by Kirchhoff’s law, the currents and voltages are related by Ii D
X
P P Fi D j gi j F j/ j g i j ! We then exploit this connection to provide insights about random walks in terms of known results about resistor networks and vice versa. Solving Resistor Networks
g i j (Vi Vj ) ;
(1)
j
where g ij is the conductance of link ij, and the sum runs over all links ij. This equation simply states that the current flowing into a node by an external current source equals the current flowing out of the node along the adjoining resistors. The right-hand side of Eq. (1) is a discrete Laplacian operator. Partly for this reason, Kirchhoff’s law has a natural connection to random walks. At nodes where the external current is zero, the node voltages in Eq. (1) satisfy P 1X j g i j Vj ! Vj : (2) Vi D P z j gi j j
The last step applies if all the conductances are identical; here z is the coordination number of the network. Thus for steady current flow, the voltage at each unforced node equals the weighted average of the voltages at the neighboring sites. This condition defines V i as a harmonic function with respect to the weight function g ij . An important general question is the role of spatial disorder on current flows in networks. One important example is the random resistor network, where the resistors of a lattice are either present with probability p or absent with probability 1 p [54]. Here the analysis tools for regular lattice networks are no longer applicable, and one must turn to qualitative and numerical approaches to understand the current-carrying properties of the system. A major goal of this article is to outline the essential role that spatial disorder has on the current-carrying properties of a resistor network by such approaches. A final issue that we will discuss is the deep relation between resistor networks and random walks [26,63]. Consider a resistor network in which the positive terminal of a battery (voltage V D 1) is connected to a set of boundary nodes, defined to be BC ), and where a disjoint set of boundary nodes B are at V D 0. Now suppose that a random walk hops between nodes of the same geometrical network in which the probability of hopping from P node i to node j in a single step is g i j / k g i k , where k is one of the neighbors of i and the boundary sets are absorbing. For this random walk, we can ask: what is the probability F i for a walk to eventually be absorbed on BC when it starts at node i? We shall show in Sect. “Random Walks and Resistor Networks” that F i satisfies Eq. (2):
Fourier Transform The translational invariance of an infinite lattice resistor network with identical bond conductances g i j D 1 cries out for applying Fourier transform methods to determine node voltages. Let’s study the problem mentioned previously: what is the voltage at any node of the network when a unit current enters at some point? Our discussion is specifically for the square lattice; the extension to other lattices is straightforward. For the square lattice, we label each site i by its x; y coordinates. When a unit current is injected at r0 D (x0 ; y0 ), Eq. (1) becomes ıx;x 0 ı y;y0 D V(x C 1; y) C V(x 1; y) C V(x; y C 1) C V(x; y 1) 4V(x; y) ;
(3)
which clearly exposes the second difference operator of the discrete Laplacian. To find the node voltages, we deP fine V(k) D r V(r) e ikr and then we Fourier transform Eq. (3) to convert this infinite set of difference equations into the single algebraic equation V(k) D
e ikr0 : 4 2(cos k x C cos k y )
(4)
Now we calculate V(r) by inverting the Fourier transform 1 V(r) D (2)2
Z Z
eik(rr0 ) dk : 4 2(cos k x C cos k y )
(5)
Formally, at least, the solution is trivial. However, the integral in the inverse Fourier transform, known as a Watson integral [96], is non-trivial, but considerable understanding has gradually been developed for evaluating this type of integral [6,16,94,95,96]. For a unit input current at the origin and a unit sink of current at r0 , the resistance between these two points is V (0) V(r0 ), and Eq. (5) gives R D V(0) V(r0 ) Z Z (1 e ikr0 ) 1 dk : D (2)2 4 2(cos k x C cos k y )
(6)
Tables for the values of R for a set of closely-separated input and output points are given in [6,94]. As some specific examples, for r0 D (1; 0), R D 1/2, thus reproducing
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
the symmetry argument result. For two points separated by a diagonal, r0 D (1; 1), R D 2/. For r0 D (2; 0), R D 24/. Finally, for two points separated by a knight’s move, r0 D (2; 1), R D 4/ 1/2.
method is prohibitively inefficient for larger networks because the number of spanning trees grows exponentially with network size. Potts Model Connection
Direct Matrix Solution Another way to solve Eq. (1), is to recast Kirchhoff’s law as the matrix equation Ii D
N X
G i j Vj ;
i D 1; 2; : : : ; N
(7)
jD1
where the elements of the conductance matrix are: (P k¤i g i k ; i D j Gi j D i ¤ j: g i j ; The conductance matrix is an example of a tree matrix, as G has the property that the sum of any row or any column equals zero. An important consequence of this tree property is that all cofactors of G are identical and are equal to the spanning tree polynomial [43]. This polynomial is obtained by enumerating all possible tree graphs (graphs with no closed loops) on the original electrical network that includes each node of the network. The weight of each spanning tree is simply the product of the conductances for each bond in the tree. Inverting Eq. (7), one obtains the voltage V i at each node i in terms of the external currents I j ( j D 1; 2; : : : ; N) and the conductances gij . Thus the two-point resistance Rij between two arbitrary (not necessarily connected) nodes i and j is then given by R i j D (Vi Vj ) / I, where the network is subject to a specified external current; for example, for the two-point geometry, I i D 1, I j D 1, and I k D 0 for k ¤ i; j. Formally, the two-point resistance can be written as [99] jG (i j) j ; Ri j D jG ( j) j
(8)
where jG ( j) j is the determinant of the conductance matrix with the jth row and column removed and jG (i j) j is the determinant with the ith and jth rows and columns removed. There is a simple geometric interpretation for this conductance matrix inversion. The numerator is just the spanning tree polynomial for the original network, while the denominator is the spanning tree polynomial for the network with the additional constraint that nodes i and j are identified as a single point. This result provides a concrete prescription to compute the conductance of an arbitrary network. While useful for small networks, this
The matrix solution of the resistance has an alternative and elegant formulation in terms of the spin correlation function of the q-state Potts model of ferromagnetism in the q ! 0 limit [88,99]. This connection between a statistical mechanical model in a seemingly unphysical limit and an enumerative geometrical problem is one of the unexpected charms of statistical physics. Another such example is the n-vector model, in which ferromagnetically interacting spins “live” in an n-dimensional spin space. In the limit n ! 0 [20], the spin correlation functions of this model are directly related to all self-avoiding walk configurations. In the q-state Potts model, each site i of a lattice is occupied by a spin si that can assume one of q discrete values. The Hamiltonian of the system is X H D J ıs i ;s j ; i; j
where the sum is over all nearest-neighbor interacting spin pairs, and ıs i ;s j is the Kronecker delta function (ıs i ;s j D 1 if s i D s j and ıs i ;s j D 0 otherwise). Neighboring aligned spin pairs have energy J, while spin pairs in different states have energy zero. One can view the spins as pointing from the center to a vertex of a q-simplex, and the interaction energy is proportional to the dot product of two interacting spins. The partition function of a system of N spins is X ˇ P Jı ZN D e i; j s i ;s j ; (9) fsg
where the sum is over all 2N spin states fsg. To make the connection to resistor networks, notice that: (i) the exponential factor associated with each link ij in the partition function takes the values 1 or eˇ J , and (ii) the exponential of the sum can be written as the product XY ZN D (1 C vıs i ;s j ) ; (10) fs i g i; j
with v D tanh ˇJ. We now make a high-temperature (small-v) expansion by multiplying out the product in (10) to generate all possible graphs on the lattice, in which each bond carries a weight vıs i ;s j . Summing over all states, the spins in each disjoint cluster must be in the same state, and the last sum over the common state of all spins leads to
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each cluster being weighted by a factor of q. The partition function then becomes X ZN D qN c v N b ; (11) graphs
where N c is the number of distinct clusters and N b is the total number of bonds in the graph. It was shown by Kasteleyn and Fortuin [47] that the limit q D 1 corresponds to the percolation problem when one chooses v D p / (1 p), where p is the bond occupation probability in percolation. Even more striking [34], if one chooses v D ˛q1/2 , where ˛ is a constant, then lim q!0 Z N /q(NC1)/2 D ˛ N1 TN , where T N is again the spanning tree polynomial; in the case where all interactions between neighboring spins have the same strength, then the polynomial reduces to the number of spanning trees on the lattice. It is because of this connection to spanning trees that the resistor network and Potts model are intimately connected [99]. In a similar vein, one can show that the correlation function between two spins at nodes i and j in the Potts model is simply related to the conductance between these same two nodes when the interactions J ij between the spins at nodes i and j are equal to the conductances g ij between these same two nodes in the corresponding resistor network [99]. -Y and Y- Transforms In elementary courses on circuit theory, one learns how to combine resistors in series and parallel to reduce the complexity of an electrical circuit. For two resistors with resistances R1 and R2 in series, the net resistance is R D R1 C R2 , while for resistors in parallel, the net resistance 1 is R D R11 C R22 . These rules provide the resistance of a network that contains only series and parallel connections. What happens if the network is more complicated? One useful way to simplify such a network is by the Y and Y- transforms that was apparently first discovered by Kennelly in 1899 [50] and applied extensively since then [33,35,61,62,81]. The basic idea of the -Y transform is illustrated in Fig. 2. Any triangular arrangement of resistors R12 , R13 , and R23 within a larger circuit can be replaced by a star, with resistances R1 , R2 , and R3 , such that all resistances between any two points among the three vertices in the triangle and the star are the same. The conditions that all two-point resistances are the same are: (R1 C R2 ) D
1 R12
1 1
C (R13 C R23 ) a12
C cyclic permutations :
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 2 Illustration of the -Y and Y- transforms
Solving for R1 ; R2 , and R3 gives R1 D 12 (a12 a23 C a13 ) + cyclic permutations; the explicit result in terms of the Rij is: R1 D
R12 R13 R12 C R13 C R23
C cyclic permutations; (12)
as well as the companion result for the conductances G i D R1 i : G1 D
G12 G13 C G12 G23 C G13 G23 Ccyclic permutations: G23
These relations allow one to replace any triangle by a star to reduce an electrical network. However, sometimes we need to replace a star by a triangle to simplify a network. To construct the inverse Y- transform, notice that the -Y transform gives the resistance ratios R1 /R2 D R13 /R23 + cyclic permutations, from which R13 D R12 (R3 /R2 ) and R23 D R12 (R3 /R1 ). Substituting these last two results in Eq. (12), we eliminate R13 and R23 and thus solve for R12 in terms of the Ri : R12 D
R1 R2 C R1 R3 C R2 R3 Ccyclic permutations; (13) R3
and similarly for G i j D R1 i j . To appreciate the utility of the -Y and Y- transforms, the reader is invited to apply them on the Wheatstone bridge. When employed judiciously and repeatedly, these transforms systematically reduce planar lattice circuits to a single bond, and thus provide a powerful approach to calculate the conductance of large networks near the percolation threshold. We will return to this aspect of the problem in Sect. “Conductance Exponent”. Effective Medium Theory Effective medium theory (EMT) determines the macroscopic conductance of a random resistor network by a homogenization procedure [10,53,54,55,57,74] that is reminiscent of the Curie–Weiss effective field theory of mag-
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
netism. The basic idea in EMT is to replace the random network by an effective homogeneous medium in which the conductance of each resistor is determined self-consistently to optimally match the conductances of the original and homogenized systems. EMT is quite versatile and has been applied, for example, to estimate the dielectric constant of dielectric composites and the conductance of conducting composites. Here we focus on the conductance of random resistor networks, in which each resistor (with conductance g 0 ) is present with probability p and absent with probability 1 p. The goal is to determine the conductance as a function of p. To implement EMT, we first replace the random network by an effective homogeneous medium in which each bond has the same conductance g m (Fig. 3). If a voltage is applied across this effective medium, there will be a potential drop V m and a current I m D g m Vm across each bond. The next step in EMT is to assign one bond in the effective medium a conductance g and adjust the external voltage to maintain a fixed total current I passing through the network. Now an additional current ıi passes through the conductor g. Consequently, a current ıi must flow through one terminal of g to the other terminal via the remainder of the network (Fig. 3). This current perturbation leads to an additional voltage drop ıV across g. Thus the current-voltage relations for the marked bond and the remainder of the network are I m C ıi D g(Vm C ıV ) ıi D G ab ıV ;
(14)
where Gab is the conductance of the rest of the lattice between the terminals of the conductor g. The last step in EMT is to require that the mean value ıV averaged over the probability distribution of individual bond conductances is zero. Thus the effective medium “matches” the current-carrying properties of the original network. Solving Eq. (14) for ıV , and using the probability distribution P(g) D pı(gg0 )C(1p)ı(g) appropriate
for the random resistor network, we obtain
(g m g0 )p g m (1 p) C D0: hıVi D Vm (G ab C g0 ) G ab
It is now convenient to write Gab D ˛g m , where ˛ is a lattice-dependent constant of the order of one. With this definition, Eq. (15) simplifies to g m D go
p(1 C ˛) 1 : ˛
(16)
The value of ˛ – the proportionality constant for the conductance of the initial lattice with a single bond removed – can usually be determined by a symmetry argument of the type presented in Sect. “Introduction to Current Flows”. For example, for the triangular lattice (coordination number 6), the conductance Gab D 2g m and ˛ D 2. For the hypercubic lattice in d dimensions (coordination number z D 2d ), Gab D ((z 2)/2)g m . The main features of the effective conductance g m that arises from EMT are: (i) the conductance vanishes at a lattice-dependent percolation threshold p c D 1/(1 C ˛); for the hypercubic lattice ˛ D (z 2)/2 and the percolation threshold p c D 2/z D 21d (fortuitously reproducing the exact percolation threshold in two dimensions); (ii) the conductance varies linearly with p and vanishes linearly in p p c as p approaches pc from above. The linearity of the effective conductance away from the percolation threshold accords with numerical and experimental results. However, EMT fails near the percolation threshold, where large fluctuations arise that invalidate the underlying assumptions of EMT. In this regime, alternative methods are needed to estimate the conductance. Conduction Near the Percolation Threshold Scaling Behavior EMT provides a qualitative but crude picture of the current-carrying properties of a random resistor network. While EMT accounts for the existence of a percolation transition, it also predicts a linear dependence of the conductance on p. However, near the percolation threshold it is well known that the conductance varies non-linearly in p p c near pc [85]. This non-linearity defines the conductance exponent t by G (p p c ) t
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 3 Illustration of EMT. (left) The homogenized network with conductances gm and one bond with conductance g. (right) The equivalent circuit to the lattice
(15)
p # pc ;
(17)
and much research on random resistor networks [85] has been performed to determine this exponent. The conductance exponent generically depends only on the spatial dimension of the network and not on any other details (a notable exception, however, is when link resistances are
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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 4 (left) Realization of bond percolation on a 25 25 square lattice at p=0.505. (right) Schematic picture of the nodes (shaded circles), links and blobs picture of percolation for p & pc
broadly distributed, see [40,93]). This universality is one of the central tenets of the theory of critical phenomena [64, 83]. For percolation, the mechanism underlying universality is the absence of a characteristic length scale; as illustrated in Fig. 4, clusters on all length scales exist when a network is close to the percolation threshold. The scale of the largest cluster defines the correlation length by (p c p) as p ! p c . The divergence in also applies for p > p c by defining the correlation length as the typical size of finite clusters only (Fig. 4), thus eliminating the infinite percolating cluster from consideration. At the percolation threshold, clusters on all length scales exist, and the absence of a characteristic length implies that the singularity in the conductance should not depend on microscopic variables. The only parameter remaining upon which the conductance exponent t can depend upon is the spatial dimension d [64,83]. As typifies critical phenomena, the conductance exponent has a constant value in all spatial dimensions d > d c , where dc is the upper critical dimension which equals 6 for percolation [22]. Above this critical dimension, mean-field theory (not to be confused with EMT) gives the correct values of critical exponents. While there does not yet exist a complete theory for the dimension dependence of the conductance exponent below the critical dimension, a crude but useful nodes, links, and blobs picture of the infinite cluster [21,82,84] provides partial information. The basic idea of this picture is that for p & p c , a large system has an irregular network-like
topology that consists of quasi-linear chains that are separated by the correlation length (Fig. 4). For a macroscopic sample of linear dimension L with a bus bar-geometry, the percolating cluster above pc then consists of (L/)d1 statistically identical chains in parallel, in which each chain consists of L/ macrolinks in series, and the macrolinks consists of nested blob-like structures. The conductance of a macrolink is expected to vanish as (p p c ) , with a new unknown exponent. Although a theory for the conductance of a single macrolink, and even a precise definition of a macrolink, is still lacking, the nodes, links, and blobs picture provides a starting point for understanding the dimension dependence of the conductance exponent. Using the rules for combining parallel and series conductances, the conductance of a large resistor network of linear dimension L is then G(p; L)
d1 L (p p c ) L/
Ld2 (p p c )(d2)C :
(18)
In the limit of large spatial dimension, we expect that a macrolink is merely a random walk between nodes. Since the spatial separation between nodes is , the number of bonds in the macrolink, and hence its resistance, scales as 2 [92]. Using the mean-field result (p p c )1/2 , the resistance of the macrolink scales as (p p c )1 and thus the exponent D 1. Using the mean-field exponents
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
D 1/2 and D 1 at the upper critical dimension of d c D 6, we then infer the mean-field value of the conductance exponent t D 3 [22,91,92]. Scaling also determines the conductance of a finitesize system of linear dimension L exactly at the percolation threshold. Although the correlation length formally diverges when p p c D 0, is limited by L in a finite system of linear dimension L. Thus the only variable upon which the conductance can depend is L itself. Equivalently, deviations in p p c that are smaller than L1/ cannot influence critical behavior because can never exceed L. Thus to determine the dependence of a singular observable for a finite-size system at pc , we may replace (pp c ) by L1/ . By this prescription, the conductance at pc of a large finite-size system of linear dimension L becomes G(p c ; L) Ld2 (L1/ )(d2)C L/ :
(19)
In this finite-size scaling [85], we fix the occupation probability to be exactly at pc and study the dependence of an observable on L to determine percolation exponents. This approach provides a convenient and more accurate method to determine the conductance exponent compared to studying the dependence of the conductance of a large system as a function of p p c . Conductance Exponent In percolation and in the random resistor network, much effort has been devoted to computing the exponents that characterize basic physical observables – such as the correlation length and the conductance G – to high precision. There are several reasons for this focus on exponents. First, because of the universality hypothesis, exponents are a meaningful quantifier of phase transitions. Second, various observables near a phase transition can sometimes be related by a scaling argument that leads to a corresponding exponent relation. Such relations may provide a decisive test of a theory that can be checked numerically. Finally, there is the intellectual challenge of developing accurate numerical methods to determine critical exponents. The best such methods have become quite sophisticated in their execution. A seminal contribution was the “theorists’ experiment” of Last and Thouless [58] in which they punched holes at random in a conducting sheet of paper and measured the conductance of the sheet as a function of the area fraction of conducting material. They found that the conductance vanished faster than linearly with (p p c ); here p corresponds to the area fraction of the conductor. Until this experiment, there was a sentiment that the conductance should be related to the fraction of material in
the percolating cluster [30] – the percolation probability P(p) – a quantity that vanished slower than linearly with (p p c ). The reason for this disparity is that in a resistor network, much of the percolating cluster consists of dangling ends – bonds that carry no current – and thus make no contribution to the conductance. A natural geometrical quantity that ought to be related to the conductance is the fraction of bonds B(p) in the conducting backbone – the subset of the percolating cluster without dangling ends. However, a clear relation between the conductivity and a geometrical property of the backbone has not yet been established. Analytically, there are primary two methods that have been developed to compute the conductance exponent: the renormalization group [44,86,87,89] and low-density series expansions [1,2,32]. In the real-space version of the renormalization group, the evolution of conductance distribution under length rescaling is determined, while the momentum-space version involves a diagrammatic implementation of this length rescaling in momentum space. The latter is a perturbative approach away from mean-field theory in the variable 6 d that become exact as d ! 6. Considerable effort has been devoted to determining the conductance exponent by numerical and algorithmic methods. Typically, the conductance is computed for networks of various linear dimensions L at p D p c , and the conductance exponent is extracted from the L dependence of the conductance, which should vanish as L/ . An exact approach, but computationally impractical for large networks, is Gauss elimination to invert the conductance matrix [79]. A simple approximate method is Gauss relaxation [59,68,80,90,97] (and its more efficient variant of Gauss–Seidel relaxation [71]). This method uses Eq. (2) as the basis for an iteration scheme, in which the voltage V i at node i at the nth update step is computed from (2) using the values of V j at the (n 1)st update in the right-hand side of this equation. However, one can do much better by the conjugate gradient algorithm [27] and speeding up this method still further by Fourier acceleration methods [7]. Another computational approach is based on the node elimination method, in which the -Y and Y- transforms are used to successively eliminate bonds from the network and ultimately reduce a large network to a single bond [33,35,62]. In a different vein, the transfer matrix method has proved to be extremely accurate and efficient [24,25,70,100]. The method is based on building up the network one bond at a time and immediately calculating the conductance of the network after each bond addition. This method is most useful when applied to very long strips of transverse dimension L so that a single realization gives an accurate value for the conductance.
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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
As a result of these investigations, as well as by series expansions for the conductance, the following exponents have been found. For d D 2, where most of the computational effort has been applied, the best estimate [70] for the exponent t (using D t in d D 2 only) is t D 1:299˙0:002. One reason for the focus on two dimensions is that early estimates for t were tantalizingly close to the correlation length exponent that is now known to exactly equal 4/3 [23]. Another such connection was the Alexander–Orbach conjecture [5], which predicted t D 91/72 D 1:2638 : : :, but again is incompatible with the best numerical estimate for t. In d D 3, the best available numerical estimate for t appears to be t D 2:003 ˙ 0:047 [12,
36], while the low concentration series method gives an equally precise result of t D 2:02 ˙ 0:05 [1,2]. These estimates are just compatible with the rigorous bound that t 2 in d D 3 [37,38]. In greater than three dimensions, these series expansions give t D 2:40 ˙ 0:03 for d D 4 and t D 2:74 ˙ 0:03 for d D 5, and the dimension dependence is consistent with t D 3 when d reaches 6. Voltage Distribution in Random Networks Multifractal Scaling While much research has been devoted to understanding the critical behavior of the conductance, it was re-
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 5 The voltage distribution on an L L square lattice random resistor network at the percolation threshold for a L D 4 (exact), b L D 10, and c L D 130. The latter two plots are based on simulation data. For L D 4, a number of peaks, that correspond to simple rational fractions of the unit potential drop, are indicated. Also shown are the average voltage over all realizations, Vav , the most probable voltage, Vmp , and the average of the minimum voltage in each realization, hVmin i. [Reprinted from Ref. [19]]
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
alized that the distribution of voltages across each resistor of the network was quite rich and exhibited multifractal scaling [17,19,72,73]. Multifractality is a generalization of fractal scaling in which the distribution of an observable is sufficiently broad that different moments of the distribution scale independently. Such multifractal scaling arises in phenomena as diverse as turbulence [45,66], localization [13], and diffusion-limited aggregation [41,42]. All these diverse examples showed scaling properties that were much richer than first anticipated. To make the discussion of multifractality concrete, consider the example of the Maxwell–Boltzmann velocity distribution of a one-dimensional ideal gas r m 2 1 2 2 P(v) D ev /2v th ; emv /2k B T q 2 kB T 2 2v th where kB is Boltzmann’s constant,pm is the particle mass, T is the temperature, and v th D kB T/m is the characteristic thermal velocity. The even integer moments of the velocity distribution are n p(n) h(v 2 )n i / v th2 v th2 : Thus a single velocity scale, vth , characterizes all positive moments of the velocity distribution. Alternatively, the exponent p(n) is linear in n. This linear dependence of successive moment exponents characterizes single-parameter scaling. The new feature of multifractal scaling is that a wide range of scales characterizes the voltage distribution (Fig. 5). As a consequence, the moment exponent p(n) is a non-linear function of n. One motivation for studying the voltage distribution is its relation to basic aspects of electrical conduction. If a voltage V D 1 is applied across a resistor network, then the conductance G and the total current flow I are equal: I D G. Consider now the power dissipated through the network P D IV D GV 2 ! G. We may also compute the dissipated power by adding up these losses in each resistor to give X X X PDGD g i j Vi2j ! Vi2j D V 2 N(V) : (20) ij
ij
tribution, we define the family of exponents p(k) for the scaling dependence of the voltage distribution at p D p c by X M(k) N(V)V k Lp(k)/ : (21) V
Since M(2) is just the network conductance, p(2) D . Other moments of the voltage distribution also have simple interpretations. For example, hV 4 i is related to the magnitude of the noise in the network [9,73], while hV k i for k ! 1 weights the bonds with the highest currents, or the “hottest” bonds of the network, most strongly, and they help understand the dynamics of fuse networks of failure [4,18]. On the other hand, negative moments weight low-current bonds more strongly and emphasize the low-voltage tail of the distribution. For example, M(1) characterizes hydrodynamic dispersion [56], in which passive tracer particles disperse in a network due to a multiplicity of network paths. In hydrodynamics dispersion, the transit time across each bond is proportional to the inverse of the current in the bond, while the probability for tracer to enter a bond is proportional to the entering current. As a result, the kth moment of the transit time distribution varies as M(k C 1), so that the quantity that quantifies dispersion, ht 2 i hti2 , scales as M(1). A simple fractal model [3,11,67] of the conducting backbone (Fig. 6) illustrates the multifractal scaling of the voltage distribution near the percolation threshold [17]. To obtain the Nth-order structure, each bond in the (N 1)st iteration is replaced by the first-order structure. The resulting fractal has a hierarchical embedding of links and blobs that captures the basic geometry of the percolating backbone. Between successive generations, the length scale changes by a factor of 3, while the number of bonds
V
Here g i j D 1 is the conductance of resistor ij, and V ij is the corresponding voltage drop across this bond. In the last equality, N(V) is the number of resistors with a voltage drop V. Thus the conductance is just the second moment of the distribution of voltage drops across each bond in the network. From the statistical physics perspective it is natural to study other moments of the voltage distribution and the voltage distribution itself. Analogous to the velocity dis-
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 6 The first few iterations of a hierarchical model
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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
changes by a factor of 4. Defining the fractal dimension df as the scaling relation between mass (M D 4 N ) and the length scale (` D 3 N ) via M `d f , gives a fractal dimension d f D ln 4/ ln 3. Now let’s determine the distribution of voltage drops across the bonds. If a unit voltage is applied at the opposite ends of a first-order structure (N D 1) and each bond is a 1 ohm resistor, then the two resistors in the central bubble each have a voltage drop of 1/5, while the two resistors at the ends have a voltage drop 2/5. In an Nth-order hierarchy, the voltage of any resistor is the product of these two factors, with number of times each factor occurs dependent on the level of embedding of a resistor within the blobs. It is a simple exercise to show that the voltage distribution is [17] ! N N N(V( j)) D 2 ; (22) j where the voltage V(j) can take the values 2 j /5 N (with j D 0; 1; : : : ; N). Because j varies logarithmically in V, the voltage distribution is log binomial [75]. Using this distribution in Eq. (21), the moments of the voltage distribution are " #N 2(1 C 2 k ) M(k) D : (23) 5k In particular, the average voltage, M(1)/M(0) Vav equals ((3/2)/5) N , which ispvery different from the most probable voltage, Vmp D ( 2/5) N as N ! 1. The underlying multiplicativity of the bond voltages is the ultimate source of the large disparity between the average and most probable values. To calculate the moment exponent p(k), we first need to relate the iteration index N to a physical length scale. For percolation, the appropriate relation is based on Coniglio’s theorem [15], which is a simple but profound statement about the structure of the percolating cluster. This theorem states that the number of singly-connected bonds in a system of linear dimension L, Ns , varies as L1/ . Singly-connected bonds are those that would disconnect the network if they were cut. An equivalent form of the theorem is Ns D @p0 /@p, where p0 is the probability that a spanning cluster exists in the system. This relation reflects the fact that when p is decreased slightly, p0 changes only if a singly-connected bond happens to be deleted. In the Nth-order hierarchy, the number of such singlyconnected links is simply 2N . Equating these two gives an effective linear dimension, L D 2 N . Using this relation in (23), the moment exponent p(k) is iı h p(k) D k 1 C k ln (5/4) ln(1 C 2k ) ln 2 : (24)
Because each p(k) is independent, the moments of the voltage distribution are characterized by an infinite set of exponents. Equation (24) is in excellent agreement with numerical data for the voltage distribution in two-dimensional random resistor networks at the percolation threshold [19]. A similar multifractal behavior was also found for the voltage distribution of the resistor network at the percolation threshold in three dimensions [8]. Maximum Voltage An important aspect of the voltage distribution, both because of its peculiar scaling properties [27] and its application to breakdown problems [18,27], is the maximum voltage in a network. The salient features of this maximum voltage are: (i) logarithmic scaling as a function of system size [14,27,28,60,65], and (ii) non-monotonic dependence on the resistor concentration p [46]. The former property is a consequence of the expected size of the largest defect in the network that gives maximal local currents. Here, we use the terms maximum local voltage and maximum local current interchangeably because they are equivalent. To find the maximal current, we first need to identify the optimal defects that lead to large local currents. A natural candidate is an ellipse [27,28] with major and minor axes a and b (continuum), or its discrete analog of a linear crack (hyperplanar crack in greater than two dimensions) in which n resistors are missing (Fig. 7). Because current has to detour around the defect, the local current at the ends of the defect is magnified. For the continuum problem, the current at the tip of the ellipse is Itip D I0 (1Ca / b), where I 0 is the current in the unperturbed system [27]. For the maximum current in the lattice system, one must integrate the continuum current over a one lattice spacing and identify a/b with n [60]. This approach gives the maximal current at the tip of a crack Imax / (1 C n1/2 ) in two dimensions and as Imax / (1 C n1/2(d1) ) in d dimensions. Next, we need to find the size of the largest defect, which is an extreme-value statistics exercise [39]. For a linear crack, each broken bond occurs with probability 1 p, so that the probability for a crack of length n is (1 p)n ean , with a D ln(1 p). In a network of d , we estimate the size of the largest defect by volume R 1 Lan d L n max e dx D 1; that is, there exists of the order of one defect of size nmax or larger in the network [39]. This estimate gives nmax varying as ln L. Combining this result with the current at the tip of a crack of length n, the largest current in a system of linear dimension L scales as (ln L)1/2(d1) . A more thorough analysis shows, however, that a single crack is not quite optimal. For a continuum two-com-
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 7 Defect configurations in two dimensions. a An ellipse and its square lattice counterpart, b a funnel, with the region of good conductor shown shaded, and a 2-slit configuration on the square lattice
ponent network with conductors of resistance 1 with probability p and with resistance r > 1 with probability 1 p, the configuration that maximizes the local current is a funnel [14,65]. For a funnel of linear dimension `, the maximum current at the apex of the funnel is proportional to `1 , where D (4/) tan1 (r1/2 ) [14,65]. The probability to find a funnel of linear dimension ` now scales as 2 eb` (exponentially in its area), with b a constant. By the same extreme statistics reasoning given above, the size of the largest funnel in a system of linear dimension L then scales as (ln L)1/2 , and the largest expected current correspondingly scales as (ln L)(1)/2 . In the limit r ! 1, where one component is an insulator, the optimal discrete configuration in two dimensions becomes two parallel slits, each of length n, between which a single resistor remains [60]. For this two-slit configuration, the maximum current is proportional to n in two dimensions, rather than n1/2 for the single crack. Thus the maximal current in a system of linear dimension L scales as ln L rather than as a fractional power of ln L. The p dependence of the maximum voltage is intriguing because it is non-monotonic. As p, the fraction of occupied bonds, decreases from 1, less total current flows (for a fixed overall voltage drop) because the conductance is decreasing, while local current in a funnel is enhanced because such defects grow larger. The competition between these two effects leads to Vmax attaining its
peak at ppeak above the percolation threshold that only slowly approaches pc as L ! 1. An experimental manifestation of this non-monotonicity in Vmax occurred in a resistor-diode network [77], where the network reproducibly burned (solder connections melting and smoking) when p ' 0:77, compared to a percolation threshold of p c ' 0:58. Although the directionality constraint imposed by diodes enhances funneling, similar behavior should occur in a random resistor network. The non-monotonic p dependence of Vmax can be understood within the quasi-one-dimensional “bubble” model [46] that captures the interplay between local funneling and overall current reduction as p decreases (Fig. 8). Although this system looks one-dimensional, it can be engineered to reproduce the percolation properties of a system in greater than one dimension by choosing the length L to scale exponentially with the width w. The probability for a spanning path in this structure is L p0 D 1 (1 p)w ! exp[L epw ] L; w ! 1 ;
(25)
which suddenly changes from 0 to 1 – indicative of percolation – at a threshold that lies strictly within (0,1) as L ! 1 and L ew . In what follows, we take L D 2w , which gives p c D 1/2. To determine the effect of bottlenecking, we appeal to the statement of Coniglio’s theorem [15], @p0 /@p equals the average number of singly-connected bonds in the system. Evaluating @p0 /@p in Eq. (25) at the percolation threshold of p c D 1/2 gives @p0 D w C O(ew ) ln L : @p
(26)
Thus at pc there are w ln L bottlenecks. However, current focusing due to bottlenecks is substantially diluted because the conductance, and hence the total current through the network, is small at pc . What is needed is a single bottleneck of width 1. One such bottleneck ensures the total current flow is still substantial, while the narrowing to width 1 endures that the focusing effect of the bottleneck is maximally effective.
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 8 The bubble model: a chain of L bubbles in series, each consisting of w bonds in parallel. Each bond is independently present with probability p
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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
Clearly, a single bottleneck of width 1 occurs above the percolation threshold. Thus let’s determine when a such an isolated bottleneck of width 1 first appears as a function of p. The probability that a single non-empty bubble contains at least two bonds is (1 qw w pqw1 ) / (1 qw ), where q D 1 p. Then the probability P1 (p) that the width of the narrowest bottleneck has width 1 in a chain of L bubbles is L w pqw1 P1 (p) D 1 1 1 qw p pw : (27) 1 exp Lw e q(1 qw ) The subtracted term is the probability that L non-empty bubbles contain at least two bonds, and then P1 (p) is the complement of this quantity. As p decreases from 1, P1 (p) sharply increases from 0 to 1 when the argument of the outer exponential becomes of the order of 1; this change occurs at pˆ p c C O(ln(ln L) / ln L). At this point, a bottleneck of width 1 first appears and therefore Vmax also occurs for this value of p. Random Walks and Resistor Networks The Basic Relation We now discuss how the voltages at each node in a resistor network and the resistance of the network are directly related to first-passage properties of random walks [31,63,76, 98]. To develop this connection, consider a random walk on a finite network that can hop between nearest-neighbor sites i to j with probability pij in a single step. We divide the boundary points of the network into two disjoint classes, BC and B , that we are free to choose; a typical situation is the geometry shown in Fig. 9. We now ask: starting at an arbitrary point i, what is the probability that the walk eventually reaches the boundary set BC without first reaching any node in B ? This quantity is termed the exit probability EC (i) (with an analogous definition for the exit probability E (i) D 1 EC (i) to B ). We obtain the exit probability EC (i) by summing the probabilities for all walk trajectories that start at i and reach a site in BC without touching any site in B (and similarly for E (i)). Thus X E˙ (i) D P p ˙ (i) ; (28)
E˙ (i) D
X
p i j E˙ ( j) :
(29)
j
Thus E˙ (i) is a harmonic function because it equals a weighted average of E˙ at neighboring points, with weighting function pij . This is exactly the same relation obeyed by the node voltages in Eq. (2) for the corresponding resistor network when we identify the single-step hopP ping probabilities pij with the conductances g i j / j g i j . We thus have the following equivalence: Let the boundary sets BC and B in a resistor network be fixed at voltages 1 and 0 respectively, with g ij the conductance of the bond between sites i and j. Then the voltage at any interior site i coincides with the probability for a random walk, which starts at i, to reach BC before reaching B , when the hopping probability from P i to j is p i j D g i j / j g i j . If all the bond conductances are the same – corresponding to single - step hopping probabilities in the equivalent random walk being identical – then Eq. (29) is just the discrete Laplace equation. We can then exploit this correspondence between conductances and hopping probabilities to infer non-trivial results about random walks and about resistor networks from basic electrostatics. This correspondence can also be extended in a natural way to general random walks with a spatially-varying bias and diffusion coefficient, and to continuous media. The consequences of this equivalence between random walks and resistor networks is profound. As an example [76], consider a diffusing particle that is initially at distance r0 from the center of a sphere of radius a < r0 in otherwise empty d-dimensional space. By the correspondence with electrostatics, the probability that this particle eventually hits the sphere is simply the electrostatic potential at r0 ; E (r0 ) D (a/r0 )d2 !
p˙
where P p ˙ (i) denotes the probability of a path from i to B˙ that avoids B . The sum over all these restricted paths can be decomposed into the outcome after one step, when the walk reaches some intermediate site j, and the sum over all path remainders from j to B˙ . This decomposition gives
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 9 a Lattice network with boundary sites BC or B . b Corresponding resistor network in which each rectangle is a 1 ohm resistor. The sites in BC are all fixed at potential V D 1, and sites in B are all grounded
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
fact, Eq. (31) gives the fundamental result
Network Resistance and Pólya’s Theorem An important extension of the relation between exit probability and node voltages is to infinite resistor networks. This extension provides a simple connection between the classic recurrence/transience transition of random walks on a given network [31,63,76,98] and the electrical resistance of this same network [26]. Consider a symmetric random walk on a regular lattice in d spatial dimensions. Suppose that the walk starts at the origin at t D 0. What is the probability that the walk eventually returns to its starting point? The answer is strikingly simple: For d 2, a random walk is certain to eventually return to the origin. This property is known as recurrence. For d > 2, there is a non-zero probability that the random walk will never return to the origin. This property is known as transience. Let’s now derive the transience and recurrence properties of random walks in terms of the equivalent resistor network problem. Suppose that the voltage V at the boundary sites BC is set to one. Then by Kirchhoff’s law, the total current entering the network is ID
X X X (1 Vj )gC j D (1 Vj )pC j gCk : (30) j
j
k
Here g C j is the conductance of the resistor between BC P and a neighboring site j, and pC j D gC j / j gC j . Because the voltage V j also equals the probability for the corresponding random walk to reach BC without reaching B , the term Vj pC j is just the probability that a random walk starts at BC , makes a single step to one of the sites j adjacent to BC (with hopping probability pij ), and then returns to BC without reaching B . We therefore deduce that X ID (1 Vj )gC j j
D
X k
D
X
gCk
escape probability Pescape D P
G : k g Ck
(32)
Suppose now that a current I is injected at a single point of an infinite network, with outflow at infinity (Fig. 10). Thus the probability for a random walk to never return to its starting point, is simply proportional to the conductance G from this starting point to infinity of the same network. Thus a subtle feature of random walks, namely, the escape probability, is directly related to currents and voltages in an equivalent resistor network. Part of the reason why this connection is so useful is that the conductance of the infinite network for various spatial dimensions can be easily determined, while a direct calculation of the return probability for a random walk is more difficult. In one dimension, the conductance of an infinitely long chain of identical resistors is clearly zero. Thus Pescape D 0 or, equivalently, Preturn D 1. Thus a random walk in one dimension is recurrent. As alluded to at the outset of Sect. “Introduction to Current Flows”, the conductance between one point and infinity in an infinite resistor lattice in general spatial dimension is somewhat challenging. However, to merely determine the recurrence or transience of a random walk, we only need to know if the return probability is zero or greater than zero. Such a simple question can be answered by a crude physical estimate of the network conductance. To estimate the conductance from one point to infinity, we replace the discrete lattice by a continuum medium
X (1 Vj )pC j j
gCk (1 return probability )
k
D
X
gCk escape probability :
(31)
k
Here “escape” means that the random walk reaches the set B without returning to a node in BC . On the other hand, the current and the voltage drop across the network are related to the conductance G between the two boundary sets by I D GV D G. From this
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 10 Decomposition of a conducting medium into concentric shells, each of which consists of fixed-conductance blocks. A current I is injected at the origin and flows radially outward through the medium
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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
of constant conductance. We then estimate the conductance of the infinite medium by decomposing it into a series of concentric shells of fixed thickness dr. A shell at radius r can be regarded as a parallel array of r d1 volume elements, each of which has a fixed conductance. The conductance of one such shell is proportional to its surface area, and the overall resistance is the sum of these shell resistances. This reasoning gives Z1 R
Rshell (r)dr Z1
dr r d1
( D
1 (Pescape
P
for d 2 j
gC j
)1
for d > 2 :
(33)
The above estimate gives an easy solution to the recurrence/transience transition of random walks. For d 2, the conductance to infinity is zero because there are an insufficient number of independent paths from the origin to infinity. Correspondingly, the escape probability is zero and the random walk is recurrent. The case d D 2 is more delicate because the integral in Eq. (33) diverges only logarithmically at the upper limit. Nevertheless, the conductance to infinity is still zero and the corresponding random walk is recurrent (but just barely). For d > 2, the conductance between a single point and infinity in an infinite homogeneous resistor network is non zero and therefore the escape probability of the corresponding random walk is also non zero – the walk is now transient. There are many amusing ramifications of the recurrence of random walks and we mention two such properties. First, for d 2, even though a random walk eventually returns to its starting point, the mean time for this event is infinite! This divergence stems from a power-law tail in the time dependence of the first-passage probability [31,76], namely, the probability that a random walk returns to the origin for the first time. Another striking aspect of recurrence is that because a random walk returns to its starting point with certainty, it necessarily returns an infinite number of times. Future Directions There is a good general understanding of the conductance of resistor networks, both far from the percolation threshold, where effective medium theory applies, and close to percolation, where the conductance G vanishes as (p p c ) t . Many advancements in numerical techniques have been developed to determine the conductance accurately and thereby obtain precise values for the conductance exponent, especially in two dimensions. In spite of this progress, we still do not yet have the right way, if it
exists at all, to link the geometry of the percolation cluster or the conducting backbone to the conductivity itself. Furthermore, many exponents of two-dimensional percolation are known exactly. Is it possible that the exact approaches developed to determine percolation exponents can be extended to give the exact conductance exponent? Finally, there are aspects about conduction in random networks that are worth highlighting. The first falls under the rubric of directed percolation [51]. Here each link in a network has an intrinsic directionality that allows current to flow in one direction only – a resistor and diode in series. Links are also globally oriented; on the square lattice for example, current can flow rightward and upward. A qualitative understanding of directed percolation and directed conduction has been achieved that parallels that of isotropic percolation. However, there is one facet of directed conduction that is barely explored. Namely, the state of the network (the bonds that are forward biased) must be determined self consistently from the current flows. This type of non-linearity is much more serious when the circuit elements are randomly oriented. These questions about the coupling between the state of the network and its conductance are central when the circuit elements are intrinsically non-linear [49,78]. This is a topic that seems ripe for new developments. Bibliography 1. Adler J (1985) Conductance Exponents From the Analysis of Series Expansions for Random Resistor Networks. J Phys A Math Gen 18:307–314 2. Adler J, Meir Y, Aharony A, Harris AB, Klein L (1990) Low-Concentration Series in General Dimension. J Stat Phys 58:511– 538 3. Aharony A, Feder J (eds) (1989) Fractals in Physics. Phys D 38:1–398 4. Alava MJ, Nukala PKVV, Zapperi S (2006) Statistical Models for Fracture. Adv Phys 55:349–476 5. Alexander S, Orbach R (1982) Density of States of Fractals: Fractons. J Phys Lett 43:L625–L631 6. Atkinson D, van Steenwijk FJ (1999) Infinite Resistive Lattice. Am J Phys 67:486–492 7. Batrouni GG, Hansen A, Nelkin M (1986) Fourier Acceleration of Relaxation Processes in Disordered Systems. Phys Rev Lett 57:1336–1339 8. Batrouni GG, Hansen A, Larson B (1996) Current Distribution in the Three-Dimensional Random Resistor Network at the Percolation Threshold. Phys Rev E 53:2292–2297 9. Blumenfeld R, Meir Y, Aharony A, Harris AB (1987) Resistance Fluctuations in Randomly Diluted Networks. Phys Rev B 35:3524–3535 10. Bruggeman DAG (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann Phys (Leipzig) 24:636–679. [Engl Trans: Computation of Different Physical Constants of Hetero-
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33. Fogelholm R (1980) The Conductance of Large Percolation Network Samples. J Phys C 13:L571–L574 34. Fortuin CM, Kasteleyn PW (1972) On the Random Cluster Model. I. Introduction and Relation to Other Models. Phys 57:536–564 35. Frank DJ, Lobb CJ (1988) Highly Efficient Algorithm for Percolative Transport Studies in Two Dimensions. Phys Rev B 37:302–307 36. Gingold DB, Lobb CJ (1990) Percolative Conduction in Three Dimensions. Phys Rev B 42:8220–8224 37. Golden K (1989) Convexity in Random Resistor Networks. In: Kohn RV, Milton GW (eds) Random Media and Composites. SIAM, Philadelphia, pp 149–170 38. Golden K (1990) Convexity and Exponent Inequalities for Conduction Near Percolation. Phys Rev Lett 65:2923–2926 39. Gumbel EJ (1958) Statistics of Extremes. Columbia University Press, New York 40. Halperin BI, Feng S, Sen PN (1985) Differences Between Lattice and Continuum Percolation Transport Exponents. Phys Rev Lett 54:2391–2394 41. Halsey TC, Jensen MH, Kadanoff LP, Procaccia I, Shraiman BI (1986) Fractal Measures and Their Singularities: The Characterization of Strange Sets. Phys Rev A 33:1141–1151 42. Halsey TC, Meakin P, Procaccia I (1986) Scaling Structure of the Surface Layer of Diffusion-Limited Aggregates. Phys Rev Lett 56:854–857 43. Harary F (1969) Graph Theory. Addison Wesley, Reading, MA 44. Harris AB, Kim S, Lubensky TC (1984) " Expansion for the Conductance of a Random Resistor Network. Phys Rev Lett 53:743–746 45. Hentschel HGE, Procaccia I (1983) The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors. Phys D 8:435–444 46. Kahng B, Batrouni GG, Redner S (1987) Logarithmic Voltage Anomalies in Random Resistor Networks. J Phys A: Math Gen 20:L827–834 47. Kasteleyn PW, Fortuin CM (1969) Phase Transitions in Lattice Systems with Random Local Properties. J Phys Soc Japan (Suppl) 26:11–14 48. Keller JB (1964) A Theorem on the Conductance of a Composite Medium. J Math Phys 5:548–549 49. Kenkel SW, Straley JP (1982) Percolation Theory of Nonlinear Circuit Elements. Phys Rev Lett 49:767–770 50. Kennelly AE (1899) The Equivalence of Triangles and ThreePointed Stars in Conducting Networks. Electr World Eng 34:413–414 51. Kinzel W (1983) Directed Percolation in Percolation Structures and Processes. In: Deutscher G, Zallen R, Adler J, Hilger A, Bristol UK, Redner S (eds) Annals of the Israel Physical Society, vol 5, Percolation and Conduction in Random Resistor-Diode Networks, ibid, pp 447–475 52. Kirchhoff G (1847) Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann Phys Chem 72:497– 508. [English translation by O’Toole JB, Kirchhoff G (1958) On the solution of the equations obtained from the investigation of the linear distribution of galvanic currents. IRE Trans Circuit Theory CT5:4–8.] 53. Kirkpatrick S (1971) Classical Transport in Disordered Media: Scaling and Effective-Medium Theories. Phys Rev Lett 27:1722–1725
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76. Redner S (2001) A Guide to First-Passage Processes. Cambridge University Press, New York 77. Redner S, Brooks JS (1982) Analog Experiments and Computer Simulations for Directed Conductance. J Phys A Math Gen 15:L605–L610 78. Roux S, Herrmann HJ (1987) Disorder-Induced Nonlinear Conductivity. Europhys Lett 4:1227–1231 79. Sahimi M, Hughes BD, Scriven LE, Davis HT (1983) Critical Exponent of Percolation Conductance by Finite-Size Scaling. J Phys C 16:L521–L527 80. Sarychev AK, Vinogradoff AP (1981) Drop Model of Infinite Cluster for 2d Percolation. J Phys C 14:L487–L490 81. Senturia SB, Wedlock BD (1975) Electronic Circuits and Applications. Wiley, New York, pp 75 82. Skal AS, Shklovskii BI (1975) Topology of the Infinite Cluster of The Percolation Theory and its Relationship to the Theory of Hopping Conduction. Fiz Tekh Poluprov 8:1586–1589 [Engl. transl.: Sov Phys-Semicond 8:1029–1032] 83. Stanley HE (1971) Introduction to Phase Transition and Critical Phenomena. Oxford University Press, Oxford, UK 84. Stanley HE (1977) Cluster Shapes at the Percolation Threshold: An Effective Cluster Dimensionality and its Connection with Critical-Point Exponents. J Phys A Math Gen 10: L211–L220 85. Stauffer D, Aharony A (1994) Introduction to Percolation Theory, 2nd edn. Taylor & Francis, London, Bristol, PA 86. Stenull O, Janssen HK, Oerding K (1999) Critical Exponents for Diluted Resistor Networks. Phys Rev E 59:4919–4930 87. Stephen M (1978) Mean-Field Theory and Critical Exponents for a Random Resistor Network. Phys Rev B 17:4444–4453 88. Stephen MJ (1976) Percolation problems and the Potts model. Phys Lett A 56:149–150 89. Stinchcombe RB, Watson BP (1976) Renormalization Group Approach for Percolation Conductance. J Phys C 9:3221–3247 90. Straley JP (1977) Critical Exponents for the Conductance of Random Resistor Lattices. Phys Rev B 15:5733–5737 91. Straley JP (1982) Random Resistor Tree in an Applied Field. J Phys C 10:3009–3014 92. Straley JP (1982) Threshold Behaviour of Random Resistor Networks: A Synthesis of Theoretical Approaches. J Phys C 10:2333–2341 93. Trugman SA, Weinrib A (1985) Percolation with a Threshold at Zero: A New Universality Class. Phys Rev B 31:2974–2980 94. van der Pol B, Bremmer H (1955) Operational Calculus Based on the Two-Sided Laplace Integral. Cambridge University Press, Cambridge, UK 95. Venezian G (1994) On the resistance between two points on a grid. Am J Phys 62:1000–1004 96. Watson GN (1939) Three Triple Integrals. Oxford Ser 2. Quart J Math 10:266–276 97. Webman I, Jortner J, Cohen MH (1975) Numerical Simulation of Electrical Conductance in Microscopically Inhomogeneous Materials. Phys Rev B 11:2885–2892 98. Weiss GH (1994) Aspects and Applications of the Random Walk. Elsevier Science Publishing Co, New York 99. Wu FY (1982) The Potts Model. Rev Mod Phys 54:235–268 100. Zabolitsky JG (1982) Monte Carlo Evidence Against the Alexander–Orbach Conjecture for Percolation Conductance. Phys Rev B 30:4077–4079
Fractal and Multifractal Time Series
Fractal and Multifractal Time Series JAN W. KANTELHARDT Institute of Physics, Martin-Luther-University Halle-Wittenberg, Halle, Germany Article Outline Glossary Definition of the Subject Introduction Fractal and Multifractal Time Series Methods for Stationary Fractal Time Series Analysis Methods for Non-stationary Fractal Time Series Analysis Methods for Multifractal Time Series Analysis Statistics of Extreme Events in Fractal Time Series Simple Models for Fractal and Multifractal Time Series Future Directions Acknowledgment Bibliography Glossary Time series One dimensional array of numbers (x i ); i D 1; : : : ; N, representing values of an observable x usually measured equidistant (or nearly equidistant) in time. Complex system A system consisting of many non-linearly interacting components. It cannot be split into simpler sub-systems without tampering with the dynamical properties. Scaling law A power law with a scaling exponent (e. g. ˛) describing the behavior of a quantity F (e. g., fluctuation, spectral power) as function of a scale parameter s (e. g., time scale, frequency) at least asymptotically: F(s) s˛ . The power law should be valid for a large range of s values, e. g., at least for one order of magnitude. Fractal system A system characterized by a scaling law with a fractal, i. e., non-integer exponent. Fractal systems are self-similar, i. e., a magnification of a small part is statistically equivalent to the whole. Self-affine system Generalization of a fractal system, where different magnifications s and s 0 D s H have to be used for different directions in order to obtain a statistically equivalent magnification. The exponent H is called Hurst exponent. Self-affine time series and time series becoming self-affine upon integration are commonly denoted as fractal using a less strict terminology.
Multifractal system A system characterized by scaling laws with an infinite number of different fractal exponents. The scaling laws must be valid for the same range of the scale parameter. Crossover Change point in a scaling law, where one scaling exponent applies for small scale parameters and another scaling exponent applies for large scale parameters. The center of the crossover is denoted by its characteristic scale parameter s in this article. Persistence In a persistent time series, a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. A fractal scaling law holds at least for a limited range of scales. Short-term correlations Correlations that decay sufficiently fast that they can be described by a characteristic correlation time scale; e. g., exponentially decaying correlations. A crossover to uncorrelated behavior is observed on larger scales. Long-term correlations Correlations that decay sufficiently slow that a characteristic correlation time scale cannot be defined; e. g., power-law correlations with an exponent between 0 and 1. Power-law scaling is observed on large time scales and asymptotically. The term long-range correlations should be used if the data is not a time series. Non-stationarities If the mean or the standard deviation of the data values change with time, the weak definition of stationarity is violated. The strong definition of stationarity requires that all moments remain constant, i. e., the distribution density of the values does not change with time. Non-stationarities like monotonous, periodic, or step-like trends are often caused by external effects. In a more general sense, changes in the dynamics of the system also represent non-stationarities. Definition of the Subject Data series generated by complex systems exhibit fluctuations on a wide range of time scales and/or broad distributions of the values. In both equilibrium and nonequilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders of magnitude. Such scaling laws allow for a characterization of the data and the generating complex system by fractal (or multifractal) scaling exponents, which can serve as characteristic fingerprints of the systems in comparisons with other systems and with models. Fractal scaling behavior has been observed, e. g., in many data series from experimental physics, geophysics, medicine, physiology, and
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even social sciences. Although the underlying causes of the observed fractal scaling are often not known in detail, the fractal or multifractal characterization can be used for generating surrogate (test) data, modeling the time series, and deriving predictions regarding extreme events or future behavior. The main application, however, is still the characterization of different states or phases of the complex system based on the observed scaling behavior. For example, the health status and different physiological states of the human cardiovascular system are represented by the fractal scaling behavior of the time series of intervals between successive heartbeats, and the coarsening dynamics in metal alloys are represented by the fractal scaling of the time-dependent speckle intensities observed in coherent X-ray spectroscopy. In order to observe fractal and multifractal scaling behavior in time series, several tools have been developed. Besides older techniques assuming stationary data, there are more recently established methods differentiating truly fractal dynamics from fake scaling behavior caused by non-stationarities in the data. In addition, short-term and long-term correlations have to be clearly distinguished to show fractal scaling behavior unambiguously. This article describes several methods originating from statistical physics and applied mathematics, which have been used for fractal and multifractal time series analysis in stationary and non-stationary data. Introduction The characterization and understanding of complex systems is a difficult task, since they cannot be split into simpler subsystems without tampering with the dynamical properties. One approach in studying such systems is the recording of long time series of several selected variables (observables), which reflect the state of the system in a dimensionally reduced representation. Some systems are characterized by periodic or nearly periodic behavior, which might be caused by oscillatory components or closed-loop regulation chains. However, in truly complex systems such periodic components are usually not limited to one or two characteristic frequencies or frequency bands. They rather extend over a wide spectrum, and fluctuations on many time scales as well as broad distributions of the values are found. Often no specific lower frequency limit – or, equivalently, upper characteristic time scale – can be observed. In these cases, the dynamics can be characterized by scaling laws which are valid over a wide (possibly even unlimited) range of time scales or frequencies; at least over orders of magnitude. Such dynamics are usually denoted as fractal or multifractal, depending on the
question if they are characterized by one scaling exponent or by a multitude of scaling exponents. The first scientist who applied fractal analysis to natural time series was Benoit B. Mandelbrot [1,2,3], who included early approaches by H.E. Hurst regarding hydrological systems [4,5]. For extensive introductions describing fractal scaling in complex systems, we refer to [6,7,8, 9,10,11,12,13]. In the last decade, fractal and multifractal scaling behavior has been reported in many natural time series generated by complex systems, including Geophysics time series (recordings of temperature, precipitation, water runoff, ozone levels, wind speed, seismic events, vegetational patterns, and climate dynamics), Medical and physiological time series (recordings of heartbeat, respiration, blood pressure, blood flow, nerve spike intervals, human gait, glucose levels, and gene expression data), DNA sequences (they are not actually time series), Astrophysical time series (X-ray light sources and sunspot numbers), Technical time series (internet traffic, highway traffic, and neutronic power from a reactor), Social time series (finance and economy, language characteristics, fatalities in conflicts), as well as Physics data (also going beyond time series), e. g., surface roughness, chaotic spectra of atoms, and photon correlation spectroscopy recordings. If one finds that a complex system is characterized by fractal (or multifractal) dynamics with particular scaling exponents, this finding will help in obtaining predictions on the future behavior of the system and on its reaction to external perturbations or changes in the boundary conditions. Phase transitions in the regulation behavior of a complex system are often associated with changes in their fractal dynamics, allowing for a detection of such transitions (or the corresponding states) by fractal analysis. One example for a successful application of this approach is the human cardiovascular system, where the fractality of heartbeat interval time series was shown to reflect certain cardiac impairments as well as sleep stages [14,15]. In addition, one can test and iteratively improve models of the system until they reproduce the observed scaling behavior. One example for such an approach is climate modeling, where the models were shown to need input from volcanos and solar radiation in order to reproduce the long-term correlated (fractal) scaling behavior [16] previously found in observational temperature data [17]. Fractal (or multifractal) scaling behavior certainly cannot be assumed a priori, but has to be established. Hence,
Fractal and Multifractal Time Series
there is a need for refined analysis techniques, which help to differentiate truly fractal dynamics from fake scaling behavior caused, e. g., by non-stationarities in the data. If conventional statistical methods are applied for the analysis of time series representing the dynamics of a complex system [18,19], there are two major problems. (i) The number of data series and their durations (lengths) are usually very limited, making it difficult to extract significant information on the dynamics of the system in a reliable way. (ii) If the length of the data is extended using computer-based recording techniques or historical (proxy) data, non-stationarities in the signals tend to be superimposed upon the intrinsic fluctuation properties and measurement noise. Non-stationarities are caused by external or internal effects that lead to either continuous or sudden changes in the average values, standard deviations or regulation mechanism. They are a major problem for the characterization of the dynamics, in particular for finding the scaling properties of given data. Fractal and Multifractal Time Series Fractality, Self-Affinity, and Scaling The topic of this article is the fractality (and/or multifractality) of time series. Since fractals and multifractals in general are discussed in many other articles of the encyclopedia, the concept is not thoroughly explained here. In particular, we refer to the articles Fractal Geometry, A Brief Introduction to and Fractals and Multifractals, Introduction to for the formalism describing fractal and multifractal structures, respectively. In a strict sense, most time series are one dimensional, since the values of the considered observable are measured in homogeneous time intervals. Hence, unless there are missing values, the fractal dimension of the support is D(0) D 1. However, there are rare cases where most of the values of a time series are very small or even zero, causing a dimension D(0) < 1 of the support. In these cases, one has to be very careful in selecting appropriate analysis techniques, since many of the methods presented in this article are not accurate for such data; the wavelet transform modulus maxima technique (see Subsect. “Wavelet Transform Modulus Maxima (WTMM) Method”) is the most advanced applicable method. Even if the fractal dimension of support is one, the information dimension D(1) and the correlation dimension D(2) can be studied. As we will see in Subsect. “The Structure Function Approach and Singularity Spectra”, D(2) is in fact explicitly related to all exponents studied in monofractal time series analysis. However, usually a slightly different approach is employed based on the no-
tion of self-affinity instead of (multi-) fractality. Here, one takes into account that the time axis and the axis of the measured values x(t) are not equivalent. Hence, a rescaling of time t by a factor a may require rescaling of the series values x(t) by a different factor aH in order to obtain a statistically similar (i. e., self-similar) picture. In this case the scaling relation x(t) ! a H x(at)
(1)
holds for an arbitrary factor a, describing the data as selfaffine (see, e. g., [6]). The Hurst exponent H (after the water engineer H.E. Hurst [4]) characterizes the type of self affinity. Figure 1a shows several examples of self-affine time series with different H. The trace of a random walk (Brownian motion, third line in Fig. 1a), for example, is characterized by H D 0:5, implying that the position axis must be rescaled by a factor of two if the time axis is rescaled by a factor of four. Note that self-affine series are often denoted as fractal even though they are not fractal in the strict sense. In this article the term “fractal” will be used in the more general sense including all data, where a Hurst exponent H can be reasonably defined. The scaling behavior of self-affine data can also be characterized by looking at their mean-square displacement. Since the mean-square displacement of a random walker is known to increase linearly in time, hx 2 (t)i t, deviations from this law will indicate the presence of selfaffine scaling. As we will see in Subsect. “Fluctuation Analysis (FA)”, one can thus retrieve the Hurst (or self-affinity) exponent H by studying the scaling behavior of the mean-square displacement, or the mean-square fluctuations hx 2 (t)i t 2H . Persistence, Long- and Short-Term Correlations Self-affine data are persistent in the sense that a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. For the trace of a random walk, persistence on all time scales is trivial, since a later position is just a former one plus some random increment(s). The persistence holds for all time scales, where the self-affinity relation (1) holds. However, the degree of persistence can also vary on different time scales. Weather is a typical example: while the weather tomorrow or in one week is probably similar to the weather today (due to a stable general weather condition), persistence is much harder to be seen on longer time scales. Considering the increments x i D x i x i1 of a selfaffine series, (x i ), i D 1; : : : ; N with N values measured equidistant in time, one finds that the x i can be either
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Fractal and Multifractal Time Series, Figure 1 a Examples of self-affine series x i characterized by different Hurst exponents H = 0.9, 0.7, 0.5, 0.3 (from top to bottom). The data has been generated by Fourier filtering using the same seed for the random number generator. b Differentiated series xi of the data from a; the xi are characterized by positive long-term correlations (persistence) with = 0.2 and 0.6 (first and second line), uncorrelated behavior (third line), and anti-correlations (bottom line), respectively
persistent, independent, or anti-persistent. Examples for all cases are shown in Fig. 1b. In our example of the random walk with H D 0:5 (third line in the figure), the increments (steps) are fully independent of each other. Persistent and anti-persistent increments, where a positive increment is likely to be followed by another positive or negative increment, respectively, are also leading to persistent P integrated series x i D ijD1 x j . For stationary data with constant mean and standard deviation the auto-covariance function of the increments, ˛ ˝ C(s) D x i x iCs D
Ns 1 X x i x iCs Ns
(2)
iD1
can be studied to determine the degree of persistence. If C(s) is divided by the variance h(x i )2 i, it becomes the auto-correlation function; both are identical if the data are normalized with unit variance. If the x i are uncorrelated (as for the random walk), C(s) is zero for s > 0. Shortrange correlations of the increments x i are usually described by C(s) declining exponentially, C(s) exp(s/t )
(3)
with a characteristic decay time t . Such behavior is typical for increments generated by an auto-regressive (AR) process x i D cx i1 C " i
(4)
with random uncorrelated offsets "i and c D exp(1/t ). Figure 2a shows the auto-correlation function for one configuration of an AR process with t D 48.
Fractal and Multifractal Time Series, Figure 2 Comparison of the autocorrelation functions C(s) (decreasing functions) and fluctuation functions F2 (s) (increasing functions) for short-term correlated data (top panel) and long-term correlated data ( D 0:4, bottom panel). The asymptotic slope H ˛ D 0:5 of F2 (s) clearly indicates missing long-term correlations, while H ˛ D 1 /2 > 0:5 indicates long-term correlations. The difference is much harder to observe in C(s), where statistical fluctuations and negative values start occurring above s 100. The data have been generated by an AR process Eq. (4) and Fourier filtering, respectively. The dashed lines indicate the theoretical curves
R1 For so-called long-range correlations 0 C(s)ds diverges in the limit of infinitely long series (N ! 1). In practice, this means that t cannot be defined because it increases with increasing N. For example, C(s) declines as a power-law
Fractal and Multifractal Time Series
C(s) / s
(5)
with an exponent 0 < < 1. Figure 2b shows C(s) for one configuration with D 0:4. This type of behavior can be modeled by the Fourier filtering technique (see Subsect. “Fourier Filtering”). Long-term correlated, i. e. persistent, behavior of the x i leads to self-affine scaling behavior of the xi , characterized by H D 1 /2, as will be shown below. Crossovers and Non-Stationarities in Time Series Short-term correlated increments x i characterized by a finite characteristic correlation decay time t lead to a crossover in the scaling behavior of the integrated sePi ries x i D jD1 x j , see Fig. 2a for an example. Since the position of the crossover might be numerically different from t , we denote it by s here. Time series with a crossover are not self-affine and there is no unique Hurst exponent H characterizing them. While H > 0:5 is observed on small time scales (indicating correlations in the increments), the asymptotic behavior (for large time scales s t and s ) is always characterized by H D 0:5, since all correlations have decayed. Many natural recordings are characterized by pronounced short-term correlations in addition to scaling long-term correlations. For example, there are short-term correlations due to particular general weather situations in temperature data and due to respirational effects in heartbeat data. Crossovers in the scaling behavior of complex time series can also be caused by different regulation mechanisms on fast and slow time scales. Fluctuations of river runoff, for example, show different scaling behavior on time scales below and above approximately one year. Non-stationarities can also cause crossovers in the scaling behavior of data if they are not properly taken into account. In the most strict sense, non-stationarities are variations in the mean or the standard deviation of the data (violating weak stationarity) or the distribution of the data values (violating strong stationarity). Non-stationarities like monotonous, periodic or step-like trends are often caused by external effects, e. g., by the greenhouse warming and seasonal variations for temperature records, different levels of activity in long-term physiological data, or unstable light sources in photon correlation spectroscopy. Another example for non-stationary data is a record consisting of segments with strong fluctuations alternating with segments with weak fluctuations. Such behavior will cause a crossover in scaling at the time scale corresponding to the typical duration of the homogeneous segments. Different mechanisms of regulation during different time segments – like, e. g., different heartbeat reg-
ulation during different sleep stages at night – can also cause crossovers; they are regarded as non-stationarities here, too. Hence, if crossovers in the scaling behavior of data are observed, more detailed studies are needed to find out the cause of the crossovers. One can try to obtain homogenous data by splitting the original series and employing methods that are at least insensitive to monotonous (polynomially shaped) trends. To characterize a complex system based on time series, trends and fluctuations are usually studied separately (see, e. g., [20] for a discussion). Strong trends in data can lead to a false detection of long-range statistical persistence if only one (non-detrending) method is used or if the results are not carefully interpreted. Using several advanced techniques of scaling time series analysis (as described in Sect. “Methods for Non-stationary Fractal Time Series Analysis”) crossovers due to trends can be distinguished from crossovers due to different regulation mechanisms on fast and slow time scales. The techniques can thus assist in gaining insight into the scaling behavior of the natural variability as well as into the kind of trends of the considered time series. It has to be stressed that crossovers in scaling behavior must not be confused with multifractality. Even though several scaling exponents are needed, they are not applicable for the same regime (i. e., the same range of time scales). Real multifractality, on the other hand, is characterized by different scaling behavior of different moments over the full range of time scales (see next section). Multifractal Time Series Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. As discussed in the previous section, there might exist crossover (time-) scales s separating regimes with different scaling exponents. In other cases, the scaling behavior is more complicated, and different scaling exponents are required for different parts of the series. In even more complicated cases, such different scaling behavior can be observed for many interwoven fractal subsets of the time series. In this case a multitude of scaling exponents is required for a full description of the scaling behavior in the same range of time scales, and a multifractal analysis must be applied. Two general types of multifractality in time series can be distinguished: (i) Multifractality due to a broad probability distribution (density function) for the values of the time series, e. g. a Levy distribution. In this case the multifractality cannot be removed by shuffling the series. (ii) Multifractality due to different long-term correlations
467
468
Fractal and Multifractal Time Series
of the small and large fluctuations. In this case the probability density function of the values can be a regular distribution with finite moments, e. g., a Gaussian distribution. The corresponding shuffled series will exhibit nonmultifractal scaling, since all long-range correlations are destroyed by the shuffling procedure. Randomly shuffling the order of the values in the time series is the easiest way of generating surrogate data; however, there are more advanced alternatives (see Sect. “Simple Models for Fractal and Multifractal Time Series”). If both kinds of multifractality are present, the shuffled series will show weaker multifractality than the original series. A multifractal analysis of time series will also reveal higher order correlations. Multifractal scaling can be observed if, e. g., three or four-point correlations scale differently from the standard two-point correlations studied by classical autocorrelation analysis (Eq. (2)). In addition, multifractal scaling is observed if the scaling behavior of small and large fluctuations is different. For example, extreme events might be more or less correlated than typical events.
C(s) / s with a correlation exponent 0 < < 1 (see Eqs. (3) and (5), respectively). As illustrated by the two examples shown in Fig. 2, a direct calculation of C(s) is usually not appropriate due to noise superimposed on the data x˜ i and due to underlying non-stationarities of unknown origin. Non-stationarities make the definition of C(s) problematic, because the average hxi is not welldefined. Furthermore, C(s) strongly fluctuates around zero on large scales s (see Fig. 2b), making it impossible to find the correct correlation exponent . Thus, one has to determine the value of indirectly.
Methods for Stationary Fractal Time Series Analysis
The spectral exponent ˇ and the correlation exponent
can thus be obtained by fitting a power-law to a double logarithmic plot of the power spectrum S( f ). An example is shown in Fig. 3. The relation (6) can be derived from the Wiener–Khinchin theorem (see, e. g., [25]). If, instead of x˜ i D x i the integrated runoff time series is Fourier P transformed, i. e., x˜ i D x i X D ijD1 x j , the resulting power spectrum scales as S( f ) f 2ˇ .
In this section we describe four traditional approaches for the fractal analysis of stationary time series, see [21,22,23] for comparative studies. The main focus is on the determination of the scaling exponents H or , defined in Eqs. (1) and (5), respectively, and linked by H D 1 /2 in long-term persistent data. Methods taking non-stationarities into account will be discussed in the next chapter.
Spectral Analysis If the time series is stationary, we can apply standard spectral analysis techniques (Fourier transform) and calculate the power spectrum S( f ) of the time series (x˜ i ) as a function of the frequency f to determine self-affine scaling behavior [24]. For long-term correlated data characterized by the correlation exponent , we have S( f ) f ˇ
with ˇ D 1 :
(6)
Autocorrelation Function Analysis We consider a record (x i ) of i D 1; : : : ; N equidistant measurements. In most applications, the index i will correspond to the time of the measurements. We are interested in the correlation of the values xi and x iCs for different time lags, i. e. correlations over different time scales s. In order to remove a constant offset in the data, the mean PN ˜ i x i hxi. Alhxi D N1 iD1 x i is usually subtracted, x ternatively, the correlation properties of increments x˜ i D x i D x i x i1 of the original series can be studied (see also Subsect. “Persistence, Long- and Short-Term Correlations”). Quantitatively, correlations between x˜ -values separated by s steps are defined by the (auto-) covariance function C(s) D hx˜ i x˜ iCs i or the (auto-) correlation function C(s)/hx˜ 2i i, see also Eq. (2). As already mentioned in Subsect. “Persistence, Longand Short-Term Correlations”, the x˜ i are short-term correlated if C(s) declines exponentially, C(s) exp(s/t ), and long-term correlated if C(s) declines as a power-law
Fractal and Multifractal Time Series, Figure 3 Spectral analysis of a fractal time series characterized by longterm correlations with D 0:4 (ˇ D 0:6). The expected scaling behavior (dashed line indicating the slope –ˇ) is observed only after binning of the spectrum (circles). The data has been generated by Fourier filtering
Fractal and Multifractal Time Series
Spectral analysis, however, does not yield more reliable results than auto-correlation analysis unless a logarithmic binning procedure is applied to the double logarithmic plot of S( f ) [21], see also Fig. 3. I. e., the average of log S( f ) is calculated in successive, logarithmically wide bands from a n f0 to a nC1 f0 , where f 0 is the minimum frequency, a > 1 is a factor (e. g., a D 1:1), and the index n is counting the bins. Spectral analysis also requires stationarity of the data. Hurst’s Rescaled-Range Analysis The first method for the analysis of long-term persistence in time series based on random walk theory has been proposed by the water construction engineer Harold Edwin Hurst (1880–1978), who developed it while working in Egypt. His so-called rescaled range analysis (R/S analysis) [1,2,4,5,6] begins with splitting of the time series (x˜ i ) into non-overlapping segments of size (time scale) s (first step), yielding Ns D int(N/s) segments altogether. In the second step, the profile (integrated data) is calculated in each segment D 0; : : : ; Ns 1, Y ( j) D
j X
(x˜sCi hx˜sCi is )
iD1
D
j X
x˜sCi
iD1
s jX x˜sCi : s
(7)
By the subtraction of the local averages, piecewise constant trends in the data are eliminated. In the third step, the differences between minimum and maximum value (ranges) R (s) and the standard deviations S (s) in each segment are calculated,
(8)
jD1
Finally, the rescaled range is averaged over all segments to obtain the fluctuation function F(s), FRS (s) D
N s 1 R (s) 1 X sH N s D0 S (s)
for s 1 ;
Fluctuation Analysis (FA) The standard fluctuation analysis (FA) [8,26] is also based on random walk theory. For a time series (x˜ i ), i D 1; : : : ; N, with zero mean, we consider the global profile, i. e., the cumulative sum (cf. Eq. (7)) Y( j) D
j X
x˜ i ;
j D 0; 1; 2; : : : ; N ;
(10)
iD1
iD1
R (s) D maxsjD1 Y ( j) minsjD1 Y ( j); v u X u1 s Y2 ( j) : S (s) D t s
that H actually characterizes the self-affinity of the profile function (7), while ˇ and refer to the original data. The values of H, that can be obtained by Hurst’s rescaled range analysis, are limited to 0 < H < 2, and significant inaccuracies are to be expected close to the bounds. Since H can be increased or decreased by one if Pj ˜ i ) or differentiated the data is integrated (x˜ j ! iD1 x (x˜ i ! x˜ i x˜ i1 ), respectively, one can always find a way to calculate H by rescaled range analysis provided the data is stationary. While values H < 1/2 indicate long-term anticorrelated behavior of the data x˜ i , H > 1/2 indicates longterm positively correlated behavior. For power-law correlations decaying faster than 1/s, we have H D 1/2 for large s values, like for uncorrelated data. Compared with spectral analysis, Hurst’s rescaled range analysis yields smoother curves with less effort (no binning procedure is necessary) and works also for data with piecewise constant trends.
(9)
where H is the Hurst exponent already introduced in Eq. (1). One can show [1,24] that H is related to ˇ and
by 2H 1 C ˇ D 2 (see also Eqs. (6) and (14)). Note that 0 < < 1, so that the right part of the equation does not hold unless 0:5 < H < 1. The relationship does not hold in general for multifractal data. Note also
and study how the fluctuations of the profile, in a given time window of size s, increase with s. The procedure is illustrated in Fig. 4 for two values of s. We can consider the profile Y( j) as the position of a random walker on a linear chain after j steps. The random walker starts at the origin and performs, in the ith step, a jump of length x˜ i to the bottom, if x˜ i is positive, and to the top, if x˜ i is negative. To find how the square-fluctuations of the profile scale with s, we first divide the record of N elements into N s D int(N/s) non-overlapping segments of size s starting from the beginning (see Fig. 4) and another N s non-overlapping segments of size s starting from the end of the considered series. This way neither data at the end nor at the beginning of the record is neglected. Then we determine the fluctuations in each segment . In the standard FA, we obtain the fluctuations just from the values of the profile at both endpoints of each segment D 1; : : : ; Ns , 2 (; s) D [Y(s) Y((1)s)]2 ; FFA
(11)
(see Fig. 4) and analogous for D N sC1 ; : : : ; 2N s , 2 (; s) D [Y(N(N s )s)Y(N(1N s )s)]2 : (12) FFA
469
470
Fractal and Multifractal Time Series
differentiation of the data, the same rules apply as listed for H in the previous subsection. The results of FA become statistically unreliable for scales s larger than one tenth of the length of the data, i. e. the analysis should be limited by s < N/10. Methods for Non-stationary Fractal Time Series Analysis Wavelet Analysis
Fractal and Multifractal Time Series, Figure 4 Illustration of the fluctuation analysis (FA) and the detrended fluctuation analysis (DFA). For two segment durations (time scales) s D 100 (a) and 200 (b), the profiles Y( j) (blue lines; defined in Eq. (11), the values used for fluctuation analysis in Eq. (12) (green circles), and least-square quadratic fits to the profiles (red lines) are shown
The origins of wavelet analysis come from signal theory, where frequency decompositions of time series were studied [27,28]. Like the Fourier transform, the wavelet transform of a signal x(t) is a convolution integral to be replaced by a summation in case of a discrete time series (x˜ i ); i D 1; : : : ; N, Z 1 1 L (; s) D x(t) [(t )/s] dt s 1 N
2 (; s) over all subsequences to obtain Then we average FFA the mean fluctuation F2 (s),
" F2 (s) D
#1/2 2N 1 Xs 2 F (; s) s˛ : 2N s D1 FA
(13)
By definition, F2 (s) can be viewed as the root-mean-square displacement of the random walker on the chain, after s steps (the reason for the index 2 will become clear later). For uncorrelated xi values, we obtain Fick’s diffusion law F2 (s) s1/2 . For the relevant case of long-term correlations, in which C(s) follows the power-law behavior of Eq. (5), F2 (s) increases by a power law, F2 (s) s ˛
with ˛ H ;
(14)
where the fluctuation exponent ˛ is identical with the Hurst exponent H for mono-fractal data and related to
and ˇ by 2˛ D 1 C ˇ D 2 :
(15)
The typical behavior of F2 (s) for short-term correlated and long-term correlated data is illustrated in Fig. 2. The relation (15) can be derived straightforwardly by inserting Eqs. (10), (2), and (5) into Eq. (11) and separating sums over products x˜ i x˜ j with identical and different i and j, respectively. The range of the ˛ values that can be studied by standard FA is limited to 0 < ˛ < 1, again with significant inaccuracies close to the bounds. Regarding integration or
D
1X x˜ i s
[(i )/s] :
(16)
iD1
Here, (t) is a so-called mother wavelet, from which all daughter wavelets ;s (t) D ((t )/s) evolve by shifting and stretching of the time axis. The wavelet coefficients L (; s) thus depend on both time position and scale s. Hence, the local frequency decomposition of the signal is described with a time resolution appropriate for the considered frequency f D 1/s (i. e., inverse time scale). All wavelets (t) must have zero mean. They are often chosen to be orthogonal to polynomial trends, so that the analysis method becomes insensitive to possible trends in the data. Simple examples are derivatives of a Gaus(n) sian, Gauss (t) D dn /(dt n ) exp(x 2 /2), like the Mexican (2) (1) hat wavelet Gauss and the Haar wavelet, Haar (t) D C1 if 0 t < 1, 1 if 1 t < 2, and 0 otherwise. It is straightforward to construct higher order Haar wavelets that are orthogonal to linear, quadratic and cubic trends, (2) e. g., Haar (t) D 1 for t 2 [0; 1) [ [2; 3), 2 for t 2 [1; 2), (3) (t) D 1 for t 2 [0; 1), 3 for and 0 otherwise, or Haar t 2 [1; 2), +3 for t 2 [2; 3), 1 for t 2 [3; 4), and 0 otherwise. Discrete Wavelet Transform (WT) Approach A detrending fractal analysis of time series can be easily implemented by considering Haar wavelet coefficients of the profile Y( j), Eq. (10) [17,30]. In this case the convolution (16) corresponds to the addition and subtraction of mean values of Y( j) within segments of size s. Hence, P defining Y¯ (s) D 1s sjD1 Y(s C j), the coefficients can
Fractal and Multifractal Time Series
be written as FWT1 (; s) L FWT2 (; s) L
(0) Haar
(s; s) D Y¯ (s) Y¯C1 (s) ;
(1) Haar
(17)
(s; s)
D Y¯ (s) 2Y¯C1 (s) C Y¯C2 (s) ;
Detrended Fluctuation Analysis (DFA) (18)
and FWT3 (; s) L
(2) Haar
(s; s)
D Y¯ (s) 3Y¯C1 (s) C 3Y¯C2 (s) Y¯C3 (s)
results become statistically unreliable for scales s larger than one tenth of the length of the data, just as for FA.
(19)
for constant, linear and quadratic detrending, respectively. The generalization for higher orders of detrending is ob2 vious. The resulting mean-square fluctuations FWTn (; s) are averaged over all to obtain the mean fluctuation F2 (s), see Eq. (13). Figure 5 shows typical results for WT analysis of long-term correlated, short-term correlated and uncorrelated data. Regarding trend-elimination, wavelet transform WT0 corresponds to standard FA (see Subsect. “Fluctuation Analysis (FA)”), and only constant trends in the profile are eliminated. WT1 is similar to Hurst’s rescaled range analysis (see Subsect. “Hurst’s Rescaled-Range Analysis”): linear trends in the profile and constant trends in the data are eliminated, and the range of the fluctuation exponent ˛ H is up to 2. In general, WTn determines the fluctuations from the nth derivative, this way eliminating trends described by (n 1)st-order polynomials in the data. The
In the last 14 years Detrended Fluctuation Analysis (DFA), originally introduced by Peng et al. [31], has been established as an important method to reliably detect longrange (auto-) correlations in non-stationary time series. The method is based on random walk theory and basically represents a linear detrending version of FA (see Subsect. “Fluctuation Analysis (FA)”). DFA was later generalized for higher order detrending [15], separate analysis of sign and magnitude series [32] (see Subsect. “Sign and Magnitude (Volatility) DFA”), multifractal analysis [33] (see Subsect. “Multifractal Detrended Fluctuation Analysis (MFDFA)”), and data with more than one dimension [34]. Its features have been studied in many articles [35,36,37,38,39,40]. In addition, several comparisons of DFA with other methods for stationary and nonstationary time-series analysis have been published, see, e. g., [21,23,41,42] and in particular [22], where DFA is compared with many other established methods for short data sets, and [43], where it is compared with recently suggested improved methods. Altogether, there are about 600 papers applying DFA (till September 2008). In most cases positive auto-correlations were reported leaving only a few exceptions with anti-correlations, see, e. g., [44,45,46]. Like in the FA method, one first calculates the global profile according to Eq. (10) and divides the profile into N s D int(N/s) non-overlapping segments of size s starting from the beginning and another N s segments starting from the end of the considered series. DFA explicitly deals with monotonous trends in a detrending procedure. This m ( j) within is done by estimating a polynomial trend y;s each segment by least-square fitting and subtracting this trend from the original profile (‘detrending’), m ( j) : Y˜s ( j) D Y( j) y;s
Fractal and Multifractal Time Series, Figure 5 Application of discrete wavelet transform (WT) analysis on uncorrelated data (black circles), long-term correlated data ( D 0:8; ˛ D 0:6, red squares), and short-term correlated data (summation of three AR processes, green diamonds). Averages of F2 (s) averaged over 20 series with N D 216 points and divided by s1/2 are shown, so that a horizontal line corresponds to uncorrelated behavior. The blue open triangles show the result for one selected extreme configuration, where it is hard to decide about the existence of long-term correlations (figure after [29])
(20)
The degree of the polynomial can be varied in order to eliminate constant (m D 0), linear (m D 1), quadratic (m D 2) or higher order trends of the profile function [15]. Conventionally the DFA is named after the order of the fitting polynomial (DFA0, DFA1, DFA2, . . . ). In DFAm, trends of order m in the profile Y( j) and of order m 1 in the original record x˜ i are eliminated. The variance of the detrended profile Y˜s ( j) in each segment yields the meansquare fluctuations, s
2 (; s) D FDFAm
1 X ˜2 Ys ( j) : s jD1
(21)
471
472
Fractal and Multifractal Time Series
Fractal and Multifractal Time Series, Figure 6 Application of Detrended Fluctuation Analysis (DFA) on the data already studied in Fig. 5 (figure after [29])
2 As for FA and discrete wavelet analysis, the FDFAm (; s) are averaged over all segments to obtain the mean fluctuations F2 (s), see Eq. (13). Calculating F2 (s) for many s, the fluctuation scaling exponent ˛ can be determined just as with FA, see Eq. (14). Figure 6 shows typical results for DFA of the same long-term correlated, short-term correlated and uncorrelated data studied already in Fig. 5. We note that in studies that include averaging over many records (or one record cut into many separate pieces by the elimination of some unreliable intermediate data points) the averaging procedure (13) must be performed for all data. Taking the square root is always the final step after all averaging is finished. It is not appropriate to calculate F2 (s) for parts of the data and then average the F2 (s) values, since such a procedure will bias the results towards smaller scaling exponents on large time scales. If F2 (s) increases for increasing s by F2 (s) s ˛ with 0:5 < ˛ < 1, one finds that the scaling exponent ˛ H is related to the correlation exponent by ˛ D 1 /2 (see Eq. (15)). A value of ˛ D 0:5 thus indicates that there are no (or only short-range) correlations. If ˛ > 0:5 for all scales s, the data are long-term correlated. The higher ˛, the stronger the correlations in the signal are. ˛ > 1 indicates a non-stationary local average of the data; in this case, FA fails and yields only ˛ D 1. The case ˛ < 0:5 corresponds to long-term anti-correlations, meaning that large values are most likely to be followed by small values and vice versa. ˛ values below 0 are not possible. Since the maximum value for ˛ in DFAm is mC1, higher detrending orders should be used for very non-stationary data with large ˛. Like in FA and Hurst’s analysis, ˛ will decrease or increase by one upon additional differentiation or integration of the data, respectively.
Small deviations from the scaling law (14), i. e. deviations from a straight line in a double logarithmic plot, occur for small scales s, in particular for DFAm with large detrending order m. These deviations are intrinsic to the usual DFA method, since the scaling behavior is only approached asymptotically. The deviations limit the capability of DFA to determine the correct correlation behavior in very short records and in the regime of small s. DFA6, e. g., is only defined for s 8, and significant deviations from the scaling law F2 (s) s ˛ occur even up to s 30. They will lead to an over-estimation of the fluctuation exponent ˛, if the regime of small s is used in a fitting procedure. An approach for correction of this systematic artefact in DFA is described in [35]. The number of independent segments of length s is larger in DFA than in WT, and the fluctuations in FA are larger than in DFA. Hence, the analysis has to be based on s values lower than smax D N/4 for DFA compared with smax D N/10 for FA and WT. The accuracy of scaling exponents ˛ determined by DFA was recently studied as a function of the length N of the data [43] (fitting range s 2 [10; N/2] was used). The results show that statistical standard errors of ˛ (one standard deviation) are approximately 0.1 for N D 500, 0.05 for N D 3000, and reach 0.03 for N D 10,000. Findings of long-term correlations with ˛ D 0:6 in data with only 500 points are thus not significant. A generalization of DFA for two-dimensional data (or even higher dimensions d) was recently suggested [34]. The generalization works well when tested with synthetic surfaces including fractional Brownian surfaces and multifractal surfaces. In the 2D procedure, a double cumulative sum (profile) is calculated by summing over both directional indices analogous with Eq. (10), Y(k; l) D Pk Pl ˜ i; j . This surface is partitioned into squares iD1 jD1 x of size s s with indices and , in which polynomials 2 like y;;s (i; j) D ai 2 C b j2 C ci j C di C e j C f are fitted. The fluctuation function F2 (s) is again obtained by calculating the variance of the profile from the fits. Detection of Trends and Crossovers with DFA Frequently, the correlations of recorded data do not follow the same scaling law for all time scales s, but one or sometimes even more crossovers between different scaling regimes are observed (see Subsect. “Crossovers and Non-stationarities in Time Series”). Time series with a well-defined crossover at s and vanishing correlations above s are most easily generated by Fourier filtering (see Subsect. “Fourier Filtering”). The power spectrum S( f ) of an uncorrelated random series is multiplied by ( f / f )ˇ
Fractal and Multifractal Time Series
with ˇ D 2˛ 1 for frequencies f > f D 1/s only. The series obtained by inverse Fourier transform of this modified power spectrum exhibits power-law correlations on time scales s < s only, while the behavior becomes uncorrelated on larger time scales s > s . The crossover from F2 (s) s ˛ to F2 (s) s 1/2 is clearly visible in double logarithmic plots of the DFA fluctuation function for such short-term correlated data. However, it (m) occurs at times s that are different from the original s used for the generation of the data and that depend on the detrending order m. This systematic deviation is most significant in the DFAm with higher m. Extensive numerical simulations (see Fig. 3 in [35]) show that the ratios of (m) /s are 1.6, 2.6, 3.6, 4.5, and 5.4 for DFA1, DFA2, . . . , s DFA5, with an error bar of approximately 0.1. Note, however, that the precise value of this ratio will depend on the (m) method used for fitting the crossover times s (and the method used for generating the data if generated data is analyzed). If results for different orders of DFA shall be (m) compared, an observed crossover s can be systematically corrected dividing by the ratio for the corresponding DFAm. If several orders of DFA are used in the procedure, several estimates for the real s will be obtained, which can be checked for consistency or used for an error approximation. A real crossover can thus be well distinguished from the effects of non-stationarities in the data, which lead to a different dependence of an apparent crossover on m. The procedure is also required if the characteristic time scale of short-term correlations shall be studied with DFA. If consistent (corrected) s values are obtained based on DFAm with different m, the existence of a real characteristic correlation time scale is positively confirmed. Note that lower detrending orders are advantageous in this case, (m) might besince the observed crossover time scale s come quite large and nearly reach one forth of the total series length (N/4), where the results become statistically inaccurate. We would like to note that studies showing scaling long-term correlations should not be based on DFA or variants of this method alone in most applications. In particular, if it is not clear whether a given time series is indeed long-term correlated or just short-term correlated with a fairly large crossover time scale, results of DFA should be compared with other methods. For example, one can employ wavelet methods (see, e. g., Subsect. “Discrete Wavelet Transform (WT) Approach”). Another option is to remove short-term correlations by considering averaged series for comparison. For a time series with daily observations and possible short-term correlations up to two years, for example, one might consider the series of two-
year averages and apply DFA together with FA, binned power spectra analysis, and/or wavelet analysis. Only if these methods still indicate long-term correlations, one can be sure that the data are indeed long-term correlated. As discussed in Subsect. “Crossovers and Non-stationarities in Time Series”, records from real measurements are often affected by non-stationarities, and in particular by trends. They have to be well distinguished from the intrinsic fluctuations of the system. To investigate the effect of trends on the DFAm fluctuation functions, one can generate artificial series (x˜ i ) with smooth monotonous trends by adding polynomials of different power p to the original record (x i ), x˜ i D x i C Ax p
with
x D i/N :
(22)
For the DFAm, such trends in the data can lead to an artificial crossover in the scaling behavior of F2 (s), i. e., the slope ˛ is strongly increased for large time scales s. The position of this artificial crossover depends on the strength A and the power p of the trend. Evidently, no artificial crossover is observed, if the detrending order m is larger than p and p is integer. The order p of the trends in the data can be determined easily by applying the different DFAm. If p is larger than m or p is not an integer, an artificial crossover is observed, the slope ˛trend in the large s regime strongly depends on m, and the position of the artificial crossover also depends strongly on m. The artificial crossover can thus be clearly distinguished from real crossovers in the correlation behavior, which result in identical slopes ˛ and rather similar crossover positions for all detrending orders m. For more extensive studies of trends with non-integer powers we refer to [35,36]. The effects of periodic trends are also studied in [35]. If the functional form of the trend in given data is not known a priori, the fluctuation function F2 (s) should be calculated for several orders m of the fitting polynomial. If m is too low, F2 (s) will show a pronounced crossover to a regime with larger slope for large scales s [35,36]. The maximum slope of log F2 (s) versus log s is m C 1. The crossover will move to larger scales s or disappear when m is increased, unless it is a real crossover not due to trends. Hence, one can find m such that detrending is sufficient. However, m should not be larger than necessary, because shifts of the observed crossover time scales and deviations on short scales s increase with increasing m. Sign and Magnitude (Volatility) DFA To study the origin of long-term fractal correlations in a time series, the series can be split into two parts which are analyzed separately. It is particularly useful to split the
473
474
Fractal and Multifractal Time Series
series of increments, x i D x i x i1 , i D 1; : : : ; N, into a series of signs x˜ i D s i D signx i and a series of magnitudes x˜ i D m i D jx i j [32,47,48]. There is an extensive interest in the magnitude time series in economics [49,50]. These data, usually called volatility, represent the absolute variations in stock (or commodity) prices and are used as a measure quantifying the risk of investments. While the actual prices are only short-term correlated, long-term correlations have been observed in volatility series [49,50]. Time series having identical distributions and longrange correlation properties can exhibit quite different temporal organizations of the magnitude and sign sub-series. The DFA method can be applied independently to both of these series. Since in particular the signs are often rather strongly anti-correlated and DFA will give incorrect results if ˛ is too close to zero, one often studies integrated sign and magnitude series. As mentioned above, integraP tion x˜ i ! ijD1 x˜ j increases ˛ by one. Most published results report short-term anti-correlations and no long-term correlations in the sign series, i. e., ˛sign < 1/2 for the non-integrated signs si (or ˛sign < 3/2 for the integrated signs) on low time scales and ˛sign ! 1/2 asymptotically for large s. The magnitude series, on the other hand, are usually either uncorrelated ˛magn D 1/2 (or 3/2) or positively long-term correlated ˛magn > 1/2 (or 3/2). It has been suggested that findings of ˛magn > 1/2 are related with nonlinear properties of the data and in particular multifractality [32,47,48], if ˛ < 1:5 in standard DFA. Specifically, the results suggest that the correlation exponent of the magnitude series is a monotonically increasing function of the multifractal spectrum (i. e., the singularity spectrum) width of the original series (see Subsect. “The Structure Function Approach and Singularity Spectra”). On the other hand, the sign series mainly relates to linear properties of the original series. At small time scales s < 16 the standard ˛ is approximately the average of ˛sign and ˛magn , if integrated sign and magnitude series are analyzed. For ˛ > 1:5 in the original series, the integrated magnitude and sign series have approximately the same two-point scaling exponents [47]. An analytical treatment is presented in [48]. Further Detrending Approaches A possible drawback of the DFA method is the occurrence of abrupt jumps in the detrended profile Y˜s ( j) (Eq. (20)) at the boundaries between the segments, since the fitting polynomials in neighboring segments are not related. A possible way to avoid these jumps would be the calculation of F2 (s) based on polynomial fits in overlapping windows. However, this is rather time consuming due to the polynomial fit in each segment and is consequently not
done in most applications. To overcome the problem of jumps several modifications and extensions of the FA and DFA methods have been suggested in recent years. These methods include The detrended moving average technique [51,52,53], which we denote by the backward moving average (BMA) technique (following [54]), The centered moving average (CMA) method [54], an essentially improved version of BMA, The modified detrended fluctuation analysis (MDFA) [55], which is essentially a mixture of old FA and DFA, The continuous DFA (CDFA) technique [56,57], which is particularly useful for the detection of crossovers, The Fourier DFA [58], A variant of DFA based on empirical mode decomposition (EMD) [59], A variant of DFA based on singular value decomposition (SVD) [60,61], and A variant of DFA based on high-pass filtering [62]. Detrended moving average techniques will be thoroughly described and discussed in the next section. A study comparing DFA with CMA and MDFA can be found in [43]. For studies comparing DFA and BMA, see [63,64]; note that [64] also discusses CMA. The method we denote as modified detrended fluctuation analysis (MDFA) [55], eliminates trends similar to the DFA method. A polynomial is fitted to the profile function Y( j) in each segment and the deviation between the profile function and the polynomial fit is calculated, p Y˜s ( j) D Y( j) y;s ( j) (Eq. (20)). To estimate correlations in the data, this method uses a derivative of Y˜s ( j), obtained for each segment , by Y˜s ( j) D Y˜s ( jCs/2)Y˜s ( j). Hence, the fluctuation function (compare with Eqs. (13) and (21)) is calculated as follows: 31/2 2 N X 2 1 (23) Y˜s ( j C s/2) Y˜s ( j) 5 : F2 (s) D 4 N jD1
As in case of DFA, MDFA can easily be generalized to remove higher order trends in the data. Since the fitting polynomials in adjacent segments are not related, Y˜s ( j) shows abrupt jumps on their boundaries as well. This leads to fluctuations of F2 (s) for large segment sizes s and limits the maximum usable scale to s < N/4 as for DFA. The detection of crossovers in the data, however, is more exact with MDFA (compared with DFA), since no correction of the estimated crossover time scales seems to be needed [43]. The Fourier-detrended fluctuation analysis [58] aims to eliminate slow oscillatory trends which are found espe-
Fractal and Multifractal Time Series
cially in weather and climate series due to seasonal influences. The character of these trends can be rather periodic and regular or irregular, and their influence on the detection of long-range correlations by means of DFA was systematically studied previously [35]. Among other things it has been shown that slowly varying periodic trends disturb the scaling behavior of the results much stronger than quickly oscillating trends and thus have to be removed prior to the analysis. In the case of periodic and regular oscillations, e. g., in temperature fluctuations, one simply removes the low frequency seasonal trend by subtracting the daily mean temperatures from the data. Another way, which the Fourier-detrended fluctuation analysis suggests, is to filter out the relevant frequencies in the signals’ Fourier spectrum before applying DFA to the filtered signal. Nevertheless, this method faces several difficulties especially its limitation to periodic and regular trends and the need for a priori knowledge of the interfering frequency band. To study correlations in data with quasi-periodic or irregular oscillating trends, empirical mode decomposition (EMD) was suggested [59]. The EMD algorithm breaks down the signal into its intrinsic mode functions (IMFs) which can be used to distinguish between fluctuations and background. The background, estimated by a quasi-periodic fit containing the dominating frequencies of a sufficiently large number of IMFs, is subtracted from the data, yielding a slightly better scaling behavior in the DFA curves. However, we believe that the method might be too complicated for wide-spread applications. Another method which was shown to minimize the effect of periodic and quasi-periodic trends is based on singular value decomposition (SVD) [60,61]. In this approach, one first embeds the original signal in a matrix whose dimension has to be much larger than the number of frequency components of the periodic or quasi-periodic trends obtained in the power spectrum. Applying SVD yields a diagonal matrix which can be manipulated by setting the dominant eigenvalues (associated with the trends) to zero. The filtered matrix finally leads to the filtered data, and it has been shown that subsequent application of DFA determines the expected scaling behavior if the embedding dimension is sufficiently large. None the less, the performance of this rather complex method seems to decrease for larger values of the scaling exponent. Furthermore SVD-DFA assumes that trends are deterministic and narrow banded. The detrending procedure in DFA (Eq. (20)) can be regarded as a scale-dependent high-pass filter since (low-frequency) fluctuations exceeding a specific scale s are eliminated. Therefore, it has been suggested to obtain the de-
trended profile Y˜s ( j) for each scale s directly by applying digital high-pass filters [62]. In particular, Butterworth, Chebyshev-I, Chebyshev-II, and an elliptical filter were suggested. While the elliptical filter showed the best performance in detecting long-range correlations in artificial data, the Chebyshev-II filter was found to be problematic. Additionally, in order to avoid a time shift between filtered and original profile, the average of the directly filtered signal and the time reversed filtered signal is considered. The effects of these complicated filters on the scaling behavior are, however, not fully understood. Finally, a continuous DFA method has been suggested in the context of studying heartbeat data during sleep [56, 57]. The method compares unnormalized fluctuation functions F2 (s) for increasing length of the data. I. e., one starts with a very short recording and subsequently adds more points of data. The method is particularly suitable for the detection of change points in the data, e. g., physiological transitions between different activity or sleep stages. Since the main objective of the method is not the study of scaling behavior, we do not discuss it in detail here. Centered Moving Average (CMA) Analysis Particular attractive modifications of DFA are the detrended moving average (DMA) methods, where running averages replace the polynomial fits. The first suggested version, the backward moving average (BMA) method [51,52,53], however, suffers from severe problems, because an artificial time shift of s between the original signal and the moving average is introduced. This time shift leads to an additional contribution to the detrended profile Y˜s ( j), which causes a larger fluctuation function F2 (s) in particular for small scales in the case of long-term correlated data. Hence, the scaling exponent ˛ is systematically underestimated [63]. In addition, the BMA method preforms even worse for data with trends [64], and its slope is limited by ˛ < 1 just as for the non-detrending method FA. It was soon recognized that the intrinsic error of BMA can be overcome by eliminating the artificial time shift. This leads to the centered moving average (CMA) method [54], where Y˜s ( j) is calculated as 1 Y˜s ( j) D Y( j) s
(s1)/2 X
Y( j C i) ;
(24)
iD(s1)/2
replacing Eq. (20) while Eq. (21) and the rest of the DFA procedure described in Subsect. “Detrended Fluctuation Analysis (DFA)” stay the same. Unlike DFA, the CMA method cannot easily be generalized to remove linear and higher order trends in the data.
475
476
Fractal and Multifractal Time Series
It was recently proposed [43] that the scaling behavior of the CMA method is more stable than for DFA1 and MDFA1, suggesting that CMA could be used for reliable computation of ˛ even for scales s < 10 (without correction of any systematic deviations needed in DFA for this regime) and up to smax D N/2. The standard errors in determining the scaling exponent ˛ by fitting straight lines to the double logarithmic plots of F2 (s) have been studied in [43]; they are comparable with DFA1 (see end of Subsect. “Detrended Fluctuation Analysis (DFA)”). Regarding the determination of crossovers, CMA is comparable to DFA1. Ultimately, the CMA seems to be a good alternative to DFA1 when analyzing the scaling properties in short data sets without trends. Nevertheless for data with possible unknown trends we recommend the application of standard DFA with several different detrending polynomial orders in order to distinguish real crossovers from artificial crossovers due to trends. In addition, an independent approach (e. g., wavelet analysis) should be used to confirm findings of long-term correlations (see also Subsect. “Detection of Trends and Crossovers with DFA”). Methods for Multifractal Time Series Analysis This section describes the multifractal characterization of time series, for an introduction, see Subsect. “Multifractal Time Series”. The simplest type of multifractal analysis is based upon the standard partition function multifractal formalism, which has been developed for the multifractal characterization of normalized, stationary measures [6,12,65,66]. Unfortunately, this standard formalism does not give correct results for non-stationary time series that are affected by trends or that cannot be normalized. Thus, in the early 1990s an improved multifractal formalism has been developed, the wavelet transform modulus maxima (WTMM) method [67,68,69,70,71], which is based on wavelet analysis and involves tracing the maxima lines in the continuous wavelet transform over all scales. An important alternative is the multifractal DFA (MFDFA) algorithm [33], which does not require the modulus maxima procedure, and hence involves little more effort in programming than the conventional DFA. For studies comparing methods for detrending multifractal analysis (MFDFA and WTMM, see [33,72,73]. The Structure Function Approach and Singularity Spectra In the general multifractal formalism, one considers a normalized measure (t), t 2 [0; 1], and defines the box R tCs/2 ˜ s (t) D ts/2 (t 0 ) dt 0 in neighborhoods of probabilities
(scale) length s 1 around t. The multifractal approach is then introduced by the partition function Z q (s) D
1/s1 X
˜ s [( C 1/2)s] s (q) q
for s 1 ; (25)
D0
where (q) is the Renyi scaling exponent and q is a real parameter that can take positive as well as negative values. Note that (q) is sometimes defined with opposite sign (see, e. g., [6]). A record is called monofractal (or selfaffine), when the Renyi scaling exponent (q) depends linearly on q; otherwise it is called multifractal. The generalized multifractal dimensions D(q) (see also Subsect. “Multifractal Time Series”) are related to (q) by D(q) D (q)/(q 1), such that the fractal dimension of the support is D(0) D (0) and the correlation dimension is D(2) D (2). In time series, a discrete version has to be used, and the considered data (x i ), i D 1; : : : ; N may usually include negative values. Hence, setting N s D int(N/s) and P X(; s) D siD1 xsCi for D 0; : : : ; N s 1 we can define [6,12], NX s 1
Z q (s) D
jX(; s)j q s (q)
for s > 1 :
(26)
D0
Inserting the profile Y( j) and FFA (; s) from Eqs. (10) and (11), respectively, we obtain Z q (s) D
NX s 1
˚
[Y(( C 1)s) Y(s)]2
q/2
D0
D
Ns X
jFFA (; s)j :
(27)
D1
Comparing Eq. (27) with (13), we see that this multifractal approach can be considered as a generalized version of the fluctuation analysis (FA) method, where the exponent 2 is replaced by q. In particular we find (disregarding the summation over the second partition of the time series)
F2 (s)
1/2 1 Z2 (s) s [1C(2)]/2 Ns ) 2˛ D 1 C (2) D 1 C D(2):
(28)
We thus see that all methods for (mono-)fractal time analysis (discussed in Sect. “Methods for Stationary Fractal Time Series Analysis”and Sect. “Methods for Non-stationary Fractal Time Series Analysis”) in fact study the correlation dimension D(2) D 2˛ 1 D ˇ D 1 (see Eq. (15)).
Fractal and Multifractal Time Series
It is straightforward to define a generalized (multifractal) Hurst exponent h(q) for the scaling behavior of the qth moments of the fluctuations [65,66],
1/q 1 Fq (s) D Z2 (s) s [1C(q)]/q D s h(q) Ns 1 C (q) ) h(q) D q
d (q) dq
and
f (˛) D q˛ (q) :
(29)
(30)
Here, ˛ is the singularity strength or Hölder exponent (see also Fractals and Multifractals, Introduction to in the encyclopedia), while f (˛) denotes the dimension of the subset of the series that is characterized by ˛. Note that ˛ is not the fluctuation scaling exponent in this section, although the same letter is traditionally used for both. Using Eq. (29), we can directly relate ˛ and f (˛) to h(q), ˛ D h(q)C qh0 (q)
and
Z(q; s) D
j max X
jL ( j ; s)j q :
(32)
jD1
with h(2) D ˛ H. In the following, we will use only h(2) for the standard fluctuation exponent (denoted by ˛ in the previous chapters), and reserve the letter ˛ for the Hölder exponent. Another way to characterize a multifractal series is the singularity spectrum f (˛), that is related to (q) via a Legendre transform [6,12], ˛D
up the qth power of the maxima,
The reason for the maxima procedure is that the absolute wavelet coefficients jL (; s)j can become arbitrarily small. The analyzing wavelet (x) must always have positive values for some x and negative values for other x, since it has to be orthogonal to possible constant trends. Hence there are always positive and negative terms in the sum (16), and these terms might cancel. If that happens, jL (; s)j can become close to zero. Since such small terms would spoil the calculation of negative moments in Eq. (32), they have to be eliminated by the maxima procedure. In fluctuation analysis, on the other hand, the calculation of the variances F 2 (; s), e. g. in Eq. (11), involves only positive terms under the summation. The variances cannot become arbitrarily small, and hence no maximum procedure is required for series with compact support. In addition, the variances will increase if the segment length s is increased, because the fit will usually be worse for a longer segment. In the WTMM method, in contrast, the absolute
f (˛) D q[˛ h(q)]C1: (31)
Wavelet Transform Modulus Maxima (WTMM) Method The wavelet transform modulus maxima (WTMM) method [67,68,69,70,71] is a well-known method to investigate the multifractal scaling properties of fractal and selfaffine objects in the presence of non-stationarities. For applications, see e. g. [74,75]. It is based upon the wavelet transform with continuous basis functions as defined in Subsect. “Wavelet Analysis”, Eq. (16). Note that in this case the series x˜ i are analyzed directly instead of the profile Y( j) defined in Eq. (10). Using wavelets orthogonal to mth order polynomials, the corresponding trends are eliminated. Instead of averaging over all wavelet coefficients L (; s), one averages, within the modulo-maxima method, only the local maxima of jL (; s)j. First, one determines for a given scale s, the positions j of the local maxima of jW(; s)j as a function of , so that jL ( j 1; s)j < jL ( j ; s)j jL ( j C1; s)j for j D 1; : : : ; jmax . This maxima procedure is demonstrated in Fig. 7. Then one sums
Fractal and Multifractal Time Series, Figure 7 Example of the wavelet transform modulus maxima (WTMM) method, showing the original data (top), its continuous wavelet transform (gray scale coded amplitude of wavelet coefficients, middle), and the extracted maxima lines (bottom) (figure taken from [68])
477
478
Fractal and Multifractal Time Series
wavelet coefficients jL (; s)j need not increase with increasing scale s, even if only the local maxima are considered. The values jL (; s)j might become smaller for increasing s since just more (positive and negative) terms are included in the summation (16), and these might cancel even better. Thus, an additional supremum procedure has been introduced in the WTMM method in order to keep the dependence of Z(q; s) on s monotonous. If, for a given scale s, a maximum at a certain position j happens to be smaller than a maximum at j0 j for a lower scale s 0 < s, then L ( j ; s) is replaced by L ( j0 ; s 0 ) in Eq. (32). Often, scaling behavior is observed for Z(q; s), and scaling exponents ˆ (q) can be defined that describe how Z(q; s) scales with s, Z(q; s) sˆ (q) :
(33)
The exponents ˆ (q) characterize the multifractal properties of the series under investigation, and theoretically they are identical with the (q) defined in Eq. (26) [67,68,69,71] and related to h(q) by Eq. (29). Multifractal Detrended Fluctuation Analysis (MFDFA) The multifractal DFA (MFDFA) procedure consists of five steps [33]. The first three steps are essentially identical to the conventional DFA procedure (see Subsect. “Detrended Fluctuation Analysis (DFA)” and Fig. 4). Let us assume that (x˜ i ) is a series of length N, and that this series is of compact support. The support can be defined as the set of the indices j with nonzero values x˜ j , and it is compact if x˜ j D 0 for an insignificant fraction of the series only. The value of x˜ j D 0 is interpreted as having no value at this j. Note that we are not discussing the fractal or multifractal features of the plot of the time series in a two-dimensional graph (see also the discussion in Subsect. “Fractality, SelfAffinity, and Scaling”), but analyzing time series as onedimensional structures with values assigned to each point. Since real time series always have finite length N, we explicitly want to determine the multifractality of finite series, and we are not discussing the limit for N ! 1 here (see also Subsect. “The Structure Function Approach and Singularity Spectra”). Step 1 Calculate the profile Y( j), Eq. (10), by integrating the time series. Step 2 Divide the profile Y( j) into N s D int(N/s) nonoverlapping segments of equal length s. Since the length N of the series is often not a multiple of the considered time scale s, the same procedure can be repeated starting from the opposite end. Thereby, 2N s segments are obtained altogether.
Step 3 Calculate the local trend for each of the 2N s segments by a least-square fit of the profile. Then determine the variance by Eqs. (20) and (21) for each segment D 1; : : : ; 2Ns . Again, linear, quadratic, cubic, or higher order polynomials can be used in the fitting procedure, and the corresponding methods are thus called MFDFA1, MFDFA2, MFDFA3, . . . ) [33]. In (MF-)DFAm [mth order (MF-)DFA] trends of order m in the profile (or, equivalently, of order m 1 in the original series) are eliminated. Thus a comparison of the results for different orders of DFA allows one to estimate the type of the polynomial trend in the time series [35,36]. Step 4 Average over all segments to obtain the qth order fluctuation function ( ) 1/q 2N q/2 1 Xs 2 F (; s) : (34) Fq (s) D 2N s D1 DFAm This is the generalization of Eq. (13) suggested by the relations derived in Subsect. “The Structure Function Approach and Singularity Spectra”. For q D 2, the standard DFA procedure is retrieved. One is interested in how the generalized q dependent fluctuation functions Fq (s) depend on the time scale s for different values of q. Hence, we must repeat steps 2 to 4 for several time scales s. It is apparent that Fq (s) will increase with increasing s. Of course, Fq (s) depends on the order m. By construction, Fq (s) is only defined for s m C 2. Step 5 Determine the scaling behavior of the fluctuation functions by analyzing log-log plots Fq (s) versus s for each value of q. If the series x˜ i are long-range powerlaw correlated, Fq (s) increases, for large values of s, as a power-law, Fq (s) s h(q)
with
h(q) D
1 C (q) : q
(35)
For very large scales, s > N/4, Fq (s) becomes statistically unreliable because the number of segments N s for the averaging procedure in step 4 becomes very small. Thus, scales s > N/4 should be excluded from the fitting procedure determining h(q). Besides that, systematic deviations from the scaling behavior in Eq. (35), which can be corrected, occur for small scales s 10. The value of h(0), which corresponds to the limit h(q) for q ! 0, cannot be determined directly using the averaging procedure in Eq. (34) because of the diverging exponent. Instead, a logarithmic averaging procedure has to be employed, ( ) 2N 1 Xs 2 F0 (s) D exp ln F (; s) s h(0) : (36) 4N s D1
Fractal and Multifractal Time Series
Note that h(0) cannot be defined for time series with fractal support, where h(q) diverges for q ! 0. For monofractal time series with compact support, h(q) is independent of q, since the scaling behavior of the 2 variances FDFAm (; s) is identical for all segments , and the averaging procedure in Eq. (34) will give just this identical scaling behavior for all values of q. Only if small and large fluctuations scale differently, there will be a significant dependence of h(q) on q. If we consider positive values of q, the segments with large variance F 2 (; s) (i. e. large deviations from the corresponding fit) will dominate the average Fq (s). Thus, for positive values of q, h(q) describes the scaling behavior of the segments with large fluctuations. On the contrary, for negative values of q, the 2 segments with small variance FDFAm (; s) will dominate the average Fq (s). Hence, for negative values of q, h(q) describes the scaling behavior of the segments with small fluctuations. Figure 8 shows typical results obtained for Fq (s) in the MFDFA procedure. Usually the large fluctuations are characterized by a smaller scaling exponent h(q) for multifractal series than the small fluctuations. This can be understood from the following arguments. For the maximum scale s D N the fluctuation function Fq (s) is independent of q, since the sum in Eq. (34) runs over only two identical segments. For smaller scales s N the averaging procedure runs over several segments, and the average value Fq (s) will be dominated by the F 2 (; s) from the segments with small (large) fluctuations if q < 0 (q > 0). Thus, for s N, Fq (s) with
Fractal and Multifractal Time Series, Figure 8 Multifractal detrended fluctuation analysis (MFDFA) of data from the binomial multifractal model (see Subsect. “The Extended Binomial Multifractal Model”) with a D 0:75. Fq (s) is plotted versus s for the q values given in the legend; the slopes of the curves correspond to the values of h(q). The dashed lines have the slopes of the theoretical slopes h(˙1) from Eq. (42). 100 configurations have been averaged
q < 0 will be smaller than Fq (s) with q > 0, while both become equal for s D N. Hence, if we assume an homogeneous scaling behavior of Fq (s) following Eq. (35), the slope h(q) in a log-log plot of Fq (s) with q < 0 versus s must be larger than the corresponding slope for Fq (s) with q > 0. Thus, h(q) for q < 0 will usually be larger than h(q) for q > 0. However, the MFDFA method can only determine positive generalized Hurst exponents h(q), and it already becomes inaccurate for strongly anti-correlated signals when h(q) is close to zero. In such cases, a modified (MF-)DFA technique has to be used. The most simple way to analyze such data is to integrate the time series before the MFDFA procedure. Following the MFDFA procedure as described above, we obtain a generalized fluctuation ˜ functions described by a scaling law with h(q) D h(q) C 1. The scaling behavior can thus be accurately determined even for h(q) which are smaller than zero for some values of q. The accuracy of h(q) determined by MFDFA certainly depends on the length N of the data. For q D ˙10 and data with N D 10,000 and 100,000, systematic and statistical error bars (standard deviations) up to h(q) ˙0:1 and ˙0:05 should be expected, respectively [33]. A difference of h(10) h(C10) D 0:2, corresponding to an even larger width ˛ of the singularity spectrum f (˛) defined in Eq. (30) is thus not significant unless the data was longer than N D 10,000 points. Hence, one has to be very careful when concluding multifractal properties from differences in h(q). As already mentioned in the introduction, two types of multifractality in time series can be distinguished. Both of them require a multitude of scaling exponents for small and large fluctuations: (i) Multifractality of a time series can be due to a broad probability density function for the values of the time series, and (ii) multifractality can also be due to different long-range correlations for small and large fluctuations. The most easy way to distinguish between these two types is by analyzing also the corresponding randomly shuffled series [33]. In the shuffling procedure the values are put into random order, and thus all correlations are destroyed. Hence the shuffled series from multifractals of type (ii) will exhibit simple random behavior, hshuf (q) D 0:5, i. e. non-multifractal scaling. For multifractals of type (i), on the contrary, the original h(q) dependence is not changed, h(q) D hshuf (q), since the multifractality is due to the probability density, which is not affected by the shuffling procedure. If both kinds of multifractality are present in a given series, the shuffled series will show weaker multifractality than the original one.
479
480
Fractal and Multifractal Time Series
Fractal and Multifractal Time Series, Figure 9 Illustration for the definition of return intervals rq between extreme events above two quantiles (thresholds) q1 and q2 (figure by Jan Eichner)
Comparison of WTMM and MFDFA The MFDFA results turn out to be slightly more reliable than the WTMM results [33,72,73]. In particular, the MFDFA has slight advantages for negative q values and short series. In the other cases the results of the two methods are rather equivalent. Besides that, the main advantage of the MFDFA method compared with the WTMM method lies in the simplicity of the MFDFA method. However, contrary to WTMM, MFDFA is restricted to studies of data with full one-dimensional support, while WTMM is not. Both WTMM and MFDFA have been generalized for higher dimensional data, see [34] for higher dimensional MFDFA and, e. g., [71] for higher dimensional WTMM. Studies of other generalizations of detrending methods like the discrete WT approach (see Subsect. “Discrete Wavelet Transform (WT) Approach”) and the CMA method (see Subsect. “Centered Moving Average (CMA) Analysis”) are currently under investigation [76]. Statistics of Extreme Events in Fractal Time Series The statistics of return intervals between well defined extremal events is a powerful tool to characterize the temporal scaling properties of observed time series and to derive quantities for the estimation of the risk for hazardous events like floods, very high temperatures, or earthquakes. It was shown recently that long-term correlations represent a natural mechanism for the clustering of the hazardous events [77]. In this section we will discuss the most important consequences of long-term correlations and fractal scaling of time series upon the statistics of extreme events [77,78,79,80,81]. Corresponding work regarding multifractal data [82] is not discussed here. Return Intervals Between Extreme Events To study the statistics of return intervals we consider again a time series (x i ), i D 1; : : : ; N with fractal scaling behavior, sampled homogeneously and normalized to zero mean
and unit variance. For describing the reoccurrence of rare events exceeding a certain threshold q, we investigate the return intervals rq between these events, see Fig. 9. The average return interval R q D hr q i increases as a function of the threshold q (see, e. g. [83]). It is known that for uncorrelated records (‘white noise’), the return intervals are also uncorrelated and distributed according to the Poisson distribution, Pq (r) D R1q exp(r/R q ). For fractal (long-term correlated) data with auto-correlations following Eq. (5), we obtain a stretched exponential [77,78,79,80,84], Pq (r) D
a exp b (r/R q ) : Rq
(37)
This behavior is shown in Fig. 10. The exponent is the correlation exponent from C(s), and the parameters a and b are independent of q. They can be determined from R (r), i. e., P (r) dr D1 the normalization conditions for P q q R and rPq (r) dr D R q . The form of the distribution (37) indicates that return intervals both well below and well above their average value Rq (which is independent of ) are considerably more frequent for long-term correlated than for uncorrelated data. It has to be noted that there are deviations from the stretched exponential law (37) for very small r (discretization effects and an additional power-law regime) and for very large r (finite size effects), see Fig. 10. The extent of the deviations from Eq. (37) depends on the distribution of the values xi of the time series. For a discussion of these effects, see [80]. Equation (37) does not quantify, however, if the return intervals themselves are arranged in a correlated or in an uncorrelated fashion, and if clustering of rare events may be induced by long-term correlations. To study this question, one has to evaluate the auto-covariance function Cr (s) D hr q (l)r q (l C s)i R2q of the return intervals. The results for model data suggests that also the return intervals are long-term power-law correlated, with the same exponent as the original record. Accordingly, large and small return intervals are not arranged in a random fash-
Fractal and Multifractal Time Series
Fractal and Multifractal Time Series, Figure 10 Normalized rescaled distribution density functions Rq Pq (r) of r values with Rq D 100 as a function of r/Rq for long-term correlated data with D 0:4 (open symbols) and D 0:2 (filled symbols; we multiplied the data for the filled symbols by a factor 100 to avoid overlapping curves). In a the original data were Gaussian distributed, in b exponentially distributed, in c power-law distributed with power 5.5, and in d log-normally distributed. All four figures follow quite well stretched exponential curves (solid lines) over several decades. For small r/Rq values a power-law regime seems to dominate, while on large scales deviations from the stretched exponential behavior are due to finite-size effects (figure by Jan Eichner)
ion but are expected to form clusters. As a consequence, the probability of finding a certain return interval r depends on the value of the preceding interval r0 , and this effect has to be taken into account in predictions and risk estimations [77,80]. The conditional distribution function Pq (rjr0 ) is a basic quantity, from which the relevant quantities in risk estimations can be derived [83]. For example, the first moment of Pq (rjr0 ) is the average value R q (r0 ) of those return intervals that directly follow r0 . By definition, R q (r0 ) is the expected waiting time to the next event, when the two events before were separated by r0 . The more general quantity is the expected waiting time q (xjr0 ) to the next event, when the time x has elapsed. For x D 0, q (0jr0 ) is identical to R q (r0 ). In general, q (xjr0 ) is related to Pq (rjr0 ) by Z 1 (r x)Pq (rjr0 )dr : (38) q (xjr0 ) D x Z 1 Pq (rjr0 )dr
Distribution of Extreme Events In this section we describe how the presence of fractal long-term correlations affects the statistics of the extreme events, i. e., maxima within time segments of fixed duration R, see Fig. 11 for illustration. By definition, extreme events are rare occurrences of extraordinary nature, such as, e. g. floods, very high temperatures, or earthquakes. In hydrological engineering such conventional extreme value statistics are commonly applied to decide what building projects are required to protect river-side areas against typical floods that occur, for example, once in 100 years. Most of these results are based on statistically independent values xi and hold only in the limit R ! 1. However, both of these assumptions are not strictly fulfilled for correlated fractal scaling data. In classical extreme value statistics one assumes that records (x i ) consist of i.i.d. data, described by density distributions P(x), which can be, e. g., a Gaussian or an exponential distribution. One is interested in the distribution
x
For uncorrelated records, q (xjr0 )/R q D 1 (except for discreteness effects that lead to q (xjr0 )/R q > 1 for x > 0, see [85]). Due to the scaling of Pq (rjr0 ), also q (xjr0 )/R q scales with r0 /R q and x/R q . Small and large return intervals are more likely to be followed by small and large ones, respectively, and hence q (0jr0 )/R q D R q (r0 )/R q is well below (above) one for r0 /R q well below (above) one. With increasing x, the expected residual time to the next event increases. Note that only for an infinite long-term correlated record, the value of q (xjr0 ) will increase indefinitely with x and r0 . For real (finite) records, there exists a maximum return interval which limits the values of x, r0 and q (xjr0 ).
Fractal and Multifractal Time Series, Figure 11 Illustration for the definition of maxima mR within periods of R D 365 values (figure by Jan Eichner)
481
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Fractal and Multifractal Time Series
density function PR (m) of the maxima (m j ) determined in segments of length R in the original series (x i ), see Fig. 11. Note that all maxima are also elements of the original data. The corresponding integrated maxima distribution G R (m) is defined as Z m G R (m) D 1 E R (m) D PR (m0 )dm0 : (39)
previous maximum [81]. The last item implies that conditional mean maxima and conditional maxima distributions should be considered for improved extreme event predictions. Simple Models for Fractal and Multifractal Time Series
1
Fourier Filtering Since G R (m) is the probability of finding a maximum smaller than m, E R (m) denotes the probability of finding a maximum that exceeds m. One of the main results of traditional extreme value statistics states that for independently and identically distributed (i.i.d.) data (x i ) with Gaussian or exponential distribution density function P(x) the integrated distribution G R (m) converges to a double exponential (Fisher–Tippet–Gumbel) distribution (often labeled as Type I) [86,87,88,89,90], i. e., G R (m) ! G
m u ˛
h m u i D exp exp (40) ˛
for R ! 1, where ˛ is the scale parameter and u the location parameter. By the method of moments those pap rameters are given by ˛ D 6/ R and u D m R n e ˛ with the Euler constant n e D 0:577216 [89,91,92,93]. Here mR and R denote the (R-dependent) mean maximum and the standard deviation, respectively. Note that different asymptotics will be reached for broader distributions of data (x i ) that belong to other domains of attraction [89]. For example, for data following a power-law distribution (or Pareto distribution), P(x) D (x/x0 )k , G R (m) converges to a Fréchet distribution, often labeled as Type II. For data following a distribution with finite upper endpoint, for example the uniform distribution P(x) D 1 for 0 x 1, G R (m) converges to a Weibull distribution, often labeled as Type III. We do not consider the latter two types of asymptotics here. Numerical studies of fractal model data have recently shown that the distribution P(x) of the original data has a much stronger effect upon the convergence towards the Gumbel distribution than the long-term correlations in the data. Long-term correlations just slightly delay the convergence of G R (m) towards the Gumbel distribution (40). This can be observed very clearly in a plot of the integrated and scaled distribution G R (m) on logarithmic scale [81]. Furthermore, it was found numerically that (i) the maxima series (m j ) exhibit long-term correlations similar to those of the original data (x i ), and most notably (ii) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the
Fractal scaling with long-term correlations can be introduced most easily into time series by the Fourier-filtering technique, see, e. g., [94,95,96]. The Fourier filtering technique is not limited to the generation of long-term correlated data characterized by a power-law auto-correlation function C(s) x with 0 < < 1. All values of the scaling exponents ˛ D h(2) H or ˇ D 2˛ 1 can be obtained, even those that cannot be found directly by the fractal analysis techniques described in Sect. “Methods for Stationary Fractal Time Series Analysis” and Sect. “Methods for Non-stationary Fractal Time Series Analysis” (e. g. ˛ < 0). Note, however, that Fourier filtering will always yield Gaussian distributed data values and that no nonlinear or multifractal properties can be achieved (see also Subsect. “Multifractal Time Series”, Subsect. “Sign and Magnitude (Volatility) DFA”, and Sect. “Methods for Multifractal Time Series Analysis”). In Subsect. “Detection of Trends and Crossovers with DFA”, we have briefly described a modification of Fourier filtering for obtaining reliable short-term correlated data. For the generation of data characterized by fractal scaling with ˇ D 2˛ 1 [94,95] we start with uncorrelated Gaussian distributed random numbers xi from an i.i.d. generator. Transforming a series of such numbers into frequency space with discrete Fourier transform or FFT (fast Fourier transform, for suitable series lengths N) yields a flat power spectrum, since random numbers correspond to white noise. Multiplying the (complex) Fourier coefficients by f ˇ /2 , where f / 1/s is the frequency, will rescale the power spectrum S( f ) to follow Eq. (6), as expected for time series with fractal scaling. After transforming back to the time domain (using inverse Fourier transform or inverse FFT) we will thus obtain the desired longterm correlated data x˜i . The final step is the normalization of this data. The Fourier filtering method can be improved using modified Bessel functions instead of the simple factors f ˇ /2 in modifying the Fourier coefficients [96]. This way problems with the divergence of the autocorrelation function C(s) at s D 0 can be avoided. An alternative method to the Fourier filtering technique, the random midpoint displacement method, is
Fractal and Multifractal Time Series
based on the construction of self-affine surfaces by an iterative procedure, see, e. g. [6]. Starting with one interval with constant values, the intervals are iterative split in the middle and the midpoint is displaced by a random offset. The amplitude of this offset is scaled according to the length of the interval. Since the method generates a selfaffine surface xi characterized by a Hurst exponent H, the differentiated series x i can be used as long-term correlated or anti-correlated random numbers. Note, however, that the correlations do not persist for the whole length of the data generated this way. Another option is the use of wavelet synthesis, the reverse of wavelet analysis described in Subsect. “Wavelet Analysis”. In that method, the scaling law is introduced by setting the magnitudes of the wavelet coefficients according to the corresponding time scale s. The Schmitz–Schreiber Method When long-term correlations in random numbers are introduced by the Fourier-filtering technique (see previous section), the original distribution P(x) of the time series values xi is always modified such that it becomes closer to a Gaussian. Hence, no series (x i ) with broad distributions of the values and fractal scaling can be generated. In these cases an iterative algorithm introduced by Schreiber and Schmitz [98,99] must be applied. The algorithm consists of the following steps: First one creates a Gaussian distributed long-term correlated data set with the desired correlation exponent by standard Fourier-filtering [96]. The power spectrum SG ( f ) D FG ( f )FG ( f ) of this data set is considered as reference spectrum (where f denotes the frequency in Fourier space and the FG ( f ) are the complex Fourier coefficients). Next one creates an uncorrelated sequence of random numbers (x ref i ), following a desired distribution P(x). The (complex) Fourier transform F( f ) of the (x ref i ) is now divided by its absolute value and multiplied by the square root of the reference spectrum, p F( f ) SG ( f ) : (41) Fnew( f ) D jF( f )j After the Fourier back-transformation of Fnew ( f ), the new sequence (x new i ) has the desired correlations (i. e. the desired ), but the shape of the distribution has changed towards a (more or less) Gaussian distribution. In order to enforce the desired distribution, we exchange the (x new i ) by the (x ref ), such that the largest value of the new set is i replaced by the largest value of the reference set, the second largest of the new set by the second largest of the reference set and so on. After this the new sequence has the desired distribution and is clearly correlated. However, due
to the exchange algorithm the perfect long-term correlations of the new data sequence were slightly altered again. So the procedure is repeated: the new sequence is Fourier transformed followed by spectrum adjustment, and the exchange algorithm is applied to the Fourier back-transformed data set. These steps are repeated several times, until the desired quality (or the best possible quality) of the spectrum of the new data series is achieved. The Extended Binomial Multifractal Model The multifractal cascade model [6,33,65] is a standard model for multifractal data, which is often applied, e. g., in hydrology [97]. In the model, a record xi of length N D 2n max is constructed recursively as follows. In generation n D 0, the record elements are constant, i. e. x i D 1 for all i D 1; : : : ; N. In the first step of the cascade (generation n D 1), the first half of the series is multiplied by a factor a and the second half of the series is multiplied by a factor b. This yields x i D a for i D 1; : : : ; N/2 and x i D b for i D N/2 C 1; : : : ; N. The parameters a and b are between zero and one, 0 < a < b < 1. One need not restrict the model to b D 1 a as is often done in the literature [6]. In the second step (generation n D 2), we apply the process of step 1 to the two subseries, yielding x i D a2 for i D 1; : : : ; N/4, x i D ab for i D N/4 C 1; : : : ; N/2, x i D ba D ab for i D N/2 C 1; : : : ; 3N/4, and x i D b2 for i D 3N/4 C 1; : : : ; N. In general, in step n C 1, each subseries of step n is divided into two subseries of equal length, and the first half of the xi is multiplied by a while the second half is multiplied by b. For example, in generation n D 3 the values in the eight subseries are a3 ; a2 b; a2 b; ab2 ; a2 b; ab2 ; ab2 ; b3 . After nmax steps, the final generation has been reached, where all subseries have length 1 and no more splitting is possible. We note that the final record can be written as x i D a n max n(i1) b n(i1) , where n(i) is the number of digits 1 in the binary representation of the index i, e. g. n(13) D 3, since 13 corresponds to binary 1101. For this multiplicative cascade model, the formula for (q) has been derived earlier [6,33,65]. The result is (q) D [ ln(a q C b q ) C q ln(a C b)]/ ln 2 or h(q) D
1 ln(a q C b q ) ln(a C b) C : q q ln 2 ln 2
(42)
It is easy to see that h(1) D 1 for all values of a and b. Thus, in this form the model is limited to cases where h(1), which is the exponent Hurst defined originally in the R/S method, is equal to one. In order to generalize this multifractal cascade process such that any value of h(1) is possible, one can sub-
483
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tract the offset h D ln(a C b)/ ln(2) from h(q) [100]. The constant offset h corresponds to additional longterm correlations incorporated in the multiplicative cascade model. For generating records without this offset, we rescale the power spectrum. First, we transform (FFT) the simple multiplicative cascade data into the frequency domain. Then, we multiply all Fourier coefficients by f h , where f is the frequency. This way, the slope ˇ of the power spectra S( f ) f ˇ is decreased from ˇ D 2h(2) 1 D [2 ln(a C b) ln(a2 C b2 )]/ ln 2 into ˇ 0 D 2[h(2) h] 1 D ln(a 2 C b2 )/ ln 2. Finally, backward FFT is employed to transform the signal back into the time domain. The Bi-fractal Model In some cases a simple bi-fractal model is already sufficient for modeling apparently multifractal data [101]. For bifractal records the Renyi exponents (q) are characterized by two distinct slopes ˛ 1 and ˛ 2 , ( q q q˛1 1 (43) (q) D q˛2 C q (˛1 ˛2 ) 1 q > q or ( (q) D
q˛1 C q (˛2 ˛1 ) 1
q q
q˛2 1
q > q
:
(44)
If this behavior is translated into the h(q) picture using Eq. (29), we obtain that h(q) exhibits a plateau from q D 1 up to a certain q and decays hyperbolically for q > q , ( q q ˛1 ; (45) h(q) D 1 q (˛1 ˛2 ) q C ˛2 q > q or vice versa, ( q (˛2 ˛1 ) 1q C ˛1 h(q) D ˛2
q q q > q
:
(46)
Both versions of this bi-fractal model require three parameters. The multifractal spectrum is degenerated to two single points, thus its width can be defined as ˛ D ˛1 ˛2 . Future Directions The most straightforward future direction is to analyze more types of time series from other complex systems than those listed in Sect. “Introduction” to check for the presence of fractal scaling and in particular long-term correlations. Such applications may include (i) data that are not
recorded as a function of time but as a function of another parameter and (ii) higher dimensional data. In particular, the inter-relationship between fractal time series and spatially fractal structures can be studied. Studies of fields with fractal scaling in time and space have already been performed in Geophysics. In some cases studying new types of data will require dealing with more difficult types of non-stationarities and transient behavior, making further development of the methods necessary. In many studies, detrending methods have not been applied yet. However, discovering fractal scaling in more and more systems cannot be an aim on its own. Up to now, the reasons for observed fractal or multifractal scaling are not clear in most applications. It is thus highly desirable to study causes for fractal and multifractal correlations in time series, which is a difficult task, of course. One approach might be based on modeling and comparing the fractal aspects of real and modeled time series by applying the methods described in this article. The fractal or multifractal characterization can thus be helpful in improving the models. For many applications, practically usable models which display fractal or transient fractal scaling still have to be developed. One example for a model explaining fractal scaling might be a precipitation, storage and runoff model, in which the fractal scaling of runoff time series could be explained by fractional integration of rainfall in soil, groundwater reservoirs, or river networks characterized by a fractal structure. Also studies regarding the inter-relationship between fractal scaling and complex networks, representing the structure of a complex system, are desirable. This way one could gain an interpretation of the causes for fractal behavior. Another direction of future research is regarding the linear and especially non-linear inter-relationships between several time series. There is great need for improved methods characterizing cross-correlations and similar statistical inter-relationships between several non-stationary time series. Most methods available so far are reserved to stationary data, which is, however, hardly found in natural recordings. An even more ambitious aim is the (timedependent) characterization of a larger network of signals. In such a network, the signals themselves would represent the nodes, while the (possibly directed) inter-relationships between each pair represent the links (or bonds) between the nodes. The properties of both nodes and links can vary with time or change abruptly, when the represented complex system goes through a phase transition. Finally, more work will have to be invested in studying the practical consequences of fractal scaling in time series. Studies should particularly focus on predictions of future values and behavior of time series and whole complex
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systems. This is very relevant, not only in hydrology and climate research, where a clear distinguishing of trends and natural fluctuations is crucial, but also for predicting dangerous medical events on-line in patients based on the continuous recording of time series. Acknowledgment We thank Ronny Bartsch, Amir Bashan, Mikhail Bogachev, Armin Bunde, Jan Eichner, Shlomo Havlin, Diego Rybski, Aicko Schumann, and Stephan Zschiegner for helpful discussions and contribution. This work has been supported by the Deutsche Forschungsgemeinschaft (grant KA 1676/3) and the European Union (STREP project DAPHNet, grant 018474-2). Bibliography 1. Mandelbrot BB, van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10: 422 2. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Water Resour Res 5:321–340 3. Mandelbrot BB (1999) Multifractals and 1/f noise: wild selfaffinity in physics. Springer, Berlin 4. Hurst HE (1951) Long-term storage capacity of reservoirs. Tran Amer Soc Civ Eng 116:770 5. Hurst HE, Black RP, Simaika YM (1965) Long-term storage: an experimental study. Constable, London 6. Feder J (1988) Fractals. Plenum Press, New York 7. Barnsley MF (1993) Fractals everywhere. Academic Press, San Diego 8. Bunde A, Havlin S (1994) Fractals in science. Springer, Berlin 9. Jorgenssen PET (2000) Analysis and probability: Wavelets, signals, fractals. Springer, Berlin 10. Bunde A, Kropp J, Schellnhuber HJ (2002) The science of disasters – climate disruptions, heart attacks, and market crashes. Springer, Berlin 11. Kantz H, Schreiber T (2003) Nonlinear time series analysis. Cambridge University Press, Cambridge 12. Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals. Springer, Berlin 13. Sornette D (2004) Critical phenomena in natural sciences. Springer, Berlin 14. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anti-correlations and non-Gaussian behaviour of the heartbeat. Phys Rev Lett 70: 1343 15. Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter JH, Voigt K (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736 16. Vyushin D, Zhidkov I, Havlin S, Bunde A, Brenner S (2004) Volcanic forcing improves atmosphere-ocean coupled general circulation model scaling performance. Geophys Res Lett 31:L10206 17. Koscielny-Bunde E, Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber HJ (1998) Indication of a universal persistence law governing atmospheric variability. Phys Rev Lett 81:729
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Fractal and Transfractal Scale-Free Networks HERNÁN D. ROZENFELD, LAZAROS K. GALLOS, CHAOMING SONG, HERNÁN A. MAKSE Levich Institute and Physics Department, City College of New York, New York, USA Article Outline Glossary Definition of the Subject Introduction Fractality in Real-World Networks Models: Deterministic Fractal and Transfractal Networks Properties of Fractal and Transfractal Networks Future Directions Acknowledgments Appendix: The Box Covering Algorithms Bibliography Glossary Degree of a node Number of edges incident to the node. Scale-free network Network that exhibits a wide (usually power-law) distribution of the degrees. Small-world network Network for which the diameter increases logarithmically with the number of nodes. Distance The length (measured in number of links) of the shortest path between two nodes. Box Group of nodes. In a connected box there exists a path within the box between any pair of nodes. Otherwise, the box is disconnected. Box diameter The longest distance in a box. Definition of the Subject The explosion in the study of complex networks during the last decade has offered a unique view in the structure and behavior of a wide range of systems, spanning many different disciplines [1]. The importance of complex networks lies mainly in their simplicity, since they can represent practically any system with interactions in a unified way by stripping complicated details and retaining the main features of the system. The resulting networks include only nodes, representing the interacting agents and links, representing interactions. The term ‘interactions’ is used loosely to describe any possible way that causes two nodes to form a link. Examples can be real physical links, such as the wires connecting computers in the Internet or roads connecting cities, or alternatively they may be virtual
links, such as links in WWW homepages or acquaintances in societies, where there is no physical medium actually connecting the nodes. The field was pioneered by the famous mathematician P. Erd˝os many decades ago, when he greatly advanced graph theory [27]. The theory of networks would have perhaps remained a problem of mathematical beauty, if it was not for the discovery that a huge number of everyday life systems share many common features and can thus be described through a unified theory. The remarkable diversity of these systems incorporates artificially or man-made technological networks such as the Internet and the World Wide Web (WWW), social networks such as social acquaintances or sexual contacts, biological networks of natural origin, such as the network of protein interactions of Yeast [1,36], and a rich variety of other systems, such as proximity of words in literature [48], items that are bought by the same people [16] or the way modules are connected to create a piece of software, among many others. The advances in our understanding of networks, combined with the increasing availability of many databases, allows us to analyze and gain deeper insight into the main characteristics of these complex systems. A large number of complex networks share the scale-free property [1,28], indicating the presence of few highly connected nodes (usually called hubs) and a large number of nodes with small degree. This feature alone has a great impact on the analysis of complex networks and has introduced a new way of understanding these systems. This property carries important implications in many everyday life problems, such as the way a disease spreads in communities of individuals, or the resilience and tolerance of networks under random and intentional attacks [19,20,21,31,59]. Although the scale-free property holds an undisputed importance, it has been shown to not completely determine the global structure of networks [6]. In fact, two networks that obey the same distribution of the degrees may dramatically differ in other fundamental structural properties, such as in correlations between degrees or in the average distance between nodes. Another fundamental property, which is the focus of this article, is the presence of self-similarity or fractality. In simpler terms, we want to know whether a subsection of the network looks much the same as the whole [8,14,29,66]. In spite of the fact that in regular fractal objects the distinction between self-similarity and fractality is absent, in network theory we can distinguish the two terms: in a fractal network the number of boxes of a given size that are needed to completely cover the network scales with the box size as a power law, while a self-similar network is defined as a network whose degree distribution remains invariant under renormaliza-
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tion of the network (details on the renormalization process will be provided later). This essential result allows us to better understand the origin of important structural properties of networks such as the power-law degree distribution [35,62,63]. Introduction Self-similarity is a property of fractal structures, a concept introduced by Mandelbrot and one of the fundamental mathematical results of the 20th century [29,45,66]. The importance of fractal geometry stems from the fact that these structures were recognized in numerous examples in Nature, from the coexistence of liquid/gas at the critical point of evaporation of water [11,39,65], to snowflakes, to the tortuous coastline of the Norwegian fjords, to the behavior of many complex systems such as economic data, or the complex patterns of human agglomeration [29,66]. Typically, real world scale-free networks exhibit the small-world property [1], which implies that the number of nodes increases exponentially with the diameter of the network, rather than the power-law behavior expected for self-similar structures. For this reason complex networks were believed to not be length-scale invariant or self-similar. In 2005, C. Song, S. Havlin and H. Makse presented an approach to analyze complex networks, that reveals their self-similarity [62]. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given size [62,64]. As a result, a power-law relation between the number of boxes needed to cover the network and the size of the box is found, defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena. Fractality in Real-World Networks The study of real complex networks has revealed that many of them share some fundamental common properties. Of great importance is the form of the degree distribution for these networks, which is unexpectedly wide. This means that the degree of a node may assume values that span many decades. Thus, although the majority of nodes have a relatively small degree, there is a finite probability
that a few nodes will have degree of the order of thousands or even millions. Networks that exhibit such a wide distribution P(k) are known as scale-free networks, where the term refers to the absence of a characteristic scale in the degree k. This distribution very often obeys a powerlaw form with a degree exponent , usually in the range 2 < < 4 [2], P(k) k :
(1)
A more generic property, that is usually inherent in scalefree networks but applies equally well to other types of networks, such as in Erd˝os–Rényi random graphs, is the small-world feature. Originally discovered in sociological studies [47], it is the generalization of the famous ‘six degrees of separation’ and refers to the very small network diameter. Indeed, in small-world networks a very small number of steps is required to reach a given node starting from any other node. Mathematically this is expressed by the slow (logarithmic) increase of the average diame¯ with the total number of nodes N, ter of the network, `, ¯ ` ln N, where ` is the shortest distance between two nodes and defines the distance metric in complex networks [2,12,27,67], namely, ¯
N e`/`0 ;
(2)
where `0 is a characteristic length. These network characteristics have been shown to apply in many empirical studies of diverse systems [1,2,28]. The simple knowledge that a network has the scale-free and/or small-world property already enables us to qualitatively recognize many of its basic properties. However, structures that have the same degree exponents may still differ in other aspects [6]. For example, a question of fundamental importance is whether scale-free networks are also self-similar or fractals. The illustrations of scale-free networks (see, e. g,. Figs. 1 and 2b) seem to resemble traditional fractal objects. Despite this similarity, Eq. (2) definitely appears to contradict a basic property of fractality: fast increase of the diameter with the system size. Moreover, a fractal object should be self-similar or invariant under a scale transformation, which is again not clear in the case of scale-free networks where the scale has necessarily limited range. So, how is it even possible that fractal scalefree networks exist? In the following, we will see how these seemingly contradictory aspects can be reconciled. Fractality and Self-Similarity The classical theory of self-similarity requires a power-law relation between the number of nodes N and the diameter of a fractal object ` [8,14]. The fractal dimension can
Fractal and Transfractal Scale-Free Networks
Fractal and Transfractal Scale-Free Networks, Figure 1 Representation of the Protein Interaction Network of Yeast. The colors show different subgroups of proteins that participate in different functionality classes [36]
be calculated using either box-counting or cluster-growing techniques [66]. In the first method the network is covered with NB boxes of linear size `B . The fractal dimension or box dimension dB is then given by [29]: B : NB `d B
(3)
In the second method, instead of covering the network with boxes, a random seed node is chosen and nodes centered at the seed are grown so that they are separated by a maximum distance `. The procedure is then repeated by choosing many seed nodes at random and the average “mass” of the resulting clusters, hMc i (defined as the number of nodes in the cluster) is calculated as a function of ` to obtain the following scaling: hMc i `d f ;
(4)
defining the fractal cluster dimension df [29]. If we use Eq. (4) for a small-world network, then Eq. (2) readily implies that df D 1. In other words, these networks cannot be characterized by a finite fractal dimension, and should be regarded as infinite-dimensional objects. If this were true, though, local properties in a part of the network would not be able to represent the whole system. Still, it is also well established that the scale-free nature is simi-
lar in different parts of the network. Moreover, a graphical representation of real-world networks allows us to see that those systems seem to be built by attaching (following some rule) copies of itself. The answer lies in the inherent inhomogeneity of the network. In the classical case of a homogeneous system (such as a fractal percolation cluster) the degree distribution is very narrow and the two methods described above are fully equivalent, because of this local neighborhood invariance. Indeed, all boxes in the box-covering method are statistically similar with each other as well as with the boxes grown when using the cluster-growing technique, so that Eq. (4) can be derived from Eq. (3) and dB D df . In inhomogeneous systems, though, the local environment can vary significantly. In this case, Eqs. (3) and (4) are no longer equivalent. If we focus on the box-covering technique then we want to cover the entire network with the minimum possible number of boxes NB (`B ), where the distance between any two nodes that belong in a box is smaller than `B . An example is shown in Fig. 2a using a simple 8-node network. After we repeat this procedure for different values of `B we can plot NB vs. `B . When the box-covering method is applied to real large-scale networks, such as the WWW [2] (http://www. nd.edu/~networks), the network of protein interaction
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of H. sapiens and E. coli [25,68] and several cellular networks [38,52], then they follow Eq. (3) with a clear power-law, indicating the fractal nature of these systems (Figs. 3a,b,c). On the other hand when the method is applied to other real world networks such as the Internet [24] or the Barabási–Albert network [7], they do not satisfy Eq. (3), which manifests that these networks are not fractal. The reason behind the discrepancy in the fractality of homogeneous and inhomogeneous systems can be better clarified studying the mass of the boxes. For a given `B value, the average mass of a box hMB (`B )i is hMB (`B )i
N `Bd B ; NB (`B )
(5)
as also verified in Fig. 3 for several real-world networks. On the other hand, the average performed in the cluster growing method (averaging over single boxes without tiling the system) gives rise to an exponential growth of the mass hMc (`)i e`/`1 ;
Fractal and Transfractal Scale-Free Networks, Figure 2 The renormalization procedure for complex networks. a Demonstration of the method for different `B and different stages in a network demo. The first column depicts the original network. The system is tiled with boxes of size `B (different colors correspond to different boxes). All nodes in a box are connected by a minimum distance smaller than the given `B . For instance, in the case of `B D 2, one identifies four boxes which contain the nodes depicted with colors red, orange, white, and blue, each containing 3, 2, 1, and 2 nodes, respectively. Then each box is replaced by a single node; two renormalized nodes are connected if there is at least one link between the unrenormalized boxes. Thus we obtain the network shown in the second column. The resulting number of boxes needed to tile the network, NB (`B ), is plotted in Fig. 3 vs. `B to obtain dB as in Eq. (3). The renormalization procedure is applied again and repeated until the network is reduced to a single node (third and fourth columns for different `B ). b Three stages in the renormalization scheme applied to the entire WWW. We fix the box size to `B D 3 and apply the renormalization for four stages. This corresponds, for instance, to the sequence for the network demo depicted in the second row in part a of this figure. We color the nodes in the web according to the boxes to which they belong
(6)
in accordance with the small-world effect, Eq. (2). Correspondingly, the probability distribution of the mass of the boxes MB using box-covering is very broad, while the cluster-growing technique leads to a narrow probability distribution of Mc . The topology of scale-free networks is dominated by several highly connected hubs—the nodes with the largest degree—implying that most of the nodes are connected to the hubs via one or very few steps. Therefore, the average performed in the cluster growing method is biased; the hubs are overrepresented in Eq. (6) since almost every node is a neighbor of a hub, and there is always a very large probability of including the same hubs in all clusters. On the other hand, the box covering method is a global tiling of the system providing a flat average over all the nodes, i. e. each part of the network is covered with an equal probability. Once a hub (or any node) is covered, it cannot be covered again. In conclusion, we can state that the two dominant methods that are routinely used for calculations of fractality and give rise to Eqs. (3) and (4) are not equivalent in scale-free networks, but rather highlight different aspects: box covering reveals the self-similarity, while cluster growth reveals the small-world effect. The apparent contradiction is due to the hubs being used many times in the latter method. Scale-free networks can be classified into three groups: (i) pure fractal, (ii) pure small-world and (iii) a mixture
Fractal and Transfractal Scale-Free Networks
Fractal and Transfractal Scale-Free Networks, Figure 3 Self-similar scaling in complex networks. a Upper panel: Log-log plot of the NB vs. `B revealing the self-similarity of the WWW according to Eq. (3). Lower panel: The scaling of s(`B ) vs. `B according to Eq. (9). b Same as a but for two protein interaction networks: H. sapiens and E. coli. Results are analogous to b but with different scaling exponents. c Same as a for the cellular networks of A. fulgidus, E. coli and C. elegans. d Internet. Log-log plot of NB (`B ). The solid line shows that the internet [24] is not a fractal network since it does not follow the power-law relation of Eq. (5). e Same as d for the Barabási–Albert model network [7] with m D 3 and m D 5
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between fractal and small-world. (i) A fractal network satisfies Eq. (3) at all scales, meaning that for any value of `B , the number of boxes always follows a power-law (examples are shown in Fig. 3a,b,c). (ii) When a network is a pure small-world, it never satisfies Eq. (3). Instead, NB follows an exponential decay with `B and the network cannot be regarded as fractal. Figures 3d and 3e show two examples of pure small-world networks. (iii) In the case of mixture between fractal and small-world, Eq. (3) is satisfied up to some cut-off value of `B , above which the fractality breaks down and the small-world property emerges. The smallworld property is reflected in the plot of NB vs. `B as an exponential cut-off for large `B . We can also understand the coexistence of the smallworld property and the fractality through a more intuitive approach. In a pure fractal network the length of a path between any pair of nodes scales as a power-law with the number of nodes in the network. Therefore, the diameter L also follows a power-law, L N 1/d B . If one adds a few shortcuts (links between randomly chosen nodes), many paths in the network are drastically shortened and the small-world property emerges as L LogN. In spite of this fact, for shorter scales, `B L, the network still behaves as a fractal. In this sense, we can say that globally the network is small-world, but locally (for short scales) the network behaves as a fractal. As more shortcuts are added, the cut-off in a plot of NB vs. `B appears for smaller `B , until the network becomes a pure small-world for which all paths lengths increase logarithmically with N. The reasons why certain networks have evolved towards a fractal or non-fractal structure will be described later, together with models and examples that provide additional insight into the processes involved.
The method works as follows. Start by fixing the value of `B and applying the box-covering algorithm in order to cover the entire network with boxes (see Appendix). In the renormalized network each box is replaced by a single node and two nodes are connected if there existed at least one connection between the two corresponding boxes in the original network. The resulting structure represents the first stage of the renormalized network. We can apply the same procedure to this new network, as well, resulting in the second renormalization stage network, and so on until we are left with a single node. The second column of the panels in Fig. 2a shows this step in the renormalization procedure for the schematic network, while Fig. 2b shows the results for the same procedure applied to the entire WWW for `B D 3. The renormalized network gives rise to a new probability distribution of links, P(k 0 ) (we use a prime 0 to denote quantities in the renormalized network). This distribution remains invariant under the renormalization: P(k) ! P(k 0 ) (k 0 ) :
(7)
Fig. 4 supports the validity of this scale transformation by showing a data collapse of all distributions with the same
according to (7) for the WWW. Here, we present the basic scaling relations that characterize renormalizable networks. The degree k 0 of each node in the renormalized network can be seen to scale with
Renormalization Renormalization is one of the most important techniques in modern Statistical Physics [17,39,58]. The idea behind this procedure is to continuously create smaller replicas of a given object, retaining at the same time the essential structural features, and hoping that the coarse-grained copies will be more amenable to analytic treatment. The idea for renormalizing the network emerges naturally from the concept of fractality described above. If a network is self-similar, then it will look more or less the same under different scales. The way to observe these different length-scales is based on renormalization principles, while the criterion to decide on whether a renormalized structure retains its form is the invariance of the main structural features, expressed mainly through the degree distribution.
Fractal and Transfractal Scale-Free Networks, Figure 4 Invariance of the degree distribution of the WWW under the renormalization for different box sizes, `B . We show the data collapse of the degree distributions demonstrating the self-similarity at different scales. The inset shows the scaling of k 0 D s(`B )k for different `B , from where we obtain the scaling factor s(`B ). Moreover, renormalization for a fixed box size (`B D 3) is applied, until the network is reduced to a few nodes. It was found that P(k) is invariant under these multiple renormalizations procedures
Fractal and Transfractal Scale-Free Networks
the largest degree k in the corresponding original box as 0
k ! k D s(`B )k :
(8)
This equation defines the scaling transformation in the connectivity distribution. Empirically, it was found that the scaling factor s(< 1) scales with `B with a new expod nent, dk , as s(`B ) `B k , so that d k
k 0 `B
k;
(9)
This scaling is verified for many networks, as shown in Fig. 3. The exponents , dB , and d k are not all independent from each other. The proof starts from the density balance equation n(k)dk D n0 (k 0 )dk 0 , where n(k) D N P(k) is the number of nodes with degree k and n0 (k 0 ) D N 0 P(k 0 ) is the number of nodes with degree k 0 after the renormalization (N 0 is the total number of nodes in the renormalized network). Substituting Eq. (8) leads to N 0 D s 1 N. Since the total number of nodes in the renormalized network is the number of boxes needed to cover the unrenormalized network at any given `B we have the identity N 0 D NB (`B ). Finally, from Eqs. (3) and (9) one obtains the relation between the three indexes dB : (10)
D1C dk The use of Eq. (10) yields the same exponent as that obtained in the direct calculation of the degree distribution. The significance of this result is that the scale-free properties characterized by can be related to a more fundamental length-scale invariant property, characterized by the two new indexes dB and dk . We have seen, thus, that concepts introduced originally for the study of critical phenomena in statistical physics, are also valid in the characterization of a different class of phenomena: the topology of complex networks. A large number of scale-free networks are fractals and an even larger number remain invariant under a scale-transformation. The influence of these features on the network properties will be delayed until the sixth chapter, after we introduce some algorithms for efficient numerical calculations and two theoretical models that give rise to fractal networks.
the q ! 1 limit of the Potts model. Unfortunately, in those days the importance of the power-law degree distribution and the concept of fractal and non-fractal complex networks were not known. Much work has been done on these types of hierarchical networks. For example, in 1984, M. Kaufman and R. Griffiths made use of Berker and Ostlund’s model to study the percolation phase transition and its percolation exponents [22,37,40,41]. Since the late 90s, when the importance of the powerlaw degree distribution was first shown [1] and after the finding of C. Song, S. Havlin and H. Makse [62], many hierarchical networks that describe fractality in complex networks have been proposed. These artificial models are of great importance since they provide insight into the origins and fundamental properties that give rise to the fractality and non-fractality of networks.
The Song–Havlin–Makse Model The correlations between degrees in a network [46,49, 50,54] are quantified through the probability P(k1 ; k2 ) that a node of degree k1 is connected to another node of degree k2 . In Fig. 5 we can see the degree correlation profile R(k1 ; k2 ) D P(k1 ; k2 )/Pr (k1 ; k2 ) of the cellular metabolic network of E. coli [38] (known to be a fractal network) and for the Internet at the router level [15] (a non-fractal network), where Pr (k1 ; k2 ) is obtained by randomly swapping the links without modifying the degree distribution. Figure 5 shows a dramatic difference between the two networks. The network of E. coli, that is a fractal network, presents an anti-correlation of the degrees (or dissasortativity [49,50]), which means that mostly high degree nodes are linked to low degree nodes. This property leads to fractal networks. On the other hand, the Internet exhibits a high correlation between degrees leading to a non-fractal network. With this idea in mind, in 2006 C. Song, S. Havlin and H. Makse presented a model that elucidates the way
Models: Deterministic Fractal and Transfractal Networks The first model of a scale-free fractal network was presented in 1979 when N. Berker and S. Ostlund [9] proposed a hierarchical network that served as an exotic example where renormalization group techniques yield exact results, including the percolation phase transition and
Fractal and Transfractal Scale-Free Networks, Figure 5 Degree correlation profile for a the cellular metabolic network of E. coli, and b the Internet at the router level
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Fractal and Transfractal Scale-Free Networks
Fractal and Transfractal Scale-Free Networks, Figure 6 The model grows from a small network, usually two nodes connected to each other. During each step and for every link in the system, each endpoint of a link produces m offspring nodes (in this drawing m D 3). In this case, with probability e D 1 the original link is removed and x new links between randomly selected nodes of the new generation are added. Notice that the case of x D 1 results in a tree structure, while loops appear for x > 1
new nodes must be connected to the old ones in order to build a fractal, a non-fractal network, or a mixture between fractal and non-fractal network [63]. This model shows that, indeed, the correlations between degrees of the nodes are a determinant factor for the fractality of a network. This model was later extended [32] to allow loops in the network, while preserving the self-similarity and fractality properties. The algorithm is as follows (see Fig. 6): In generation n D 0, start with two nodes connected by one link. Then, generation n C 1 is obtained recursively by attaching m new nodes to the endpoints of each link l of generation n. In addition, with probability e remove link l and add x new links connecting pairs of new nodes attached to the endpoints of l. The degree distribution, diameter and fractal dimension can be easily calculated. For example, if e D 1 (pure fractal network), the degree distribution follows a powerlaw P(k) k with exponent D 1Clog(2mC x)/logm and the fractal dimension is dB D log(2m C x)/logm. The diameter L scales, in this case, as power of the number of nodes as L N 1/d B [63,64]. Later, in Sect. “Properties of Fractal and Transfractal Networks”, several topological properties are shown for this model network.
Fractal and Transfractal Scale-Free Networks, Figure 7 (u; v)-flowers with u C v D 4( D 3). a u D 1 (dotted line) and v D 3 (broken line). b u D 2 and v D 2. The graphs may also be iterated by joining four replicas of generation n at the hubs A and B, for a, or A and C, for b
(u; v)-Flowers In 2006, H. Rozenfeld, S. Havlin and D. ben-Avraham proposed a new family of recursive deterministic scalefree networks, the (u; v)-flowers, that generalize both, the original scale-free model of Berker and Ostlund [9] and the pseudo-fractal network of Dorogovstev, Goltsev and Mendes [26] and that, by appropriately varying its two parameters u and v, leads to either fractal networks or non-fractal networks [56,57]. The algorithm to build the (u; v)-flowers is the following: In generation n D 1 one starts with a cycle graph (a ring) consisting of u C v w links and nodes (other choices are possible). Then, generation n C 1 is obtained recursively by replacing each link by two parallel paths of u and v links long. Without loss of generality, u v. Examples of (1; 3)- and (2; 2)-flowers are shown in Fig. 7. The DGM network corresponds to the special case of u D 1 and v D 2 and the Berker and Ostlund model corresponds to u D 2 and v D 2. An essential property of the (u; v)-flowers is that they are self-similar, as evident from an equivalent method of construction: to produce generation nC1, make w D uCv
Fractal and Transfractal Scale-Free Networks
copies of the net in generation n and join them at the hubs. The number of links of a (u; v)-flower of generation n is M n D (u C v)n D w n ;
(11)
and the number of nodes is Nn D
w 2 w wn C : w1 w1
(12)
The degree distribution of the (u; v)-flowers can also be easily obtained since by construction, (u; v)-flowers have only nodes of degree k D 2m , m D 1; 2; : : : ; n. As in the DGM case, (u; v)-flowers follow a scale-free degree distribution, P(k) k , of degree exponent
D1C
ln(u C v) : ln 2
(13)
Recursive scale-free trees may be defined in analogy to the flower nets. If v is even, one obtains generation n C 1 of a (u; v)-tree by replacing every link in generation n with a chain of u links, and attaching to each of its endpoints chains of v/2 links. Figure 8 shows how this works for the (1; 2)-tree. If v is odd, attach to the endpoints (of the chain of u links) chains of length (v ˙ 1)/2. The trees may be also constructed by successively joining w replicas at the appropriate hubs, and they too are self-similar. They share many of the fundamental scaling properties with (u; v)-flowers: Their degree distribution is also scale-free, with the same degree exponent as (u; v)-flowers. The self-similarity of (u; v)-flowers, coupled with the fact that different replicas meet at a single node, makes them amenable to exact analysis by renormalization techniques. The lack of loops, in the case of (u; v)-trees, further simplifies their analysis [9,13,56,57]. Dimensionality of the (u; v)-Flowers There is a vast difference between (u; v)-nets with u D 1 and u > 1. If u D 1 the diameter Ln of the nth generation flower scales linearly with n. For example, L n for the (1; 2)-flower [26] and L n D 2n for the (1; 3)-flower. It is easy to see that the diameter of the (1; v)-flower, for v odd, is L n D (v 1)n C (3 v)/2, and, in general one can show that L n (v 1)n. For u > 1, however, the diameter grows as a power of n. For example, for the (2; 2)-flower we find L n D 2n , and, more generally, the diameter satisfies L n u n . To summarize, ( Ln
(v 1)n
u D1;
un
u >1;
flowers :
(14)
Fractal and Transfractal Scale-Free Networks, Figure 8 The (1; 2)-tree. a Each link in generation n is replaced by a chain of u D 1 links, to which ends one attaches chains of v/2 D 1 links. b Alternative method of construction highlighting selfsimilarity: u C v D 3 replicas of generation n are joined at the hubs. c Generations n D 1; 2; 3
Similar results are quite obvious for the case of (u; v)-trees, where ( Ln
vn
u D1;
un
u >1;
trees :
(15)
Since N n (u C v)n (Eq. (12)), we can recast these relations as ( L
ln N
u D1;
N ln u/ ln(uCv)
u > 1:
(16)
Thus, (u; v)-nets are small world only in the case of u D 1. For u > 1, the diameter increases as a power of N, just as in finite-dimensional objects, and the nets are in fact fractal. For u > 1, the change of mass upon the rescaling of length by a factor b is N(bL) D b d B N(L) ;
(17)
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where dB is the fractal dimension [8]. In this case, N(uL) D (u C v)N(L), so dB D
ln(u C v) ; ln u
u >1:
(18)
Transfinite Fractals Small world nets, such as (1; v)nets, are infinite-dimensional. Indeed, their mass (N, or M) increases faster than any power (dimension) of their diameter. Also, note that a naive application of (4) to u ! 1 yields df ! 1. In the case of (1; v)-nets one can use their weak self-similarity to define a new measure of dimensionality, d˜f , characterizing how mass scales with diameter: `d˜f
N(L C `) D e
N(L) :
(19)
Instead of a multiplicative rescaling of length, L 7! bL, a slower additive mapping, L 7! L C `, that reflects the small world property is considered. Because the exponent d˜f usefully distinguishes between different graphs of infinite dimensionality, d˜f has been termed the transfinite fractal dimension of the network. Accordingly, objects that are self-similar and have infinite dimension (but finite transfinite dimension), such as the (1; v)-nets, are termed transfinite fractals, or transfractals, for short. For (1; v)-nets, we see that upon ‘zooming in’ one generation level the mass increases by a factor of w D 1 C v, while the diameter grows from L to L C v 1 (for flowers), or to L C v (trees). Hence their transfractal dimension is ( ln(1Cv) (1; v)-trees ; v ˜ df D ln(1Cv) (20) (1; v)-flowers : v1 There is some arbitrariness in the selection of e as the base of the exponential in the definition (19). However the base is inconsequential for the sake of comparison between dimensionalities of different objects. Also, scaling relations between various transfinite exponents hold, irrespective of the choice of base: consider the scaling relation of Eq. (10) valid for fractal scale-free nets of degree exponent [62,63]. For example, in the fractal (u; v)-nets (with u > 1) renormalization reduces lengths by a factor b D u and all degrees are reduced by a factor of 2, so b d k D 2. Thus dk D ln 2/ ln u, and since dB D ln(u C v)/ ln u and
D 1Cln(uCv)/ ln 2, as discussed above, the relation (10) is indeed satisfied. For transfractals, renormalization reduces distances by an additive length, `, and we express the self-similarity manifest in the degree distribution as ˜
˜
P0 (k) D e`d k P(e`d k k) ;
(21)
where d˜k is the transfinite exponent analogous to dk . Renormalization of the transfractal (1; v)-nets reduces the
link lengths by ` D v 1 (for flowers), or ` D v (trees), while all degrees are halved. Thus, ( ln 2 (1; v)-trees ; ˜ dk D lnv 2 (1; v)-flowers : v1 Along with (20), this result confirms that the scaling relation
D1C
d˜f d˜k
(22)
is valid also for transfractals, and regardless of the choice of base. A general proof of this relation is practically identical to the proof of (10) [62], merely replacing fractal with transfractal scaling throughout the argument. For scale-free transfractals, following m D L/` renormalizations the diameter and mass reduce to order one, ˜ and the scaling (19) implies L m`, N em`d f , so that 1 L ˜ ln N ; df in accordance with their small world property. At the same ˜ ˜ ˜ time the scaling (21) implies K e m`d k , or K N d k /d f . 1/( Using the scaling relation (22), we rederive K N 1) , which is indeed valid for scale-free nets in general, be they fractal or transfractal. Properties of Fractal and Transfractal Networks The existence of fractality in complex networks immediately calls for the question of what is the importance of such a structure in terms of network properties. In general, most of the relevant applications seem to be modified to a larger or lesser extent, so that fractal networks can be considered to form a separate network sub-class, sharing the main properties resulting from the wide distribution of regular scale-free networks, but at the same time bearing novel properties. Moreover, from a practical point of view a fractal network can be usually more amenable to analytic treatment. In this section we summarize some of the applications that seem to distinguish fractal from non-fractal networks. Modularity Modularity is a property closely related to fractality. Although this term does not have a unique well-defined definition we can claim that modularity refers to the existence of areas in the network where groups of nodes share some common characteristics, such as preferentially connecting within this area (the ‘module’) rather than to the rest of
Fractal and Transfractal Scale-Free Networks
the network. The isolation of modules into distinct areas is a complicated task and in most cases there are many possible ways (and algorithms) to partition a network into modules. Although networks with significant degree of modularity are not necessarily fractals, practically all fractal networks are highly modular in structure. Modularity naturally emerges from the effective ‘repulsion’ between hubs. Since the hubs are not directly connected to each other, they usually dominate their neighborhood and can be considered as the ‘center of mass’ for a given module. The nodes surrounding hubs are usually assigned to this module. The renormalization property of self-similar networks is very useful for estimating how modular a given network is, and especially for how this property is modified under varying scales of observation. We can use a simple definition for modularity M, based on the idea that the number of links connecting nodes within a module, Lin i , is higher than the number of link connecting nodes in different modules, Lout i . For this purpose, the boxes that result from the boxcovering method at a given length-scale `B are identified as the network modules for this scale. This partitioning assumes that the minimization of the number of boxes corresponds to an increase of modularity, taking advantage of the idea that all nodes within a box can reach each other within less than `B steps. This constraint tends to assign the largest possible number of nodes in a given neighborhood within the same box, resulting in an optimized modularity function. A definition of the modularity function M that takes advantage of the special features of the renormalization process is, thus, the following [32]: M(`B ) D
NB Lin 1 X i ; NB Lout i
(23)
iD1
where the sum is over all the boxes. The value of M through Eq. (23) for a given `B value is of small usefulness on its own, though. We can gather more information on the network structure if we measure M for different values of `B . If the dependence of M on `B has the form of a power-law, as if often the case in practice, then we can define the modularity exponent d M through M(`B ) `Bd M :
(24)
The exponent d M carries the important information of how modularity scales with the length, and separates modular from non-modular networks. The value of d M is easy
to compute in a d-dimensional lattice, since the number of d links within any module scales with its bulk, as Lin i `B and the number of links outside the module scale with the length of its interface, i. e. Lout `Bd1 . So, the resulting i scaling is M `B i. e. d M D 1. This is also the borderline value that separates non-modular structures (d M < 1) from modular ones (d M > 1). For the Song–Havlin–Makse fractal model introduced in the previous section, a module can be identified as the neighborhood around a central hub. In the simplest version with x D 1, the network is a tree, with well-defined modules. Larger values of x mean that a larger number of links are connecting different modules, creating more loops and ‘blurring’ the discreteness of the modules, so that we can vary the degree of modularity in the network. For this model, it is also possible to analytically calculate the value of the exponent d M . During the growth process at step t, the diameter in the network model increases multiplicatively as L(t C 1) D 3L(t). The number of links within a module grows with 2m C x (each node on the side of one link gives rise to m new links and x extra links connect the new nodes), while the number of links pointing out of a module is by definition proportional to x. Thus, the modularity M(`B ) of a network is proportional to (2m C x)/x. Equation (24) can then be used to calculate d M for the model: 2m C x 3d M ; x
(25)
which finally yields
dM
ln 2 mx C 1 D : ln 3
(26)
So, in this model the important quantity that determines the degree of modularity in the system is the ratio of the growth parameters m/x. Most of the real-life networks that have been measured display some sort of modular character, i. e. d M > 1, although many of them have values very close to 1. Only in a few cases we have observed exponents d M < 1. Most interesting is, though, the case of d M values much larger than 1, where a large degree of modularity is observed and this trend is more pronounced for larger length-scales. The importance of modularity as described above can be demonstrated in biological networks. There, it has been suggested that the boxes may correspond to functional modules and in protein interaction networks, for example, there may be an evolution drive of the system behind the development of its modular structure.
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Robustness Shortly after the discovery of the scale-free property, the first important application of their structure was perhaps their extreme resilience to removal of random nodes [3,10,19,21,57,60,61]. At the same time such a network was found to be quite vulnerable to an intentional attack, where nodes are removed according to decreasing order of their degree [20,30]. The resilience of a network is usually quantified through the size of the largest remaining connected cluster Smax (p), when a fraction p of the nodes has been removed according to a given strategy. At a critical point pc where this size becomes equal to Smax (pc ) ' 0, we consider that the network has been completely disintegrated. For the random removal case, this threshold is pc ' 1, i. e. practically all nodes need to be destroyed. In striking contrast, for intentional attacks pc is in general of the order of only a few percent, although the exact value depends on the system details. Fractality in networks considerably strengthens the robustness against intentional attacks, compared to nonfractal networks with the same degree exponent . In Fig. 9 the comparison between two such networks clearly shows that the critical fraction pc increases almost 4 times from pc ' 0:02 (non-fractal topology) to pc ' 0:09 (fractal topology). These networks have the same exponent, the same number of links, number of nodes, number of loops and the same clustering coefficient, differing only in whether hubs are directly connected to each other. The fractal property, thus, provides a way of increasing resistance against the network collapse, in the case of a targeted attack. The main reason behind this behavior is the dispersion of hubs in the network. A hub is usually a central node that helps other nodes to connect to the main body of the system. When the hubs are directly connected to each other, this central core is easy to destroy in a targeted attack leading to a rapid collapse of the network. On the contrary, isolating the hubs into different areas helps the network to retain connectivity for longer time, since destroying the hubs now is not similarly catastrophic, with most of the nodes finding alternative paths through other connections. The advantage of increased robustness derived from the combination of modular and fractal network character, may provide valuable hints on why most biological networks have evolved towards a fractal architecture (better chance of survival against lethal attacks). Degree Correlations We have already mentioned the importance of hub-hub correlations or anti-correlations in fractality. Generalizing
Fractal and Transfractal Scale-Free Networks, Figure 9 Vulnerability under intentional attack of a non-fractal Song– Makse–Havlin network (for e D 0) and a fractal Song–Makse– Havlin network (for e D 1). The plot shows the relative size of the largest cluster, S, and the average size of the remaining isolated clusters, hsi as a function of the removal fraction f of the largest hubs for both networks
this idea to nodes of any degree, we can ask what is the joint degree probability P(k1 ; k2 ) that a randomly chosen link connects two nodes with degree k1 and k2 , respectively. Obviously, this is a meaningful question only for networks with a wide degree distribution, otherwise the answer is more or less trivial with all nodes having similar degrees. A similar and perhaps more useful quantity is the conditional degree probability P(k1 jk2 ), defined as the probability that a random link from a node having degree k2 points to a node with degree k1 . In general, the following balance condition is satisfied k2 P(k1 jk2 )P(k2 ) D k1 P(k2 jk1 )P(k1 ) :
(27)
It is quite straightforward to calculate P(k1 jk2 ) for completely uncorrelated networks. In this case, P(k1 jk2 ) does not depend on k2 , and the probability to chose a node with degree k1 becomes simply P(k1 jk2 ) D k1 P(k1 )/hk1 i. In the case where degree-degree correlations are present, though, the calculation of this function is very difficult, even when restricting ourselves to a direct numerical evaluation, due to the emergence of huge fluctuations. We can still estimate this function, though, using again the self-similarity principle. If we consider that the function P(k1 ; k2 ) remains invariant under the network renormalization scheme described above, then it is possible to
Fractal and Transfractal Scale-Free Networks
show that [33] ( 1) k2
P(k1 ; k2 ) k1
(k1 > k2 ) ;
(28)
and similarly P(k1 jk2 ) k1( 1) k2( C1) (k1 > k2 ) ;
(29)
In the above equations we have also introduced the correlation exponent , which characterizes the degree of correlations in a network. For example, the case of uncorrelated networks is described by the value D 1. The exponent can be measured quite accurately using an appropriate quantity. For this purpose, we can introduce a measure such as R1 P(kjk2 )dk2 Eb (k) bkR 1 ; (30) bk P(k)dk which estimates the probability that a node with degree k has neighbors with degree larger than bk, and b is an arbitrary parameter that has been shown not to influence the results. It is easy to show that Eb (k)
k 1 D k ( ) : k 1
(31)
This relation allows us to estimate for a given network, after calculating the quantity Eb (k) as a function of k. The above discussion can be equally applied to both fractal and non-fractal networks. If we restrict ourselves to fractal networks, then we can develop our theory a bit further. If we consider the probability E(`B ) that the largest degree node in each box is connected directly with the other largest degree nodes in other boxes (after optimally covering the network), then this quantity scales as a power law with `B : d e
E(`B ) `B
;
(32)
where d e is a new exponent describing the probability of hub-hub connection [63]. The exponent , which describes correlations over any degree, is related to d e , which refers to correlations between hubs only. The resulting relation is D2C
de de D 2 C ( 1) : dk dB
(33)
For an infinite fractal dimension dB ! 1, which is the onset of non-fractal networks that cannot be described by the above arguments, we have the limiting case of D 2. This value separates fractal from non-fractal networks, so that fractality is indicated by > 2. Also, we have seen
that the line D 1 describes networks for which correlations are minimal. Measurements of many real-life networks have verified the above statements, where networks with > 2 having been clearly characterized as fractals with alternate methods. All non-fractal networks have values of < 2 and the distance from the D 1 line determines how much stronger or weaker the correlations are, compared to the uncorrelated case. In short, using the self-similarity principle makes it possible to gain a lot of insight on network correlations, a notoriously difficult task otherwise. Furthermore, the study of correlations can be reduced to the calculation of a single exponent , which is though capable of delivering a wealth of information on the network topological properties. Diffusion and Resistance Scale-free networks have been described as objects of infinite dimensionality. For a regular structure this statement would suggest that one can simply use the known diffusion laws for d D 1. Diffusion on scale-free structures, however, is much harder to study, mainly due to the lack of translational symmetry in the system and different local environments. Although exact results are still not available, the scaling theory on fractal networks provides the tools to better understand processes, such as diffusion and electric resistance. In the following, we describe diffusion through the average first-passage time TAB , which is the average time for a diffusing particle to travel from node A to node B. At the same time, assuming that each link in the network has an electrical resistance of 1 unit, we can describe the electrical properties through the resistance between the two nodes A and B, RAB . The connection between diffusion (first-passage time) and electric networks has long been established in homogeneous systems. This connection is usually expressed through the Einstein relation [8]. The Einstein relation is of great importance because it connects a static quantity RAB with a dynamic quantity TAB . In other words, the behavior of a diffusing particle can be inferred by simply having knowledge of a static topological property of the network. In any renormalizable network the scaling of T and R follows the form: T0 R0 w ; D `d D `B ; B T R
(34)
where T 0 (R0 ) and T(R) are the first-passage time (resistance) for the renormalized and original networks, respectively. The dynamical exponents dw and characterize the
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Fractal and Transfractal Scale-Free Networks
scaling in any lattice or network that remains invariant under renormalization. The Einstein relation relates these two exponents through the dimensionality of the substrate dB , according to: dw D C dB :
(35)
The validity of this relation in inhomogeneous complex networks, however, is not yet clear. Still, in fractal and transfractal networks there are many cases where this relation has been proved to be valid, hinting towards a wider applicability. For example, in [13,56] it has been shown that the Einstein Relation [8] in (u; v)-flowers and (u; v)-trees is valid for any u and v, that is for both fractal and transfractal networks. In general, in terms of the scaling theory we can study diffusion and resistance (or conductance) in a similar manner [32]. Because of the highly inhomogeneous character of the structure, though, we are interested in how these quantities behave as a function of the end-node degrees k1 and k2 when they are separated by a given distance `. Thus, we are looking for the full dependence of T(`; k1 ; k2 ) and R(`; k1 ; k2 ). Obviously, for lattices or networks with narrow degree distribution there is no degree dependence and those results should be a function of ` only. For self-similar networks, we can rewrite Eq. (34) above as 0 d w /d B 0 /d B R0 N N T0 ; ; (36) D D T N R N where we have taken into account Eq. (3). This approach offers the practical advantage that the variation of N 0 /N is larger than the variation of `B , so that the exponents calculation can be more accurate. To calculate these exponents, we fix the box size `B and we measure the diffusion time T and resistance R between any two points in a network before and after renormalization. If for every such pair we plot the corresponding times and resistances in T 0 vs. T and R0 vs. R plots, as shown in Fig. 10, then all these points fall in a narrow area, suggesting a constant value for the ratio T 0 /T over the entire network. Repeating this procedure for different `B values yields other ratio values. The plot of these ratios vs. N 0 /N (Fig. 11) finally exhibits a power-law dependence, verifying Eq. (36). We can then easily calculate the exponents dw and from the slopes in the plot, since the dB exponent is already known through the standard box-covering methods. It has been shown that the results for many different networks are consistent, within statistical error, with the Einstein relation [32,56]. The dependence on the degrees k1 , k2 and the distance ` can also be calculated in a scaling form using the
Fractal and Transfractal Scale-Free Networks, Figure 10 Typical behavior of the probability distributions for the resistance R vs. R0 and the diffusion time T vs. T 0 , respectively, for a given `B value. Similar plots for other `B values verify that the ratios of these quantities during a renormalization stage are roughly constant for all pairs of nodes in a given biological network
Fractal and Transfractal Scale-Free Networks, Figure 11 Average value of the ratio of resistances R/R0 and diffusion times T/T 0 , as measured for different `B values (each point corresponds to a different value of `B ). Results are presented for both biological networks, and two fractal network models with different dM values. The slopes of the curves correspond to the exponents /dB (top panel) and dw /dB (bottom panel)
self-similarity properties of fractal networks. After renormalization, a node with degree k in a given network, will d have a degree k 0 D `B k k according to Eq. (9). At the same time all distances ` are scaled down according to `0 D `/`B . This means that Eqs. (36) can be written as
R0 (`0 ; k10 ; k20 ) D `B R(`; k1 ; k2 )
(37)
w T 0 (`0 ; k10 ; k20 ) D `d T(`; k1 ; k2 ) : B
(38)
Substituting the renormalized quantities we get: d k
R0 (`1 B `; `B
d k
k 1 ; `B
k2 ) D `B R(`; k1 ; k2 ) :
(39)
Fractal and Transfractal Scale-Free Networks
The above equation holds for all values of `B , so we can 1/d select this quantity to be `B D k2 k . This constraint allows us to reduce the number of variables in the equation, with the final result: 0 1 ` k 1 d R @ 1 ; ; 1A D k2 k R(`B ; k1 ; k2 ) : (40) k2 dk k2 This equation suggests a scaling for the resistance R: 0 1 ` k d 1 (41) R(`; k1 ; k2 ) D k2 k f R @ 1 ; A ; k2 dk k2 where f R () is an undetermined function. All the above arguments can be repeated for the diffusion time, with a similar expression: 0 1 dw ` k dk 1 T(`; k1 ; k2 ) D k2 f T @ 1 ; A ; (42) k2 dk k2 where the form of the right-hand function may be different. The final result for the scaling form is Eqs. (41) and (42), which is also supported by the numerical data collapse in Fig. 12. Notice that in the case of homogeneous networks, where there is almost no k-dependence, the unknown functions in the rhs reduce to the forms f R (x; 1) D x , f T (x; 1) D x d w , leading to the well-established classical relations R ` and T `d w . Future Directions Fractal networks combine features met in fractal geometry and in network theory. As such, they present many unique aspects. Many of their properties have been well-studied and understood, but there is still a great amount of open and unexplored questions remaining to be studied. Concerning the structural aspects of fractal networks, we have described that in most networks the degree distribution P(k), the joint degree distribution P(k1 ; k2 ) and a number of other quantities remain invariant under renormalization. Are there any quantities that are not invariable, and what would their importance be? Of central importance is the relation of topological features with functionality. The optimal network covering leads to the partitioning of the network into boxes. Do these boxes carry a message other than nodes proximity? For example, the boxes could be used as an alternative definition for separated communities, and fractal methods could be used as a novel method for community detection in networks [4,5,18,51,53].
Fractal and Transfractal Scale-Free Networks, Figure 12 Rescaling of a the resistance and b the diffusion time according to Eqs. (41) and (42) for the protein interaction network of yeast (upper symbols) and the Song–Havlin–Makse model for e D 1 (lower filled symbols). The data for PIN have been vertically shifted upwards by one decade for clarity. Each symbol corresponds to a fixed ratio k1 /k2 and the different colors denote a different value for k 1 . Inset: Resistance R as a function of distance `, before rescaling, for constant ratio k1 /k2 D 1 and different k 1 values
The networks that we have presented are all static, with no temporal component, and time evolution has been ignored in all our discussions above. Clearly, biological networks, the WWW, and other networks have grown (and continue to grow) from some earlier simpler state to their present fractal form. Has fractality always been there or has it emerged as an intermediate stage obeying certain evolutionary drive forces? Is fractality a stable condition or growing networks will eventually fall into a non-fractal form? Finally, we want to know what is the inherent reason behind fractality. Of course, we have already described how hub-hub anti-correlations can give rise to fractal networks. However, can this be directly related to some underlying mechanism, so that we gain some information on the process? In general, in Biology we already have some idea on the advantages of adopting a fractal structure. Still, the question remains: why fractality exists in certain networks and not in others? Why both fractal and non-fractal networks are needed? It seems that we will be able to increase our knowledge for the network evolutionary mechanisms through fractality studies. In conclusion, a deeper understanding of the self-similarity, fractality and transfractality of complex networks will help us analyze and better understand many fundamental properties of real-world networks.
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Acknowledgments We acknowledge support from the National Science Foundation. Appendix: The Box Covering Algorithms The estimation of the fractal dimension and the self-similar features in networks have become standard properties in the study of real-world systems. For this reason, in the last three years many box covering algorithms have been proposed [64,69]. This section presents four of the main algorithms, along with a brief discussion on the advantages and disadvantages that they offer. Recalling the original definition of box covering by Hausdorff [14,29,55], for a given network G and box size `B , a box is a set of nodes where all distances ` i j between any two nodes i and j in the box are smaller than `B . The minimum number of boxes required to cover the entire network G is denoted by NB . For `B D 1, each box encloses only 1 node and therefore, NB is equal to the size of the network N. On the other hand, NB D 1 for `B `max B , is the diameter of the network plus one. where `max B The ultimate goal of a box-covering algorithm is to find the minimum number of boxes NB (`B ) for any `B . It has been shown that this problem belongs to the family of NP-hard problems [34], which means that the solution cannot be achieved in polynomial time. In other words, for a relatively large network size, there is no algorithm that can provide an exact solution in a reasonably short amount of time. This limitation requires treating the box covering problem with approximations, using for example optimization algorithms. The Greedy Coloring Algorithm The box-covering problem can be mapped into another NP-hard problem [34]: the graph coloring problem. An algorithm that approximates well the optimal solution of this problem was presented in [64]. For an arbitrary value of `B , first construct a dual network G 0 , in which two nodes are connected if the distance between them in G (the original network) is greater or equal than `B . Figure 13 shows an example of a network G which yields such a dual network G 0 for `B D 3 (upper row of Fig. 13). Vertex coloring is a well-known procedure, where labels (or colors) are assigned to each vertex of a network, so that no edge connects two identically colored vertices. It is clear that such a coloring in G 0 gives rise to a natural box covering in the original network G, in the sense that vertices of the same color will necessarily form a box since the distance between them must be less than `B . Accordingly, the minimum number of boxes NB (G) is equal
Fractal and Transfractal Scale-Free Networks, Figure 13 Illustration of the solution for the network covering problem via mapping to the graph coloring problem. Starting from G (upper left panel) we construct the dual network G0 (upper right panel) for a given box size (here `B D 3), where two nodes are connected if they are at a distance ` `B . We use a greedy algorithm for vertex coloring in G0 , which is then used to determine the box covering in G, as shown in the plot
to the minimum required number of colors (or the chromatic number) in the dual network G 0 , (G 0 ). In simpler terms, (a) if the distance between two nodes in G is greater than `B these two neighbors cannot belong in the same box. According to the construction of G 0 , these two nodes will be connected in G 0 and thus they cannot have the same color. Since they have a different color they will not belong in the same box in G. (b) On the contrary, if the distance between two nodes in G is less than `B it is possible that these nodes belong in the same box. In G 0 these two nodes will not be connected and it is allowed for these two nodes to carry the same color, i. e. they may belong to the same box in G, (whether these nodes will actually be connected depends on the exact implementation of the coloring algorithm). The algorithm that follows both constructs the dual network G 0 and assigns the proper node colors for all `B values in one go. For this implementation a two-dimenis needed, whose values sional matrix c i` of size N `max B represent the color of node i for a given box size ` D `B . 1. Assign a unique id from 1 to N to all network nodes, without assigning any colors yet. 2. For all `B values, assign a color value 0 to the node with id=1, i. e. c1` D 0. 3. Set the id value i D 2. Repeat the following until i D N. (a) Calculate the distance ` i j from i to all the nodes in the network with id j less than i.
Fractal and Transfractal Scale-Free Networks
(b) Set `B D 1 (c) Select one of the unused colors c j` i j from all nodes j < i for which ` i j `B . This is the color c i`B of node i for the given `B value. (d) Increase `B by one and repeat (c) until `B D `max B . (e) Increase i by 1. The results of the greedy algorithm may depend on the original coloring sequence. The quality of this algorithm was investigated by randomly reshuffling the coloring sequence and applying the greedy algorithm several times and in different models [64]. The result was that the probability distribution of the number of boxes NB (for all box sizes `B ) is a narrow Gaussian distribution, which indicates that almost any implementation of the algorithm yields a solution close to the optimal. Strictly speaking, the calculation of the fractal dimenB sion dB through the relation NB `d is valid only for B the minimum possible value of NB , for any given `B value, so any box covering algorithm must aim to find this minimum NB . Although there is no rule to determine when this minimum value has been actually reached (since this would require an exact solution of the NP-hard coloring problem) it has been shown [23] that the greedy coloring algorithm can, in many cases, identify a coloring sequence which yields the optimal solution. Burning Algorithms This section presents three box covering algorithms based on more traditional breadth-first search algorithm. A box is defined as compact when it includes the maximum possible number of nodes, i. e. when there do not exist any other network nodes that could be included in this box. A connected box means that any node in the box can be reached from any other node in this box, without having to leave this box. Equivalently, a disconnected box denotes a box where certain nodes can be reached by other nodes in the box only by visiting nodes outside this box. For a demonstration of these definitions see Fig. 14. Burning with the Diameter `B , and the Compact-BoxBurning (CBB) Algorithm The basic idea of the CBB algorithm for the generation of a box is to start from a given box center and then expand the box so that it includes the maximum possible number of nodes, satisfying at the same time the maximum distance between nodes in the box `B . The CBB algorithm is as follows (see Fig. 15): 1. Initially, mark all nodes as uncovered. 2. Construct the set C of all yet uncovered nodes.
Fractal and Transfractal Scale-Free Networks, Figure 14 Our definitions for a box that is a non-compact for `B D 3, i. e. could include more nodes, b compact, c connected, and d disconnected (the nodes in the right box are not connected in the box). e For this box, the values `B D 5 and rB D 2 verify the relation `B D 2rB C 1. f One of the pathological cases where this relation is not valid, since `B D 3 and rB D 2
Fractal and Transfractal Scale-Free Networks, Figure 15 Illustration of the CBB algorithm for `B D 3. a Initially, all nodes are candidates for the box. b A random node is chosen, and nodes at a distance further than `B from this node are no longer candidates. c The node chosen in b becomes part of the box and another candidate node is chosen. The above process is then repeated until the box is complete
3. Choose a random node p from the set of uncovered nodes C and remove it from C. 4. Remove from C all nodes i whose distance from p is `pi `B , since by definition they will not belong in the same box. 5. Repeat steps (3) and (4) until the candidate set is empty. 6. Repeat from step (2) until all the network has been covered.
Random Box Burning In 2006, J. S. Kim et al. presented a simple algorithm for the calculation of fractal dimension in networks [42,43,44]: 1. Pick a randomly chosen node in the network as a seed of the box. 2. Search using breath-first search algorithm until distance lB from the seed. Assign all newly burned nodes
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to the new box. If no new node is found, discard and start from (1) again. 3. Repeat (1) and (2) until all nodes have a box assigned. This Random Box Burning algorithm has the advantage of being a fast and simple method. However, at the same time there is no inherent optimization employed during the network coverage. Thus, this simple Monte-Carlo method is almost certain that will yield a solution far from the optimal and one needs to implement many different realizations and only retain the smallest number of boxes found out of all these realizations. Burning with the Radius rB , and the MaximumExcluded-Mass-Burning (MEMB) Algorithm A box of size `B includes nodes where the distance between any pair of nodes is less than `B . It is possible, though, to grow a box from a given central node, so that all nodes in the box are within distance less than a given box radius rB (the maximum distance from a central node). This way, one can still recover the same fractal properties of a network. For the original definition of the box, `B corresponds to the box diameter (maximum distance between any two nodes in the box) plus one. Thus, `B and rB are connected through the simple relation `B D 2rB C 1. In general this relation is exact for loopless configurations, but in general there may exist cases where this equation is not exact (Fig. 14). The MEMB algorithm always yields the optimal solution for non scale-free homogeneous networks, since the choice of the central node is not important. However, in inhomogeneous networks with wide-tailed degree distribution, such as scale-free networks, this algorithm fails to achieve an optimal solution because of the presence of hubs. The MEMB, as a difference from the Random Box Burning and the CBB, attempts to locate some optimal central nodes which act as the burning origins for the boxes. It contains as a special case the choice of the hubs as centers of the boxes, but it also allows for low-degree nodes to be burning centers, which sometimes is convenient for finding a solution closer to the optimal. In the following algorithm we use the basic idea of box optimization, in which each box covers the maximum possible number of nodes. For a given burning radius rB , we define the excluded mass of a node as the number of uncovered nodes within a chemical distance less than rB . First, calculate the excluded mass for all the uncovered nodes. Then, seek to cover the network with boxes of maximum excluded mass. The details of this algorithm are as follows (see Fig. 17):
Fractal and Transfractal Scale-Free Networks, Figure 16 Burning with the radius rB from a a hub node or b a non-hub node results in very different network coverage. In a we need just one box of rB D 1 while in b 5 boxes are needed to cover the same network. This is an intrinsic problem when burning with the radius. c Burning with the maximum distance `B (in this case `B D 2rB C 1 D 3) we avoid this situation, since independently of the starting point we would still obtain NB D 1
Fractal and Transfractal Scale-Free Networks, Figure 17 Illustration of the MEMB algorithm for rB D 1. Upper row: Calculation of the box centers a We calculate the excluded mass for each node. b The node with maximum mass becomes a center and the excluded masses are recalculated. c A new center is chosen. Now, the entire network is covered with these two centers. Bottom row: Calculation of the boxes d Each box includes initially only the center. Starting from the centers we calculate the distance of each network node to the closest center. e We assign each node to its nearest box
1. Initially, all nodes are marked as uncovered and noncenters. 2. For all non-center nodes (including the already covered nodes) calculate the excluded mass, and select the node p with the maximum excluded mass as the next center. 3. Mark all the nodes with chemical distance less than rB from p as covered. 4. Repeat steps (2) and (3) until all nodes are either covered or centers. Notice that the excluded mass has to be updated in each step because it is possible that it has been modified during this step. A box center can also be an already covered node, since it may lead to a larger box mass. After the above procedure, the number of selected centers coincides with the
Fractal and Transfractal Scale-Free Networks
number of boxes NB that completely cover the network. However, the non-center nodes have not yet been assigned to a given box. This is performed in the next step: 1. Give a unique box id to every center node. 2. For all nodes calculate the “central distance”, which is the chemical distance to its nearest center. The central distance has to be less than rB , and the center identification algorithm above guarantees that there will always exist such a center. Obviously, all center nodes have a central distance equal to 0. 3. Sort the non-center nodes in a list according to increasing central distance. 4. For each non-center node i, at least one of its neighbors has a central distance less than its own. Assign to i the same id with this neighbor. If there exist several such neighbors, randomly select an id from these neighbors. Remove i from the list. 5. Repeat step (4) according to the sequence from the list in step (3) for all non-center nodes. Comparison Between Algorithms The choice of the algorithm to be used for a problem depends on the details of the problem itself. If connected boxes are a requirement, MEMB is the most appropriate algorithm; but if one is only interested in obtaining the fractal dimension of a network, the greedy-coloring or the random box burning are more suitable since they are the fastest algorithms.
Fractal and Transfractal Scale-Free Networks, Figure 18 Comparison of the distribution of NB for 104 realizations of the four network covering methods presented in this paper. Notice that three of these methods yield very similar results with narrow distributions and comparable minimum values, while the random burning algorithm fails to reach a value close to this minimum (and yields a broad distribution)
As explained previously, any algorithm should intend to find the optimal solution, that is, find the minimum number of boxes that cover the network. Figure 18 shows the performance of each algorithm. The greedy-coloring, the CBB and MEMB algorithms exhibit a narrow distribution of the number of boxes, showing evidence that they cover the network with a number of boxes that is close to the optimal solution. Instead, the Random Box Burning returns a wider distribution and its average is far above the average of the other algorithms. Because of the great ease and speed with which this technique can be implemented, it would be useful to show that the average number of covering boxes is overestimated by a fixed proportionality constant. In that case, despite the error, the predicted number of boxes would still yield the correct scaling and fractal dimension. Bibliography 1. Albert R, Barabási A-L (2002) Rev Mod Phys 74:47; Barabási AL (2003) Linked: how everything is connected to everything else and what it means. Plume, New York; Newman MEJ (2003) SIAM Rev 45:167; Dorogovtsev SN, Mendes JFF (2002) Adv Phys 51:1079; Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: from biological nets to the internet and WWW. Oxford University Press, Oxford; Bornholdt S, Schuster HG (2003) Handbook of graphs and networks. Wiley-VCH, Berlin; PastorSatorras R, Vespignani A (2004) Evolution and structure of the internet. Cambridge University Press, Cambridge; Amaral LAN, Ottino JM (2004) Complex networks – augmenting the framework for the study of complex systems. Eur Phys J B 38: 147–162 2. Albert R, Jeong H, Barabási A-L (1999) Diameter of the world wide web. Nature 401:130–131 3. Albert R, Jeong H, Barabási AL (2000) Nature 406:p378 4. Bagrow JP, Bollt EM (2005) Phys Rev E 72:046108 5. Bagrow JP (2008) Stat Mech P05001 6. Bagrow JP, Bollt EM, Skufca JD (2008) Europhys Lett 81:68004 7. Barabási A-L, Albert R (1999) Sience 286:509 8. ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge 9. Berker AN, Ostlund S (1979) J Phys C 12:4961 10. Beygelzimer A, Grinstein G, Linsker R, Rish I (2005) Physica A Stat Mech Appl 357:593–612 11. Binney JJ, Dowrick NJ, Fisher AJ, Newman MEJ (1992) The theory of critical phenomena: an introduction to the renormalization group. Oxford University Press, Oxford 12. Bollobás B (1985) Random graphs. Academic Press, London 13. Bollt E, ben-Avraham D (2005) New J Phys 7:26 14. Bunde A, Havlin S (1996) Percolation I and Percolation II. In: Bunde A, Havlin S (eds) Fractals and disordered systems, 2nd edn. Springer, Heidelberg 15. Burch H, Chewick W (1999) Mapping the internet. IEEE Comput 32:97–98 16. Butler D (2006) Nature 444:528 17. Cardy J (1996) Scaling and renormalization in statistical physics. Cambridge University Press, Cambridge
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18. Clauset A, Newman MEJ, Moore C (2004) Phys Rev E 70:066111 19. Cohen R, Erez K, ben-Avraham D, Havlin S (2000) Phys Rev Lett 85:4626 20. Cohen R, Erez K, ben-Avraham D, Havlin S (2001) Phys Rev Lett 86:3682 21. Cohen R, ben-Avraham D, Havlin S (2002) Phys Rev E 66:036113 22. Comellas F Complex networks: deterministic models physics and theoretical computer science. In: Gazeau J-P, Nesetril J, Rovan B (eds) From Numbers and Languages to (Quantum) Cryptography. 7 NATO Security through Science Series: Information and Communication Security. IOS Press, Amsterdam. pp 275–293. 348 pags. ISBN 1-58603-706-4 23. Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms. MIT Press, Cambridge 24. Data from SCAN project. The Mbone. http://www.isi.edu/scan/ scan.html Accessed 2000 25. Database of Interacting Proteins (DIP) http://dip.doe-mbi.ucla. edu Accessed 2008 26. Dorogovtsev SN, Goltsev AV, Mendes JFF (2002) Phys Rev E 65:066122 ˝ P, Rényi A (1960) On the evolution of random graphs. 27. Erdos Publ Math Inst Hung Acad Sci 5:17–61 28. Faloutsos M, Faloutsos P, Faloutsos C (1999) Comput Commun Rev 29:251–262 29. Feder J (1988) Fractals. Plenum Press, New York 30. Gallos LK, Argyrakis P, Bunde A, Cohen R, Havlin S (2004) Physica A 344:504–509 31. Gallos LK, Cohen R, Argyrakis P, Bunde A, Havlin S (2005) Phys Rev Lett 94:188701 32. Gallos LK, Song C, Havlin S, Makse HA (2007) PNAS 104:7746 33. Gallos LK, Song C, Makse HA (2008) Phys Rev Lett 100:248701 34. Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, New York 35. Goh K-I, Salvi G, Kahng B, Kim D (2006) Phys Rev Lett 96:018701 36. Han J-DJ et al (2004) Nature 430:88–93 37. Hinczewski M, Berker AN (2006) Phys Rev E 73:066126 38. Jeong H, Tombor B, Albert R, Oltvai ZN, Barabási A-L (2000) Nature 407:651–654 39. Kadanoff LP (2000) Statistical physics: statics, dynamics and renormalization. World Scientific Publishing Company, Singapore
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Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to ARMIN BUNDE1 , SHLOMO HAVLIN2 1 Institut für Theoretische Physik, Gießen, Germany 2 Institute of Theoretical Physics, Bar-Ilan-University, Ramat Gan, Israel Article Outline Definition of the Subject Deterministic Fractals Random Fractal Models How to Measure the Fractal Dimension Self-Affine Fractals Long-Term Correlated Records Long-Term Correlations in Financial Markets and Seismic Activity Multifractal Records Acknowledgments Bibliography Definition of the Subject In this chapter we present some definitions related to the fractal concept as well as several methods for calculating the fractal dimension and other relevant exponents. The purpose is to introduce the reader to the basic properties of fractals and self-affine structures so that this book will be self contained. We do not give references to most of the original works, but, we refer mostly to books and reviews on fractal geometry where the original references can be found. Fractal geometry is a mathematical tool for dealing with complex systems that have no characteristic length scale. A well-known example is the shape of a coastline. When we see two pictures of a coastline on two different scales, with 1 cm corresponding for example to 0.1 km or 10 km, we cannot tell which scale belongs to which picture: both look the same, and this features characterizes also many other geographical patterns like rivers, cracks, mountains, and clouds. This means that the coastline is scale invariant or, equivalently, has no characteristic length scale. Another example are financial records. When looking at a daily, monthly or annual record, one cannot tell the difference. They all look the same. Scale-invariant systems are usually characterized by noninteger (“fractal”) dimensions. The notion of noninteger dimensions and several basic properties of fractal objects were studied as long ago as the last century by Georg Cantor, Giuseppe Peano, and David Hilbert, and in
Fractal Geometry, A Brief Introduction to, Figure 1 The Dürer pentagon after five iterations. For the generating rule, see Fig. 8. The Dürer pentagon is in blue, its external perimeter is in red, Courtesy of M. Meyer
the beginning of this century by Helge von Koch, Waclaw Sierpinski, Gaston Julia, and Felix Hausdorff. Even earlier traces of this concept can be found in the study of arithmetic-geometric averages by Carl Friedrich Gauss about 200 years ago and in the artwork of Albrecht Dürer (see Fig. 1) about 500 years ago. Georg Friedrich Lichtenberg discovered, about 230 years ago, fractal discharge patterns. He was the first to describe the observed self-similarity of the patterns: A part looks like the whole. Benoit Mandelbrot [1] showed the relevance of fractal geometry to many systems in nature and presented many important features of fractals. For further books and reviews on fractals see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Before introducing the concept of fractal dimension, we should like to remind the reader of the concept of dimension in regular systems. It is well known that in regular systems (with uniform density) such as long wires, large thin plates, or large filled cubes, the dimension d characterizes how the mass M(L) changes with the linear size L of the system. If we consider a smaller part of the system of linear size bL (b < 1), then M(bL) is decreased by a factor of bd , i. e., M(bL) D b d M(L) :
(1)
The solution of the functional equation (1) is simply M(L) D ALd . For the long wire the mass changes linearly with b, i. e., d D 1. For the thin plates we obtain d D 2, and for the cubes d D 3; see Fig. 2.
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Fractal Geometry, A Brief Introduction to, Figure 2 Examples of regular systems with dimensions d D 1, d D 2, and dD3
Next we consider fractal objects. Here we distinguish between deterministic and random fractals. Deterministic fractals are generated iteratively in a deterministic way, while random fractals are generated using a stochastic process. Although fractal structures in nature are random, it is useful to study deterministic fractals where the fractal properties can be determined exactly. By studying deterministic fractals one can gain also insight into the fractal properties of random fractals, which usually cannot be treated rigorously. Deterministic Fractals In this section, we describe several examples of deterministic fractals and use them to introduce useful fractal concepts such as fractal and chemical dimension, self similarity, ramification, and fractal substructures (minimum path, external perimeter, backbone, and red bonds).
One of the most common deterministic fractals is the Koch curve. Figure 3 shows the first n D 4 iterations of this fractal curve. By each iteration the length of the curve is increased by a factor of 4/3. The mathematical fractal is defined in the limit of infinite iterations, n ! 1, where the total length of the curve approaches infinity. The dimension of the curve can be obtained as for regular objects. From Fig. 3 we notice that, if we decrease the linear size by a factor of b D 1/3, the total length (mass) of the curve decreases by a factor of 1/4, i. e., 1 3 L D
1 4
M(L) :
d D log 4/ log 3. For such non-integer dimensions Mandelbrot coined the name “fractal dimension” and those objects described by a fractal dimension are called fractals. Thus, to include fractal structures, Eq. (1) is generalized by M(bL) D b d f M(L) ;
(2)
This feature is very different from regular curves, where the length of the object decreases proportional to the linear scale. In order to satisfy Eqs. (1) and (2) we are led to introduce a noninteger dimension, satisfying 1/4 D (1/3)d , i. e.,
(3)
and M(L) D ALd f ;
The Koch Curve
M
Fractal Geometry, A Brief Introduction to, Figure 3 The first iterations of the Koch curve. The fractal dimension of the Koch curve is df D log 4/ log 3
(4)
where df is the fractal dimension. When generating the Koch curve and calculating df , we observe the striking property of fractals – the property of self-similarity. If we examine the Koch curve, we notice that there is a central object in the figure that is reminiscent of a snowman. To the right and left of this central snowman there are two other snowmen, each being an exact reproduction, only smaller by a factor of 1/3. Each of the smaller snowmen has again still smaller copies (by 1/3) of itself to the right and to the left, etc. Now, if we take any such triplet of snowmen (consisting of 1/3m of the curve), for any m, and magnify it by 3m , we will obtain exactly the original Koch curve. This property of self-similarity or scale invariance is the basic feature of all deterministic and random fractals: if we take a part of a fractal and magnify it by the same magnification factor in all directions, the magnified picture cannot be distinguished from the original.
Fractal Geometry, A Brief Introduction to
For the Koch curve as well as for all deterministic fractals generated iteratively, Eqs. (3) and (4) are of course valid only for length scales L below the linear size L0 of the whole curve (see Fig. 3). If the number of iterations n is finite, then Eqs. (3) and (4) are valid only above a lower cut off length Lmin , Lmin D L0 /3n for the Koch curve. Hence, for a finite number of iterations, there exist two cut-off length scales in the system, an upper cut-off Lmax D L0 representing the total linear size of the fractal, and a lower cut-off Lmin . This feature of having two characteristic cutoff lengths is shared by all fractals in nature. An interesting modification of the Koch curve is shown in Fig. 4, which demonstrates that the chemical distance is an important concept for describing structural properties of fractals (for a review see, for example, [16] and Chap. 2 in [13]). The chemical distance ` is defined as shortest path on the fractal between two sites of the fractal. In analogy to the fractal dimension df that characterizes how the mass of a fractal scales with (air) distance L, we introduce the chemical dimension d` in order to characterize how the mass scales with the chemical distance `, M(b`) D b d ` M(`) ;
or M(`) D B`d ` :
(5)
From Fig. 4 we see that if we reduce ` by a factor of 5, the mass of the fractal within the reduced chemical distance is reduced by a factor of 7, i. e., M(1/5 `)D 1/7 M(`), yielding d` D log 7/ log 5 Š 1:209. Note that the chemical dimension is smaller than the fractal dimension df D log 7/ log 4 Š 1:404, which follows from M(1/4 L) D 1/7 M(L).
Fractal Geometry, A Brief Introduction to, Figure 4 The first iterations of a modified Koch curve, which has a nontrivial chemical distance metric
The structure of the shortest path between two sites represents an interesting fractal by itself. By definition, the length of the path is the chemical distance `, and the fractal dimension of the shortest path, dmin , characterizes how ` scales with (air) distance L. Using Eqs. (4) and (5), we obtain ` Ld f /d ` Ld min ;
(6)
from which follows dmin D df /d` . For our example we find that dmin D log 5/ log 4 Š 1:161. For the Koch curve, as well as for any linear fractal, one simply has d` D 1 and hence dmin D df . Since, by definition, dmin 1, it follows that d` df for all fractals. The Sierpinski Gasket, Carpet, and Sponge Next we discuss the Sierpinski fractal family: the “gasket”, the “carpet”, and the “sponge”. The Sierpinski Gasket The Sierpinski gasket is generated by dividing a full triangle into four smaller triangles and removing the central triangle (see Fig. 5). In the following iterations, this procedure is repeated by dividing each of the remaining triangles into four smaller triangles and removing the central triangles. To obtain the fractal dimension, we consider the mass of the gasket within a linear size L and compare it with the mass within 1/2 L. Since M(1/2 L) D 1/3 M(L), we have df D log 3/ log 2 Š 1:585. It is easy to see that d` D df and dmin D 1.
Fractal Geometry, A Brief Introduction to, Figure 5 The Sierpinski gasket. The fractal dimension of the Sierpinski gasket is df D log 3/ log 2
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Fractal Geometry, A Brief Introduction to, Figure 6 A Sierpinski carpet with n D 5 and k D 9. The fractal dimension of this structure is df D log 16/ log 5
The Sierpinski Carpet The Sierpinski carpet is generated in close analogy to the Sierpinski gasket. Instead of starting with a full triangle, we start with a full square, which we divide into n2 equal squares. Out of these squares we choose k squares and remove them. In the next iteration, we repeat this procedure by dividing each of the small squares left into n2 smaller squares and removing those k squares that are located at the same positions as in the first iteration. This procedure is repeated again and again. Figure 6 shows the Sierpinski carpet for n D 5 and the specific choice of k D 9. It is clear that the k squares can be chosen in many different ways, and the fractal structures will all look very different. However, since M(1/n L) D 1/(n2 k) M(L) it follows that df D log(n2 k)/ log n, irrespective of the way the k squares are chosen. Similarly to the gasket, we have d` D df and hence dmin D 1. In contrast, the external perimeter (“hull”, see also Fig. 1) of the carpet and its fractal dimension dh depend strongly on the way the squares are chosen. The hull consists of those sites of the cluster, which are adjacent to empty sites and are connected with infinity via empty sites. In our example, see Fig. 6, the hull is a fractal with the fractal dimension dh D log 9/ log 5 Š 1:365. On the other hand, if a Sierpinski gasket is constructed with the k D 9 squares chosen from the center, the external perimeter stays smooth and dh D 1. Although the rules for generating the Sierpinski gasket and carpet are quite similar, the resulting fractal structures belong to two different classes, to finitely ramified and infinitely ramified fractals. A fractal is called finitely ramified if any bounded subset of the fractal can be isolated by cutting a finite number of bonds or sites. The Sierpinski
Fractal Geometry, A Brief Introduction to, Figure 7 The Sierpinski sponge (third iteration). The fractal dimension of the Sierpinski sponge is df D log 20/ log 3
gasket and the Koch curve are finitely ramified, while the Sierpinski carpet is infinitely ramified. For finitely ramified fractals like the Sierpinski gasket many physical properties, such as conductivity and vibrational excitations, can be calculated exactly. These exact solutions help to provide insight onto the anomalous behavior of physical properties on fractals, as was shown in Chap. 3 in [13]. The Sierpinski Sponge The Sierpinski sponge shown in Fig. 7 is constructed by starting from a cube, subdividing it into 3 3 3 D 27 smaller cubes, and taking out the central small cube and its six nearest neighbor cubes. Each of the remaining 20 small cubes is processed in the same way, and the whole procedure is iterated ad infinitum. After each iteration, the volume of the sponge is reduced by a factor of 20/27, while the total surface area increases. In the limit of infinite iterations, the surface area is infinite, while the volume vanishes. Since M(1/3 L) D 1/20 M(L), the fractal dimension is df D log 20/ log 3 Š 2:727. We leave it to the reader to prove that both the fractal dimension dh of the external surface and the chemical dimension d` is the same as the fractal dimension df . Modification of the Sierpinski sponge, in analogy to the modifications of the carpet can lead to fractals, where the fractal dimension of the hull, dh , differs from df . The Dürer Pentagon Five-hundred years ago the artist Albrecht Dürer designed a fractal based on regular pentagons, where in each iteration each pentagon is divided into six smaller pentagons and five isosceles triangles, and the triangles are removed (see Fig. 8). In each triangle, the ratio of the larger side to
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 8 The first iterations of the Dürer pentagon. The fractal dimension of the Dürer pentagon is df D log 6/ log(1 C g) Fractal Geometry, A Brief Introduction to, Figure 10 The first iterations of the triadic Cantor set. The fractal dimension of this Cantor set is df D log 2/ log 3
Fig. 10). We divide a unit interval [0; 1] into three equal intervals and remove the central one. In each following iteration, each of the remaining intervals is treated in this way. In the limit of n D 1 iterations one obtains a set of points. Since M(1/3 L) D 1/2 M(L), we have df D log 2/ log 3 Š 0:631, which is smaller than one. In chaotic systems, strange fractal attractors occur. The simplest strange attractor is the Cantor set. It occurs, for example, when considering the one-dimensional logistic map x tC1 D x t (1 x t ) :
Fractal Geometry, A Brief Introduction to, Figure 9 The first iterations of the David fractal. The fractal dimension of the David fractal is df D log 6/ log 3
the smaller side is the famous proportio divina or golden p ratio, g 1/(2 cos 72ı ) (1 C 5)/2. Hence, in each iteration the sides of the pentagons are reduced by 1 C g. Since M(L/(1 C g)) D 1/6 M(L), the fractal dimension of the Dürer pentagon is df D log 6/ log(1 C g) Š 1:862. The external perimeter of the fractal (see Fig. 1) forms a fractal curve with dh D log 4/ log(1 C g). A nice modification of the Dürer pentagon is a fractal based on regular hexagons, where in each iteration one hexagon is divided into six smaller hexagons, six equilateral triangles, and a David-star in the center, and the triangles and the David-star are removed (see Fig. 9). We leave it as an exercise to the reader to show that df D log 6/ log 3 and dh D log 4/ log 3. The Cantor Set Cantor sets are examples of disconnected fractals (fractal dust). The simplest set is the triadic Cantor set (see
(7)
The index t D 0; 1; 2; : : : represents a discrete time. For 0 4 and x0 between 0 and 1, the trajectories xt are bounded between 0 and 1. The dynamical behavior of xt for t ! 1 depends on the parameter . Below 1 D 3, only one stable fixed-point exists to which xt is attracted. At 1 , this fixed-point becomes unstable and bifurcates into two new stable fixed-points. At large times, the trajectories move alternately between both fixed-points, p and the motion is periodic with period 2. At 2 D 1C 6 Š 3:449 each of the two fixed-points bifurcates into two new stable fix points and the motion becomes periodic with period 4. As is increased, further bifurcation points n occur, with periods of 2n between n and nC1 . For large n, the differences between nC1 and n become smaller and smaller, according to the law nC1 n D (n n1 )/ı, where ı Š 4:6692 is the socalled Feigenbaum constant. The Feigenbaum constant is “universal”, since it applies to all nonlinear “single-hump” maps with a quadratic maximum [17]. At 1 Š 3:569 945 6, an infinite period occurs, where the trajectories xt move in a “chaotic” way between the infinite attractor points. These attractor points define the strange attractor, which forms a Cantor set with a fractal dimension df Š 0:538 [18]. For a further discussion of strange attractors and chaotic dynamics we refer to [3,8,9].
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Fractal Geometry, A Brief Introduction to, Figure 11 Three generations of the Mandelbrot–Given fractal. The fractal dimension of the Mandelbrot–Given fractal is df D log 8/ log 3
The Mandelbrot–Given Fractal This fractal was suggested as a model for percolation clusters and its substructures (see Sect. 3.4 and Chap. 2 in [13]). Figure 11 shows the first three generations of the Mandelbrot–Given fractal [19]. At each generation, each segment of length a is replaced by 8 segments of length a/3. Accordingly, the fractal dimension is df D log 8/ log 3 Š 1:893, which is very close to df D 91/46 Š 1:896 for percolation in two dimensions. It is easy to verify that d` D df , and therefore dmin D 1. The structure contains loops, branches, and dangling ends of all length scales. Imagine applying a voltage difference between two sites at opposite edges of a metallic Mandelbrot–Given fractal: the backbone of the fractal consists of those bonds which carry the electric current. The dangling ends are those parts of the cluster which carry no current and are connected to the backbone by a single bond only. The red bonds (or singly connected bonds) are those bonds that carry the total current; when they are cut the current flow stops. The blobs, finally, are those parts of the backbone that remain after the red bonds have been removed. The backbone of this fractal can be obtained easily by eliminating the dangling ends when generating the fractal (see Fig. 12). It is easy to see that the fractal dimension of the backbone is dB D log 6/ log 3 Š 1:63. The red bonds are all located along the x axis of the figure and form a Cantor set with the fractal dimension dred D log 2/ log 3 Š 0:63. Julia Sets and the Mandelbrot Set A complex version of the logistic map (7) is z tC1 D z2t C c ;
(8)
Fractal Geometry, A Brief Introduction to, Figure 12 The backbone of the Mandelbrot–Given fractal, with the red bonds shown in bold
where both the trajectories zt and the constant c are complex numbers. The question is: if a certain c-value is given, for example c D 1:5652 i1:03225, for which initial values z0 are the trajectories zt bounded? The set of those values forms the filled-in Julia set, and the boundary points of them form the Julia set. To clarify these definitions, consider the simple case c D 0. For jz0 j > 1, zt tends to infinity, while for jz0 j < 1, zt tends to zero. Accordingly, the filled-in Julia set is the set of all points jz0 j 1, the Julia set is the set of all points jz0 j D 1. In general, points on the Julia set form a chaotic motion on the set, while points outside the Julia set move away from the set. Accordingly, the Julia set can be regarded as a “repeller” with respect to Eq. (8). To generate the Julia set, it is thus practical to use the inverted transformation p z t D ˙ z tC1 c ; (9) start with an arbitrarily large value for t C 1, and go back-
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 13 Julia sets for a c D i and b c D 0:11031 i0:67037. After [9]
ward in time. By going backward in time, even points far away from the Julia set are attracted by the Julia set. For obtaining the Julia set for a given value of c, one starts with some arbitrary value for z tC1 , for example, z tC1 D 2. To obtain zt , we use Eq. (9), and determine the sign randomly. This procedure is continued to obtain z t1 , z t2 , etc. By disregarding the initial points, e. g., the first 1000 points, one obtains a good approximation of the Julia set. The Julia sets can be connected (Fig. 13a) or disconnected (Fig. 13b) like the Cantor sets. The self-similarity of the pictures is easy to see. The set of c values that yield connected Julia sets forms the famous Mandelbrot set. It has been shown by Douady and Hubbard [20] that the Mandelbrot set is identical to that set of c values for which zt converges starting from the initial point z0 D 0. For a detailed discussion with beautiful pictures see [10] and Chaps. 13 and 14 in [3]. Random Fractal Models In this section we present several random fractal models that are widely used to mimic fractal systems in nature. We begin with perhaps the simplest fractal model, the random walk. Random Walks Imagine a random walker on a square lattice or a simple cubic lattice. In one unit of time, the random walker advances one step of length a to a randomly chosen nearest neighbor site. Let us assume that the walker is unwinding a wire, which he connects to each site along his way. The length (mass) M of the wire that connects the random walker with his starting point is proportional to the number of steps n (Fig. 14) performed by the walker. Since for a random walk in any d-dimensional space the mean end-to-end distance R is proportional to n1/2 (for a simple derivation see e. g., Chap. 3 in [13], it follows that
Fractal Geometry, A Brief Introduction to, Figure 14 a A normal random walk with loops. b A random walk without loops
M R2 . Thus Eq. (4) implies that the fractal dimension of the structure formed by this wire is df D 2, for all lattices. The resulting structure has loops, since the walker can return to the same site. We expect the chemical dimension d` to be 2 in d D 2 and to decrease with increasing d, since. Loops become less relevant. For d 4 we have d` D 1. If we assume, however, that there is no contact between sections of the wire connected to the same site (Fig. 14b), the structure is by definition linear, i. e., d` D 1 for all d. For more details on random walks and its relation to Brownian motion, see Chap. 5 in [15] and [21]. Self-Avoiding Walks Self-avoiding walks (SAWs) are defined as the subset of all nonintersecting random walk configurations. An example is shown in Fig. 15a. As was found by Flory in 1944 [22], the end-to-end distance of SAWs scales with the number of steps n as R n ;
(10)
with D 3/(d C 2) for d 4 and D 1/2 for d > 4. Since n is proportional to the mass of the chain, it follows from Eq. (4) that df D 1/. Self-avoiding walks serve as models for polymers in solution, see [23]. Subsets of SAWs do not necessarily have the same fractal dimension. Examples are the kinetic growth walk (KGW) [24] and the smart growth walk (SGW) [25], sometimes also called the “true” or “intelligent” self-avoiding walk. In the KGW, a random walker can only step on those sites that have not been visited before. Asymptotically, after many steps n, the KGW has the same fractal dimension as SAWs. In d D 2, however, the asymptotic regime is difficult to reach numerically, since the random walker can be trapped with high probability (see Fig. 15b). A related structure is the hull of a random walk in d D 2. It has been conjectured by Mandelbrot [1] that the fractal dimension of the hull is dh D 4/3, see also [26].
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Fractal Geometry, A Brief Introduction to, Figure 16 a Generation of a DLA cluster. The inner release radius is usually a little larger than the maximum distance of a cluster site from the center, the outer absorbing radius is typically 10 times this distance. b A typical off-lattice DLA cluster of 10,000 particles
Fractal Geometry, A Brief Introduction to, Figure 15 a A typical self-avoiding walk. b A kinetic growth walk after 8 steps. The available sites are marked by crosses. c A smart growth walk after 19 steps. The only available site is marked by “yes”
In the SGW, the random walker avoids traps by stepping only at those sites from which he can reach infinity. The structure formed by the SGW is more compact and characterized by df D 7/4 in d D 2 [25]. Related structures with the same fractal dimension are the hull of percolation clusters (see also Sect. “Percolation”) and diffusion fronts (for a detailed discussion of both systems see also Chaps. 2 and 7 in [13]). Kinetic Aggregation The simplest model of a fractal generated by diffusion of particles is the diffusion-limited aggregation (DLA) model, which was introduced by Witten and Sander in 1981 [27]. In the lattice version of the model, a seed particle is fixed at the origin of a given lattice and a second particle is released from a circle around the origin. This particle performs a random walk on the lattice. When it comes to a nearest neighbor site of the seed, it sticks and a cluster (aggregate) of two particles is formed. Next, a third particle is released from the circle and performs a random walk. When it reaches a neighboring site of the aggregate, it sticks and becomes part of the cluster. This procedure is repeated many times until a cluster of the desired number of sites is generated. For saving computational time it is convenient to eliminate particles that have diffused too far away from the cluster (see Fig. 16).
In the continuum (off-lattice) version of the model, the particles have a certain radius a and are not restricted to diffusing on lattice sites. At each time step, the length ( a) and the direction of the step are chosen randomly. The diffusing particle sticks to the cluster, when its center comes within a distance a of the cluster perimeter. It was found numerically that for off-lattice DLA, df D 1:71 ˙ 0:01 in d D 2 and df D 2:5 ˙ 0:1 in d D 3 [28,29]. These results may be compared with the mean field result df D (d 2 C 1)/(d C 1) [30]. For a renormalization group approach, see [31] and references therein. The chemical dimension d` is found to be equal to df [32]. Diffusion-limited aggregation serves as an archetype for a large number of fractal realizations in nature, including viscous fingering, dielectric breakdown, chemical dissolution, electrodeposition, dendritic and snowflake growth, and the growth of bacterial colonies. For a detailed discussion of the applications of DLA we refer to [5,13], and [29]. Models for the complex structure of DLA have been developed by Mandelbrot [33] and Schwarzer et al. [34]. A somewhat related model for aggregation is the cluster-cluster aggregation (CCA) [35]. In CCA, one starts from a very low concentration of particles diffusing on a lattice. When two particles meet, they form a cluster of two, which can also diffuse. When the cluster meets another particle or another cluster, a larger cluster is formed. In this way, larger and larger aggregates are formed. The structures are less compact than DLA, with df Š 1:4 in d D 2 and df Š 1:8 in d D 3. CCA seems to be a good model for smoke aggregates in air and for gold colloids. For a discussion see Chap. 8 in [13]. Percolation Consider a square lattice, where each site is occupied randomly with probability p or empty with probability 1 p.
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 17 Square lattice of size 20 20. Sites have been randomly occupied with probability p (p D 0:20, 0.59, 0.80). Sites belonging to finite clusters are marked by full circles, while sites on the infinite cluster are marked by open circles
At low concentration p, the occupied sites are either isolated or form small clusters (Fig. 17a). Two occupied sites belong to the same cluster, if they are connected by a path of nearest neighbor occupied sites. When p is increased, the average size of the clusters increases. At a critical concentration pc (also called the percolation threshold) a large cluster appears which connects opposite edges of the lattice (Fig. 17b). This cluster is called the infinite cluster, since its size diverges when the size of the lattice is increased to infinity. When p is increased further, the density of the infinite cluster increases, since more and more sites become part of the infinite cluster, and the average size of the finite clusters decreases (Fig. 17c). The percolation transition is characterized by the geometrical properties of the clusters near pc . The probability P1 that a site belongs to the infinite cluster is zero below pc and increases above pc as P1 (p pc )ˇ :
(11)
The linear size of the finite clusters, below and above pc , is characterized by the correlation length . The correlation length is defined as the mean distance between two sites on the same finite cluster and represents the characteristic length scale in percolation. When p approaches pc , increases as jp pc j ;
(12)
with the same exponent below and above the threshold. While pc depends explicitly on the type of the lattice (e. g., pc Š 0:593 for the square lattice and 1/2 for the triangular lattice), the critical exponents ˇ and are universal and depend only on the dimension d of the lattice, but not on the type of the lattice. Near pc , on length scales smaller than , both the infinite cluster and the finite clusters are self-similar. Above pc , on length scales larger than , the infinite cluster can be regarded as an homogeneous system which is composed of
Fractal Geometry, A Brief Introduction to, Figure 18 A large percolation cluster in d D 3. The colors mark the topological distance from an arbitrary center of the cluster in the middle of the page. Courtesy of M. Meyer
many unit cells of size . Mathematically, this can be summarized as ( r df ; r ; M(r) d (13) r ; r: The fractal dimension df can be related to ˇ and : df D d
ˇ :
(14)
Since ˇ and are universal exponents, df is also universal. One obtains df D 91/48 in d D 2 and df Š 2:5 in d D 3. The chemical dimension d` is smaller than df , d` Š 1:15 in d D 2 and d` Š 1:33 in d D 3. A large percolation cluster in d D 3 is shown in Fig. 18. Interestingly, a percolation cluster is composed of several fractal sub-structures such as the backbone, dangling ends, blobs, External perimeter, and the red bonds, which are all described by different fractal dimensions. The percolation model has found applications in physics, chemistry, and biology, where occupied and empty sites may represent very different physical, chemical, or biological properties. Examples are the physics of two component systems (the random resistor, magnetic or superconducting networks), the polymerization process in chemistry, and the spreading of epidemics and forest fires. For reviews with a comprehensive list of references, see Chaps. 2 and 3 of [13] and [36,37,38].
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How to Measure the Fractal Dimension One of the most important “practical” problems is to determine the fractal dimension df of either a computer generated fractal or a digitized fractal picture. Here we sketch the two most useful methods: the “sandbox” method and the “box counting” method. The Sandbox Method To determine df , we first choose one site (or one pixel) of the fractal as the origin for n circles of radii R1 < R2 < < R n , where Rn is smaller than the radius R of the fractal, and count the number of points (pixels) M1 (R i ) within each circle i. (Sometimes, it is more convenient to choose n squares of side length L1 . . . Ln instead of the circles.) We repeat this procedure by choosing randomly many other (altogether m) pixels as origins for the n circles and determine the corresponding number of points M j (R i ), j D 2; 3; : : : ; m within each circle (see Fig. 19a). We obtain the mean number of points M(R i ) within each P circle by averaging, M(R i ) D 1/m m jD1 M j (R i ), and plot M(R i ) versus Ri in a double logarithmic plot. The slope of the curve, for large values of Ri , determines the fractal dimension. In order to avoid boundary effects, the radii must be smaller than the radius of the fractal, and the centers of the circles must be chosen well inside the fractal, so that the largest circles will be well within the fractal. In order to obtain good statistics, one has either to take a very large fractal cluster with many centers of circles or many realizations of the same fractal. The Box Counting Method We draw a grid on the fractal that consists of N12 squares, and determine the number of squares S(N1 ) that are
needed to cover the fractal (see Fig. 19b). Next we choose finer and finer grids with N12 < N22 < N32 < < 2 squares and calculate the corresponding numbers of Nm squares S(N1 ) . . . S(N m ) needed to cover the fractal. Since S(N) scales as S(N) N d f ;
we obtain the fractal dimension by plotting S(N) versus 1/N in a double logarithmic plot. The asymptotic slope, for large N, gives df . Of course, the finest grid size must be larger than the pixel size, so that many pixels can fall into the smallest square. To improve statistics, one should average S(N) over many realizations of the fractal. For applying this method to identify self-similarity in real networks, see Song et al. [39]. Self-Affine Fractals The fractal structures we have discussed in the previous sections are self-similar: if we cut a small piece out of a fractal and magnify it isotropically to the size of the original, both the original and the magnification look the same. By magnifying isotropically, we have rescaled the x, y, and z axis by the same factor. There exist, however, systems that are invariant only under anisotropic magnifications. These systems are called self-affine [1]. A simple model for a self-affine fractal is shown in Fig. 20. The structure is invariant under the anisotropic magnification x ! 4x, y ! 2y. If we cut a small piece out of the original picture (in the limit of n ! 1 iterations), and rescale the x axis by a factor of four and the y axis by a factor of two, we will obtain exactly the original structure. In other words, if we describe the form of the curve in Fig. 20 by the function F(x), this function satisfies the equation F(4x) D 2F(x) D 41/2 F(x). In general, if a self-affine curve is scale invariant under the transformation x ! bx, y ! ay, we have F(bx) D aF(x) b H F(x) ;
Fractal Geometry, A Brief Introduction to, Figure 19 Illustrations for determining the fractal dimension: a the sandbox method, b the box counting method
(15)
(16)
where the exponent H D log a/ log b is called the Hurst exponent [1]. The solution of the functional equation (16) is simply F(x) D Ax H . In the example of Fig. 20, H D 1/2. Next we consider random self-affine structures, which are used as models for random surfaces. The simplest structure is generated by a one-dimensional random walk, where the abscissa is the time axis and the ordinate is the P displacement Z(t) D tiD1 e i of the walker from its starting point. Here, e i D ˙1 is the unit step made by the random walker at time t. Since different steps of the random walker are uncorrelated, he i e j i D ı i j , it follows that the
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 20 A simple deterministic model of a self-affine fractal
root mean square displacement F(t) hZ 2 (t)i1/2 D t 1/2 , and the Hurst exponent of the structure is H D 1/2. Next we assume that different steps i and j are correlated in such a way that he i e j i D bji jj , 1 > 0. To see how the Hurst exponent depends on , we have to Pt evaluate again hZ 2 (t)i D i; j he i e j i. For calculating the double sum it is convenient to introduce the Fourier transP form of ei , e! D (1/˝)1/2 ˝ l D1 e l exp(i! l), where ˝ is the number of sites in the system. It is easy to verify that hZ 2 (t)i can be expressed in terms of the power spectrum he! e! i [40]: hZ 2 (t)i D
1 X he! e! ij f (!; t)j2 ; ˝ !
(17a)
where f (!; t)
ei!(tC1) 1 : ei! 1
Since the power spectrum scales as he! e! i ! (1 ) ;
(17b)
the integration of (17a) yields, for large t, hZ 2 (t)i t 2 :
(17c)
Therefore, the Hurst exponent is H D (2 )/2. According to Eq. (17c), for 0 < < 1, hZ 2 (t)i increases faster in time than the uncorrelated random walk. The long-range correlated random walks were called fractional Brownian motion by Mandelbrot [1].
There exist several methods to generate correlated random surfaces. We shall describe the successive random additions method [41], which iteratively generates the selfaffine function Z(x) in the unit interval 0 x 1. An alternative method that is detailed in the chapter of Jan Kantelhardt is the Fourier-filtering technique and its variants. In the n D 0 iteration, we start at the edges x D 0 and x D 1 of the unit interval and choose the values of Z(0) and Z(1) from a distribution with zero mean and variance 02 D 1 (see Fig. 21). In the n D 1 iteration, we choose the midpoint x D 1/2 and determine Z(1/2) by linear interpolation, i. e., Z(1/2) D (Z(0) C Z(1))/2, and add to all so-far calculated Z values (Z(0), Z(1/2), and Z(1)) random displacements from the same distribution as before, but with a variance 1 D (1/2)H (see Fig. 21). In the n D 2 iteration we again first choose the midpoints (x D 1/4 and x D 3/4), determine their Z values by linear interpolation, and add to all so-far calculated Z values random displacements from the same distribution as before, but with a variance 2 D (1/2)2H . In general, in the nth iteration, one first interpolates the Z values of the midpoints and then adds random displacements to all existing Z values, with variance n D (1/2)nH . The procedure is repeated until the required resolution of the surface is obtained. Figure 22 shows the graphs of three random surfaces generated this way, with H D 0:2, H D 0:5, and H D 0:8. The generalization of the successive random addition method to two dimensions is straightforward (see Fig. 21). We consider a function Z(x; y) on the unit square 0
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Fractal Geometry, A Brief Introduction to, Figure 21 Illustration of the successive random addition method in d D 1 and d D 2. The circles mark those points that have been considered already in the earlier iterations, the crosses mark the new midpoints added at the present iteration. At each iteration n, first the Z values of the midpoints are determined by linear interpolation from the neighboring points, and then random displacements of variance n are added to all Z values
x; y 1. In the n D 0 iteration, we start with the four corners (x; y) D (0; 0); (1; 0); (1; 1); (0; 1) of the unit square and choose their Z values from a distribution with zero mean and variance 02 D 1 (see Fig. 21). In the n D 1 iteration, we choose the midpoint at (x; y) D (1/2; 1/2) and determine Z(1/2; 1/2) by linear interpolation, i. e., Z(1/2; 1/2) D (Z(0; 0)CZ(0; 1)CZ(1; 1)CZ(1; 0))/4. Then we add to all so far calculated Z-values (Z(0; 0), Z(0; 1), Z(1; 0), Z(1; 1) and Z(1/2; 1/2)) random displacements from the same p distribution as before, but with a variance 1 D (1/ 2)H (see Fig. 21). In the n D 2 iteration we again choose the midpoints of the five sites (0; 1/2), (1/2; 0), (1/2; 1) and (1; 1/2), determine their Z value by linear interpolation, and add to all so far calculated Z values random displacements from the same p distribution as before, but with a variance 2 D (1/ 2)2H . This procedure is repeated again and again, until the required resolution of the surface is obtained. At the end of this section we like to note that selfsimilar or self-affine fractal structures with features similar to those fractal models discussed above can be found in nature on all, astronomic as well as microscopic, length scales. Examples include clusters of galaxies (the fractal dimension of the mass distribution is about 1.2 [42]), the crater landscape of the moon, the distribution of earthquakes (see Chap. 2 in [15]), and the structure of coastlines, rivers, mountains, and clouds. Fractal cracks (see, for example, Chap. 5 in [13]) occur on length scales ranging from 103 km (like the San Andreas fault) to micrometers (like fractures in solid materials) [44]. Many naturally growing plants show fractal structures, examples range from trees and the roots of trees to cauliflower and broccoli. The patterns of blood vessels in the human body, the kidney, the lung, and some types of nerve cells have fractal features (see Chap. 3 in [15]). In materials sciences, fractals appear in polymers, gels, ionic glasses, aggregates, electro-deposition, rough interfaces and surfaces (see [13] and Chaps. 4 and 6 in [15]), as well as in fine particle systems [43]. In all these structures there is no characteristic length scale in the system besides the physical upper and lower cut-offs. The occurrence of self-similar or self-affine fractals is not limited to structures in real space as we will discuss in the next section. Long-Term Correlated Records
Fractal Geometry, A Brief Introduction to, Figure 22 Correlated random walks with H D 0:2, 0.5, and 0.8, generated by the successive random addition method in d D 1
Long-range dependencies as described in the previous section do not only occur in surfaces. Of great interest is longterm memory in climate, physiology and financial markets, the examples range from river floods [45,46,47,48,
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 23 Comparison of an uncorrelated and a long-term correlated record with D 0:4. The full line is the moving average over 30 data points
49], temperatures [50,51,52,53,54], and wind fields [55] to market volatilities [56], heart-beat intervals [57,58] and internet traffic [59]. Consider a record xi of discrete numbers, where the index i runs from 1 to N. xi may be daily or annual temperatures, daily or annual river flows, or any other set of data consisting of N successive data points. We are interested in the fluctuations of the data around their (sometimes seasonal) mean value. Without loss of generality, we assume that the mean of the data is zero and the variance equal to one. In analogy to the previous section, we call the data long-term correlated, when the corresponding autocorrelation function C x (s) decays by a power law, C x (s) D hx i x iCs i
1 N s
Ns X
x1 x iCs s ;
(18)
iD1
where denotes the correlation exponent, 0 < < 1. Such correlations are Rnamed ‘long-term’ since the mean 1 correlation time T D 0 C x (s) ds diverges in the limit of infinitely long series where N ! 1. If the xi are uncorrelated, C x (s) D 0 for s > 0. More generally, if correlations exist up to a certain correlation time sx , then C(s) > 0 for s < s x and C(s) D 0 for s > s x . Figure 23 shows parts of an uncorrelated (left) and a long-term correlated (right) record, with D 0:4; both series have been generated by the computer. The red line is the moving average over 30 data points. For the uncorrelated data, the moving average is close to zero, while for the long-term correlated data set, the moving average can have large deviations from the mean, forming some kind of mountain valley structure. This structure is a consequence of the power-law persistence. The mountains and valleys in Fig. 23b look as if they had been generated by external trends, and one might be inclined to draw a trend-
line and to extrapolate the line into the near future for some kind of prognosis. But since the data are trend-free, only a short-term prognosis utilizing the persistence can be made, and not a longer-term prognosis, which often is the aim of such a regression analysis. Alternatively, in analogy to what we described above for self-affine surfaces, one can divide the data set in K s equidistant windows of length s and determine in each window the squared sum !2 s X 2 F (s) D xi (19a) iD1
and detect how the average of this quantity over all winPK s F2 (s), scales with the window dows, F 2 (s) D 1/K s D1 size s. For long-term correlated data one can show that F 2 (s) scales as hZ 2 (t)i in the previous section, i. e. F 2 (s) s 2˛ ;
(19b)
where ˛ D 1 /2. This relation represents an alternative way to determine the correlation exponent . Since trends resemble long-term correlations and vice versa, there is a general problem to distinguish between trends and long-term persistence. In recent years, several methods have been developed, mostly based on the hierarchical detrended fluctuation analysis (DFAn) where longterm correlations in the presence of smooth polynomial trends of order n 1 can be detected [57,58,60] (see also Fractal and Multifractal Time Series). In DFAn, one considers the cumulated sum (“profile”) of the xi and divides the N data points of the profile into equidistant windows of fixed length s. Then one determines, in each window, the best fit of the profile by an nth order polynomial and determines the variance around the fit. Finally, one av2 and erages these variances to obtain the mean variance F(n)
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Fractal Geometry, A Brief Introduction to, Figure 24 DFAn analysis of six temperature records, one precipitation record and two run-off records. The black curves are the DFA0 results, while the upper red curves refer to DFA1 and the lower red curves to DFA2. The blue numbers denote the asymptotic slopes of the curves
the corresponding mean standard deviation (mean fluctuation) F(n) (s). One can show that for long-term correlated trend-free data F(n) (s) scales with the window size s as F(s) in Eq. (19b), i. e., F(n) (s) s ˛ , with ˛ D 1 /2, irrespective of the order of the polynomial n. For short-term correlated records (including the case 1), the exponent is 1/2 for s above sx . It is easy to verify that trends of order k 1 in the original data are eliminated in F(k) (s) but contribute to F(k1) ; F(k2) etc., and this allows one to determine the correlation exponent in the presence of trends. For example, in the case of a linear trend, DFA0 and DFA1 (where F(0) (s) and F(1) (s) are determined) are affected by the trend and will exaggerate the asymptotic exponents ˛, while DFA2, DFA3 etc. (where F(2) (s) and F(3) (s) etc. is determined) are not affected by the trend and will show, in a double logarithmic plot, the same value of ˛, which then gives immediately the correlation exponent . When is known this way, one can try to detect the trend, but there is no unique treatment available. In recent papers [61,62,63], different kinds of analysis have been elaborated and applied to estimate trends in the temperature records of the Northern Hemisphere and Siberian locations.
Climate Records Figure 24 shows representative results of the DFAn analysis, for temperature, precipitation and run-off data. For continental temperatures, the exponent ˛ is around 0.65, while for island stations and sea surface temperatures the exponent is considerably higher. There is no crossover towards uncorrelated behavior at larger time scales. For the precipitation data, the exponent is close to 0.55, not being significantly larger than for uncorrelated records. Figure 25 shows a summary of the exponent ˛ for a large number of climate records. It is interesting to note that while the distribution of ˛-values is quite broad for run-off, sea-surface temperature, and precipitation records, the distribution is quite narrow, located around ˛ D 0:65 for continental atmospheric temperature records. For the island records, the exponent is larger. The quite universal exponent ˛ D 0:65 for continental stations can be used as an efficient test bed for climate models [62, 64,65]. The time window accessible by DFAn is typically 1/4 of the length of the record. For instrumental records, the time window is thus restricted to about 50 years. For ex-
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 25 Distribution of fluctuation exponents ˛ for several kinds of climate records (from [53,66,67])
tending this limit, one has to take reconstructed records or model data, which range up to 2000y. Both have, of course, large uncertainties, but it is remarkable that exactly the same kind of long-term correlations can be found in these data, thus extending the time scale where long-term memory exists to at least 500y [61,62]. Clustering of Extreme Events Next we consider the consequences of long-term memory on the occurrence of rare events. Understanding (and predicting) the occurrence of extreme events is one of the major challenges in science (see, e. g., [68]). An important quantity here is the time interval between successive extreme events (see Fig. 26), and by understanding the statistics of these return intervals one aims to better understand the occurrence of extreme events. Since extreme events are, by definition, very rare and the statistics of their return intervals poor, one usually studies also the return intervals between less extreme events, where the data are above some threshold q and where the statistics is better, and hopes to find some general “scaling” relations between the return intervals at low and high thresholds, which then allows one to extrapolate the results to very large, extreme thresholds (see Fig. 26).
Fractal Geometry, A Brief Introduction to, Figure 26 Illustration of the return intervals for three equidistant threshold values q1 ; q2 ; q3 for the water levels of the Nile at Roda (near Cairo, Egypt). One return interval for each threshold (quantile) q is indicated by arrows
For uncorrelated data, the return intervals are independent of each other and their probability density function (pdf) is a simple exponential, Pq (r) D (1/R q ) exp(r/R q ). In this case, all relevant quantities can be derived from the knowledge of the mean return interval Rq . Since the return intervals are uncorrelated, a sequential ordering cannot occur. There are many cases, however, where some kind of ordering has been observed where the
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hazardous events cluster, for example in the floods in Central Europe during the middle ages or in the historic water levels of the Nile river which are shown in Fig. 26 for 663y. Even by eye one can realize that the events are not distributed randomly but are arranged in clusters. A similar clustering was observed for extreme floods, winter storms, and avalanches in Central Europe (see, e. g., Figs. 4.4, 4.7, 4.10, and 4.13 in [69], Fig. 66 in [70], and Fig. 2 in [71]). The reason for this clustering is the long-term memory. Figure 27 shows Pq (r) for long-term correlated records with ˛ D 0:4 (corresponding to D 0:8), for three values of the mean return interval Rq (which is easily obtained from the threshold q and independent of the correlations). The pdf is plotted in a scaled way, i. e., R q Pq (r) as a function of r/R q . The figure shows that all three curves collapse. Accordingly, when we know the functional form of the pdf for one value of Rq , we can easily deduce its functional form also for very large Rq values which due to its poor statistics cannot be obtained directly from the data. This scaling is a very important property, since it allows one to make predictions also for rare events which otherwise are not accessible with meaningful statistics. When the data are shuffled, the long-term correlations are destroyed and the pdf becomes a simple exponential. The functional form of the pdf is a quite natural extension of the uncorrelated case. The figure suggests that ln Pq (r) (r/R q )
(20)
i. e. simple stretched exponential behavior [72,73]. For
approaching 1, the long-term correlations tend to vanish
Fractal Geometry, A Brief Introduction to, Figure 27 Probability density function of the return intervals in long-term correlated data, for three different return periods Rq , plotted in a scaled way. The full line is a stretched exponential, with exponent D 0:4 (after [73])
and we obtain the simple exponential behavior characteristic for uncorrelated processes. For r well below Rq , however, there are deviations from the pure stretched exponential behavior. Closer inspection of the data shows that for r/R q the decay of the pdf is characterized by a power law, with the exponent 1. This overall behavior does not depend crucially on the way the original data are distributed. In the cases shown here, the data had a Gaussian distribution, but similar results have been obtained also for exponential, power-law and log-normal distributions [74]. Indeed, the characteristic stretched exponential behavior of the pdf can also be seen in long historic and reconstructed records [73]. The form of the pdf indicates that return intervals both well below and well above their average value are considerably more frequent for long-term correlated data than for uncorrelated data. The distribution does not quantify, however, if the return intervals themselves are arranged in a correlated or in an uncorrelated fashion, and if clustering of rare events may be induced by long-term correlations. To study this question, [73] and [74] have evaluated the autocorrelation function of the return intervals in synthetic long-term correlated records. They found that also the return intervals are arranged in a long-term correlated fashion, with the same exponent as the original data. Accordingly, a large return interval is more likely to be followed by a large one than by a short one, and a small return interval is more likely to be followed by a small one than by a large one, and this leads to clustering of events above some threshold q, including extreme events. As a consequence of the long-term memory, the probability of finding a certain return interval depends on the preceding interval. This effect can be easily seen in synthetic data sets generated numerically, but not so well in climate records where the statistics is comparatively poor. To improve the statistics, we now only distinguish between two kinds of return intervals, “small” ones (below the median) and “large” ones (above the median), and determine the mean RC q and R q of those return intervals following a large (C) or a small () return interval. Due to scal ing, RC q /R q and R q /R q are independent of q. Figure 28 shows both quantities (calculated numerically for longterm correlated Gaussian data) as a function of the correlation exponent . The lower dashed line is R q /R q , the /R . In the limit of vanishing longupper dashed line is RC q q term memory, for D 1, both quantities coincide, as ex pected. Figure 28 also shows RC q /R q and R q /R q for five climate records with different values of . One can see that the data agree very well, within the error bars, with the theoretical curves.
Fractal Geometry, A Brief Introduction to
Fractal Geometry, A Brief Introduction to, Figure 28 Mean of the (conditional) return intervals that either follow a return interval below the median (lower dashed line) or above the median (upper dashed line), as a function of the correlation exponent , for five long reconstructed and natural climate records. The theoretical curves are compared with the corresponding values of the climate records (from right to left): The reconstructed run-offs of the Sacramento River, the reconstructed temperatures of Baffin Island, the reconstructed precipitation record of New Mexico, the historic water levels of the Nile and one of the reconstructed temperature records of the Northern hemisphere (Mann record) (after [73])
Long-Term Correlations in Financial Markets and Seismic Activity The characteristic behavior of the return intervals, i. e. long-term correlations and stretched exponential decay, can also be observed in financial markets and seismic activity. It is well known (see, e. g. [56]) that the volatility of stocks and exchange rates is long-term correlated. Figure 29 shows that, as expected from the foregoing, also the return intervals between daily volatilities are long-term
correlated, with roughly the same exponent as the original data [75]. It has further been shown [75] that also the pdfs show the characteristic behavior predicted above. A further example where long-term correlations seem to play an important role, are earthquakes in certain bounded areas (e. g. California) in time regimes where the seismic activity is (quasi) stationary. It has been discovered recently by [76] that the magnitudes of earthquakes in Northern and Southern California, from 1995 until 1998, are long-term correlated with an exponent around
D 0:4, and that also the return intervals between the earthquakes are long-term correlated with the same exponent. For the given exponential distribution of the earthquake magnitudes (following the Gutenberg–Richter law), the long-term correlations lead to a characteristic dependence on the scaled variable r/R q which can explain, without any fit parameter, the previous results on the pdf of the return intervals by [77]. Multifractal Records Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. In some cases, there exist crossover (time-) scales sx separating regimes with different scaling exponents, e. g. long-term correlations on small scales below sx and another type of correlations or uncorrelated behavior on larger scales above sx . In other cases, the scaling behavior is more complicated, and different scaling exponents are required for different parts of the series. In even more complicated cases, such different scaling behavior can be observed for many interwoven fractal subsets of the time series. In this case a multitude of scaling exponents is required for a full description of the scaling behavior, and
Fractal Geometry, A Brief Introduction to, Figure 29 Long-term correlation exponent for the daily volatility (left) and the corresponding return intervals (right). The studied commodities are (from left to right), the S&P 500 index, six stocks (IBM, DuPont, AT&T, Kodak, General Electric, Coca-Cola) and seven currency exchange rates (US$ vs. Japanese Yen, British Pound vs. Swiss Franc, US$ vs. Swedish Krona, Danish Krone vs. Australian $, Danish Krone vs. Norwegian Krone, US$ vs. Canadian $ and US$ vs. South African $). Courtesy of Lev Muchnik
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a multifractal analysis must be applied (see, e. g., [78,79] and literature therein). To see this, it is meaningful to extend Eqs. (19a) and (19b) by considering the more general average F q (s) D
1 Ks
Ks X D1
q/2 F2 (s)
(21)
with q between 1 and C1. For q 1 the small fluctuations will dominate the sum, while for q 1 the large fluctuations are dominant. It is reasonable to assume that the q-dependent average scales with s as F q (s) s qˇ (q) ;
(22)
with ˇ(2) D ˛. Equation (22) generalizes Eq. (19b). If ˇ(q) is independent of q, we have (F q (s))1/q s ˛ , independent of q, and both large and small fluctuations scale the same. In this case, a single exponent is sufficient to characterize the record, which then is referred to as monofractal. If ˇ(q) is not identical to ˛, we have a multifractal [1,4,12]. In this case, the dependence of ˇ(q) on q characterizes the record. Instead of ˇ(q) one considers frequently the spectrum f (!) that one obtains by Legendre transform of qˇ(q), ! D d(qˇ(q))/dq, f (!) D q! qˇ(q) C 1. In the monofractal limit we have f (!) D 1. For generating multifractal data sets, one considers mostly multiplicative random cascade processes, described, e. g., in [3,4]. In this process, the data set is obtained in an iterative way, where the length of the record doubles in each iteration. It is possible to generate random cascades with vanishing autocorrelation function (Cx (s) D 0 for s 1) or with algebraically decaying autocorrelation functions (C x (s) s ). Here we focus on a multiplicative random cascade with vanishing autocorrelation function, which is particularly interesting since it can be used as a model for the arithmetic returns (Pi Pi1 )/Pi of daily stock closing prices Pi [80]. In the zeroth iteration n D 0, the data set (x i ) consists of one value, x1(nD0) D 1. In the nth iteration, the data x (n) i consist of 2n values that are obtained from (n) (n1) (n) x2l m2l 1 1 D x l
and (n) x2l D x (n1) m(n) l 2l ;
(23)
where the multipliers m are independent and identically distributed (i.i.d.) random numbers with zero mean and unit variance. The resulting pdf is symmetric with lognormal tails, with vanishing correlation function C x (s) for s 1.
It has been shown that in this case, the pdf of the return intervals decays by a power-law Pq (r)
r Rq
ı(q) ;
(24)
where the exponent ı depends explicitly on Rq and seems to converge to a limiting curve for large data sets. Despite of the vanishing autocorrelation function of the original record, the autocorrelation function of the return intervals decays by a power law with a threshold-dependent exponent [80]. Obviously, these long-term correlations have been induced by the nonlinear correlations in the multifractal data set. Extracting the return interval sequence from a data set is a nonlinear operation, and thus the return intervals are influenced by the nonlinear correlations in the original data set. Accordingly, the return intervals in data sets without linear correlations are sensitive indicators for nonlinear correlations in the data records. The power-law dependence of Pq (r) can be used for an improved risk estimation. Both power-law dependencies can be observed in economic and physiological records that are known to be multifractal [81]. Acknowledgments We like to thank all our coworkers in this field, in particular Eva Koscielny-Bunde, Mikhail Bogachev, Jan Kantelhardt, Jan Eichner, Diego Rybski, Sabine Lennartz, Lev Muchnik, Kazuko Yamasaki, John Schellnhuber and Hans von Storch. Bibliography 1. Mandelbrot BB (1977) Fractals: Form, chance and dimension. Freeman, San Francisco; Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco 2. Jones H (1991) Part 1: 7 chapters on fractal geometry including applications to growth, image synthesis, and neutral net. In: Crilly T, Earschaw RA, Jones H (eds) Fractals and chaos. Springer, New York 3. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New York 4. Feder J (1988) Fractals. Plenum, New York 5. Vicsek T (1989) Fractal growth phenomena. World Scientific, Singapore 6. Avnir D (1992) The fractal approach to heterogeneous chemistry. Wiley, New York 7. Barnsley M (1988) Fractals everywhere. Academic Press, San Diego 8. Takayasu H (1990) Fractals in the physical sciences. Manchester University Press, Manchester 9. Schuster HG (1984) Deterministic chaos – An introduction. Physik Verlag, Weinheim 10. Peitgen H-O, Richter PH (1986) The beauty of fractals. Springer, Heidelberg
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Fractal Growth Processes
Fractal Growth Processes LEONARD M. SANDER Physics Department and Michigan Center for Theoretical Physics, The University of Michigan, Ann Arbor, USA Article Outline Glossary Definition of the Subject Introduction Fractals and Multifractals Aggregation Models Conformal Mapping Harmonic Measure Scaling Theories Future Directions Bibliography Glossary Fractal A fractal is a geometric object which is self-similar and characterized by an effective dimension which is not an integer. Multifractal A multifractal measure is a non-negative real function) defined on a support (a geometric region) which has a spectrum of scaling exponents. Diffusion-limited aggregation Diffusion-limited aggregation (DLA) is a discrete model for the irreversible growth of a cluster. The rules of the model involve a sequence of random walkers that are incorporated into a growing aggregate when they wander into contact with one of the previously aggregated walkers. Dielectric breakdown model The dielectric breakdown model (DBM) is a generalization of DLA where the probability to grow is proportional to a power of the diffusive flux onto the aggregate. If the power is unity, the model is equivalent to DLA: in this version it is called Laplacian growth. Harmonic measure If a geometric object is thought of as an isolated grounded conductor of fixed charge, the distribution of electric field on its surface is the harmonic measure. The harmonic measure of a DLA cluster is the distribution of growth probability on the surface. Definition of the Subject Fractal growth processes are a class of phenomena which produce self-similar, disordered objects in the course of development far from equilibrium. The most famous of these processes involve growth which can be modeled on
the large scale by the diffusion-limited aggregation algorithm of Witten and Sander [1]. DLA is a paradigm for pattern formation modeling which has been very influential. The algorithm describes growth limited by diffusion: many natural processes fall in this category, and the salient characteristics of clusters produced by the DLA algorithm are observed in a large number of systems such as electrodeposition clusters, viscous fingering patterns, colonies of bacteria, dielectric breakdown patterns, and patterns of blood vessels. At the same time the DLA algorithm poses a very rich problem in mathematical physics. A full “solution” of the DLA problem, in the sense of a scaling theory that can predict the important features of computer simulations is still lacking. However, the problem shows many features that resemble thermal continuous phase transitions, and a number of partially successful approaches have been given. There are deep connections between DLA in two dimensions and theories such as Loewner evolution that use conformal maps. Introduction In the 1970s Benoit Mandelbrot [2] developed the idea of fractal geometry to unify a number of earlier studies of irregular shapes and natural processes. Mandelbrot focused on a particular set of such objects and shapes, those that are self-similar, i. e., where a part of the object is identical (either in shape, or for an ensemble of shapes, in distribution) to a larger piece. He dubbed these fractals. He noted the surprising ubiquity of self-similar shapes in nature. Of particular interest to Mandelbrot and his collaborators were incipient percolation clusters [3]. These are the shapes generated when, for example, a lattice is diluted by cutting bonds at random until a cluster of connected bonds just reaches across a large lattice. This model has obvious applications in physical processes such as transport in random media. The model has a non-trivial mapping to a statistical model [4] and can be treated by the methods of equilibrium statistical physics. It is likely that percolation processes account for quite a few observations of fractals in nature. In 1981 Tom Witten and the present author observed that a completely different type of process surprisingly appears to make fractals [1,5]. These are unstable, irreversible, growth processes which we called diffusion-limited aggregation (DLA). DLA is a kinetic process which is not related to equilibrium statistical physics, but rather defined by growth rules. The rules idealize growth limited by diffusion: in the model there are random walking “par-
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ticles” which attach irreversibly to a single cluster made up of previously aggregated particles. As we will see, quite a few natural processes can be described by DLA rules, and DLA-like clusters are reasonably common in nature. The Witten–Sander paper and subsequent developments unleashed a large amount of activity – the original work has been cited more than 2700 times as of this writing. The literature in this area up to 1998 was reviewed in a very comprehensive manner by T. Vicsek [6] and P. Meakin [7]. See also the chapter by the present author in [8]. For non-technical reviews see [9,10,11]. There are three major areas where self-similar shapes arising from non-equilibrium processes have been studied. The first is the related to the original DLA algorithm. The model may be seen as an idealization of solidification of an amorphous substance. The study of this simple-seeming model is quite unexpectedly rich, and quite difficult to treat theoretically. It will be our focus in this article. We will review the early work, but emphasize developments since [7]. Meakin [12] and Kolb, et al. [13] generalized DLA to consider cluster-cluster or colloid aggregation. In this process particles aggregate when they come into contact, but the clusters so formed are mobile, and themselves aggregate by attaching to each other. This is an idealization of colloid or coagulated aerosol formation. This model also produces fractals but this is not really a surprise: at each stage, clusters of similar size are aggregating, and the result is an approximately hierarchical object. This model is quite important in applications in colloid science. The interested reader should consult [6,14]. A third line of work arose from studies of ballistic aggregation [15] and the Eden model [16]. In the former case particles attach to an aggregate after moving in straight paths, and in the latter, particles are simply added to the surface of an aggregate at any available site, with equal probability. These models give rise to non-fractal clusters with random rough surfaces. The surfaces have scaling properties which are often characterized at the continuum level by a stochastic partial differential equation proposed by Kardar, Parisi, and Zhang [17]. For accounts of this area the reader can consult [18,19,20]. The remainder of this article is organized as follows: we first briefly review fractals and multifractals. Then we give details about DLA and related models, numerical methods, and applications of the models. The major development that has fueled a remarkable revival of interest in this area was the work of Hastings and Levitov [21] who related two-dimensional DLA to a certain class of conformal maps. Developments of this theme is the subject of the subsequent section. Then we discuss the ques-
tion of the distribution of growth probabilities on the cluster surface, and finally turn to scaling theories of DLA. Fractals and Multifractals The kind of object that we deal with in this article is highly irregular; see Fig. 1. We think of the object as being made up of a large number, N, of units, and to be of overall linear dimension R. In growth processes the objects are formed by adding the units according to some dynamics. We will call the units “particles”, and take their size to be a. Such patterns need not be merely random, but can have well-defined scaling properties in the regime a R. The picture is of a geometric object in two dimensions, but the concepts we introduce here are often applied to more abstract spaces. For example, a strange attractor in phase space often has the kind of scaling properties described here. In order to characterize the geometry of such complex objects, we first cover the points in question with a set of n(l) “boxes” of fixed size, l such that a l R. Clearly, for a smooth curve the product ln(l) approaches a limit (the length of the curve) as l ! 0. This is a number of order R. For a planar region with smooth boundaries l 2 n(l) approaches the area, of order R2 . The objects of ordinary geometry in d dimensions have measures given by the limit of l d n(l). For an object with many scales (a fractal), in general none of these relations hold. Rather, the product l D n(l) approaches a limit with D not necessarily an integer; D is called the (similarity) fractal dimension.
Fractal Growth Processes, Figure 1 A partial covering of a pattern with boxes. Smaller boxes reveal smaller scales of the pattern. For a pattern with many scales (like this one) there is a non-trivial scaling between the box size, l and the number of boxes
Fractal Growth Processes
Said another way, we define the fractal dimension by: D
n(l) / (R/l) :
(1)
For many simple examples of mathematical objects with non-integer values of D see [2,6]. For an infinite fractal there are no characteristic lengths. For a finite size object there is a characteristic length, the overall scale, R. This can be taken to be any measure of the size such as the radius of gyration or the extremal radius – all such lengths must be proportional. It is useful to generalize this definition in two ways. First we consider not only a geometric object, but also a measure, that is a non-negative R function defined on the points of the object such that d D 1. For the geometry, we take the measure to be uniform on the points. However, for the case of growing fractals, we could also consider the growth probability at a point. As we will see, this is very non-uniform for DLA. Second, we define a sequence of generalized dimensions. If we are interested in geometry, we denote the mass of the object covered by box i by pi . For an arbitrary measure, we define: Z p i D d ; (2) where the integral is over the box labeled i. Then, following [22,23] we define a partition function for the pi : (q) D
n X
q
pi ;
(3)
points can be found by focusing on any point, and drawing a d-dimensional disk of radius r around it. The number of other points in the disk will scale as r D 2 . For DLA, it is common to simply count the number of points within radius r of the origin, or, alternatively, the dependence of some mean radius, R on the total number of aggregated particles, N, that is N / R D 2 This method is closely related to the Grassberger–Procaccia correlation integral [24]. For a simple fractal all of the Dq are the same, and we use the symbol D. If the generalized dimensions differ, then we have a multifractal. Multifractals were introduced by Mandelbrot [25] in the context of turbulence. In the context of fractal growth processes, the clusters themselves are usually simple fractals. They are the support for a multifractal measure, the growth probability. We need another bit of formalism. It is useful to look at the fractal measure and note how the pi scale with l/R. Suppose we assume a power-law form, p i / (l/R)˛ , where there are different values of ˛ for different parts of the measure. Also, suppose that we make a histogram of the ˛, and look at the parts of the support on which we have the same scaling. It is natural to adopt a form like Eq. (1) for the size of these sets, (l/R) f (˛) . (It is natural to think of f (˛) as the fractal dimension of the set on which the measure has exponent ˛, but this is not quite right because f can be negative due to ensemble averaging.) Then it is not hard to show [23] that ˛; f (˛) are related to q; (q) by a Legendre transform:
iD1
where q is a real number. For an object with well-defined scaling properties we often find that scales with l in the following way as l/R ! 0: (q) / (R/l)(q) (R/l)(q1)D q ; (q) D (q 1)D q :
(4)
Objects with this property are called fractals if all the Dq are the same. Otherwise they are multifractals. Some of the Dq have special significance. The similarity (or box-counting) dimension mentioned above is D0 since in this case D n. If we take the limit q ! 1 we have the information dimension of dynamical systems theory: ˇ X ln p i d ˇˇ D pi : (5) D1 D ˇ dq qD1 ln l i
D2 is called the correlation dimension since p2i measures the probability that two points are close together, i. e. the number of pairs within distance l. This interpretation gives rise to a popular way to measure D2 . If we suppose that the structure is homogeneous, then the number of pairs of
f (˛) D q
d ; dq
˛D
d : dq
(6)
Aggregation Models In this section we review aggregation models of the DLA type, and their relationship to certain continuum processes. We look at practical methods of simulation, and exhibit a few applications to physical and biological processes. DLA The original DLA algorithm [1,5] was quite simple: on a lattice, declare that the point at the origin is the first member of the cluster. Then launch a random walker from a distant point and allow it to wander until it arrives at a neighboring site to the origin, and attach it to the cluster, i. e., freeze its position. Then launch another walker and let it attach to one of the two previous points, and so on. The name, diffusion-limited aggregation, refers to the fact that random walkers, i. e., diffusing particles, control the growth. DLA is a simplified view of a common physical process, growth limited by diffusion.
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Fractal Growth Processes, Figure 2 a A DLA cluster of 50,000,000 particles produced with an offlattice algorithm by E. Somfai. b A three-dimensional off-lattice cluster. Figure courtesy of R. Ball
It became evident that for large clusters the overall shape is dependent on the lattice type [26,27], that is, DLA clusters are deformed by lattice anisotropy. This is an interesting subject [26,28,29,30,31] but most modern work is on DLA clusters without anisotropy, off-lattice clusters. The off-lattice algorithm is similar to the original one: instead of a random walk on a lattice the particle is considered to have radius a. For each step of the walk the particle moves its center from the current position to a random point on its perimeter. If it overlaps a particle of the current cluster, it is backed up until it just touches the cluster, and frozen at that point. Then another walker is launched. A reasonably large cluster grown in two dimensions is shown in Fig. 2, along with a smaller three-dimensional example. Most of the work on DLA has been for two dimensions, but dimensions up to 8 have been considered [32]. Patterns like the one in Fig. 2 have been analyzed for fractal properties. Careful measurements of both D0 and D2 (using the method, mentioned above, of fitting n(r) to r D 2 ) give the same fractal dimension, D=1.71 [32,33,34]; see Fig. 3. There is some disagreement about the next digit. There have been suggestions that DLA is a mass multifractal [35], but most authors now agree that all of the Dq are the same for the mass distribution. For three dimensions D 2:5 [31,32], and for four dimensions D 3:4 [32]. However, some authors [36,37] have claimed that plane DLA is not a self-similar fractal at all, and that the fractal dimension will drift towards 2 as the number of particles increases. More recent work based on conformal maps [38] has cast doubt on this. We will return to this point below. Plane DLA clusters can be grown in restricted geometries, see Fig. 4. The shape of such clusters is an interesting problem in pattern formation [39,40,41,42]. It was long thought that DLA grown in a channel had a different fractal dimension than in radial geometry [43,44,45].
Fractal Growth Processes, Figure 3 The number of particles inside of radius r for a large DLA cluster. This plot gives an estimate of D2
However, more careful work has shown that the dimensions are the same in the two cases [34]. Laplacian Growth and DBM Suppose we ask how the cluster of Fig. 2 gets to be rough. A simple answer is that if we already have a rough shape, it is quite difficult for a random walker to penetrate a narrow channel. (Just how difficult is treated in the Sect. “Harmonic Measure”, below) The channels don’t fill in, and the shape might be preserved. But we can also ask why a a smooth outline, e. g. a disk, does not continue to grow smoothly. In fact, it is easy to test that any initial condition is soon forgotten in the growth [5]. If we start with a smooth shape it roughens immediately because of a growth instability intrinsic to diffusion-limited growth. This instability was discovered by Mullins and Sekerka [46] who used a statement of the problem of diffusion-limited growth in continuum terms: this is known as the Stefan problem (see [47,48]), and is the standard way to idealize crystallization in the diffusion-limited case. The Stefan problem goes as follows: suppose that we have a density (r; t) of particles that diffuse until they reach the growing cluster where they deposit. Then we have: @ D r 2 @t
(7)
Fractal Growth Processes
Fractal Growth Processes, Figure 4 DLA grown in a channel (a) and a wedge (b). The boxes in a are the hierarchical maps of the Sect. “Numerical Methods”. Figures due to E. Somfai
@ / vn @n
(8)
That is, should obey the diffusion equation; is the diffusion constant. The normal growth velocity, vn , of the interface is proportional to the flux onto the surface, @/@n. However the term @/@t is of order v@/@x, where v is a typical growth velocity. Now jr 2 j (v/D)j@/@nj. In the DLA case we launch one particle at a time, so that the velocity goes to zero. Hence Eq. (7) reduces to the Laplace equation, r 2 D 0
(9)
Since the cluster absorbs the particles, we should think of it as having D 0 on the surface. We are to solve an electrostatics problem: the cluster is a grounded conductor with fixed electric flux far away. We grow by an amount proportional to the electric field at each point on the surface. This is called the quasi-static or Laplacian growth regime for deterministic growth. A linear stability analysis of these equations gives the Mullins–Sekerka instability [46]. The qualitative reason for the instability is that near the tips of the cluster the contours of are compressed so that @/@n,
the growth rate, is large. Thus tips grow unstably. We expect DLA to have a growth instability. However, we can turn the argument, and use these observations to give a restatement of the DLA algorithm in continuum terms: we calculate the electric field on the surface of the aggregate, and interpret Eq. (8) as giving the distribution of the growth probability, p, at a point on the surface. We add a particle with this probability distribution, recalculate the potential using Eq. (9) and continue. This is called Laplacian growth. Simulations of Laplacian growth yield the same sort of clusters as the original discrete algorithm. (Some authors use the term Laplacian growth in a different way, to denote deterministic growth according to the Stefan model without surface tension [49].) DLA is thus closely related to one of the classic problems of mathematical physics, dendritic crystal growth in the quasistatic regime. However, it is not quite the same for several reasons: DLA is dominated by noise, whereas the Stefan problem is deterministic. Also, the boundary conditions are different [48]: for a crystal, if we interpret u as T Tm , where T is the temperature, and T m the melting temperature, we have D 0 only on a flat surface. On a curved surface we need / where is the surface stiffness, and is the curvature. The surface tension acts
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Fractal Growth Processes, Figure 5 DBM patterns for 1000 particles on a triangular lattice. a D 0:5 b D 1. This is a small DLA cluster. c D 2. d D 3
as a regularization which prevents the Mullins–Sekerka instability from producing sharp cusps [50]. In DLA the regularization is provided by the finite particle size. And, of course, crystals have anisotropy in the surface tension. There is another classic problem very similar to this, that of viscous fingering in Hele–Shaw flow [48]. This is the description of the displacement of an incompressible viscous fluid by an inviscid one: the “bubble” of inviscid fluid plays the role of the cluster, is the difference of pressures in the viscous fluid and the bubble, and the Laplace equation is the direct result of incompressibility, r v D 0 and D’Arcy’s law, v D kr where k is the permeability, and v the fluid velocity [51]. These considerations led Niemeyer, Pietronero, and Weismann [52] to a clever generalization of Laplacian growth. They were interested in dielectric breakdown with representing a real electrostatic potential. This is known to be a threshold process so that we expect that the breakdown probability is non-linear in @/@n. To generalize they chose: p/
@ @n
;
(10)
where is a non-negative real number. There are some interesting special cases for this model. For D 0 each growth site is equally likely to be used. This is the Eden model [16]. For D 1 we have the Laplacian growth version of DLA, and for larger we get a higher probability to grow at the tips so that the aggregates are more spread out (as in a real dielectric breakdown pattern like atmospheric lightning). There is a remarkable fact which was suggested numerically in [53] and confirmed more recently [54,55]: for > 4 the aggregate prefers growth at tips so much that it becomes essentially linear and non-fractal. DBM patterns for small numbers of particles are shown in Fig. 5.
A large number of variants of the basic model have been proposed such as having the random walkers perform Lévy flights, having a variable particle size, imposing a drift on the random walkers, imposing anisotropy in attachment, etc. For references on these and other variants, see [7]. There are two qualitative results from these studies that are worth mentioning here: In the presence of drift, DLA clusters cross over to (non-fractal) Eden-like clusters [56,57]. And, anisotropy deforms the shape of the clusters on the large scale, as we mentioned above. Numerical Methods In the foregoing we have talked about the algorithm for DLA, but we have not described what is actually done to compute the positions of 50,000,000 particles. This is a daunting computational task, and considerable ingenuity has been devoted to making efficient algorithms. The techniques are quite interesting in their own right, and have been applied to other types of simulation. The first observation to make is that the algorithm is an idealization of growth due to particles wandering into contact with the aggregate from far away. However, it is not necessary to start the particles far away: they arrive at the aggregate with uniform probability on a circle which just circumscribes all of the presently aggregated particles; thus we can start the particles there. As the cluster grows, the starting circle grows. This was already done in the original simulations [1,5]. However, particles can wander in and then out of the starting circle without attaching. These walkers must be followed, and could take a long time to find the cluster again. However, since there is no matter outside, it is possible to speed up the algorithm by noting that if the random walker takes large steps it will still have the same probability distribution, provided that it cannot encounter any matter. For example, if it walks onto the circumference of
Fractal Growth Processes
Fractal Growth Processes, Figure 6 A cluster is surrounded by a starting circle (gray). A random walker outside the circle can safely take steps of length of the black circle. If the walker is at any point, it can walk on a circle equal in size to the distance to the nearest point of the cluster. However, finding that circle appears to require a time-consuming search
a circle that just reaches the starting circle, it will have the correct distribution. The radius of this circle is easy to find. This observation, due to P. Meakin, is the key to what follows: see Fig. 6. We should note that the most efficient way to deal with particles that wander away is to return the particle to the starting circle in one step using the Green’s function for a point charge outside an absorbing circle. A useful algorithm to do this is given in [58]. However, the idea of taking big steps is still a good one because there is a good deal of empty space inside the starting circle. If we could take steps in this empty space (see Fig. 6) we could again speed up the algorithm. The trick is to efficiently find the largest circle centered on the random walker that has no point of the aggregate within it. One could imagine simply doing a spiral search from the current walker position. This technique has actually been used in a completely different setting, that of Kinetic Monte Carlo simulations of surface growth in materials science [59]. For the case of DLA an even more efficient method, the method of hierarchical maps, was devised [26]. It was extended and applied to higher dimensions and off-lattice in [32], and is now the standard method. One version of the idea is illustrated in Fig. 4a for the case of growth in a channel. What is shown is an adaptively refined square mesh. The cluster is covered with a square – a map. The square is subdivided into four smaller squares, and each is further divided, but only if the cluster is closer to it than half of the side of the square. The subdivision continues only up to a predefined maximum depth so that the smallest maps are a few particle diameters. All particles of the cluster will be in one of the smallest maps: a list of the particles is attached to these maps.
As the cluster grows the maps are updated. Each time a particle is added to a previously empty smallest map, the neighboring maps (on all levels) are checked to see whether they satisfy the rule. If not, they are subdivided until they do. When a walker lands, we find the smallest map containing the point. If this map is not at the maximum depth, then the particle is far away from any matter, and half the side of the map is a lower estimate of the walker’s distance from the cluster. If, on the other hand, the particle lands in a map of maximum depth, then it is close to the cluster. The particle lists of the map and of the neighboring smallest size maps can be checked to calculate the exact distance from the cluster. Either way, the particle is enclosed in an empty circle of known radius, and can be brought to the perimeter of the circle in one step. Note that updating the map means that there is only a search for the cluster if we are in the smallest map. Empirically, the computational time, T for an N particle cluster obeys T N 1:1 , and the memory is linear in N. A more recent version of the algorithm for three-dimensional growth uses a covering with balls rather than cubes [31]. For simulations of the Laplacian growth version of the situation is quite different, and simulations are much slower. The reason is that a literal interpretation of the algorithm requires that the Laplace equation be solved each time a particle is added, for example, by the relaxation or boundary-integral method. This is how corresponding simulations for viscous fingering are done [60]. For DBM clusters the growth step requires taking a power of the electric field at the surface. It is possible to make DBM clusters using random walkers [61], but the method is rather subtle. It involves estimating the growth probability at a point by figuring out the “age” of a site, i. e., the time between attachments. This can be converted into an estimate of the growth probability. This algorithm is quite fast. There is another class of methods involving conformal maps. These are slow methods: the practical limit of the size of clusters that can be grown this way is about 50,000. However, this method calculates the growth probability as it goes along, and thus is of great interest. Conformal mapping is discussed in the Sect. “Loewner Evolution and the Hastings–Levitov Scheme”, below. Selected Applications Probably the most important impact of the DLA model has been the realization that diffusion-limited growth naturally gives rise to tenuous, branched objects. Of course, in the context of crystallization, this was understood in the centuries-long quest to understand the shape of
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Fractal Growth Processes, Figure 7 a A scanning tunneling microscope picture of Rh metal islands. The color scale indicates height, and the figure is about 500 Å across. Figure courtesy of R. Clarke. b A bacteria colony on a Petri dish. The figure is a few centimeters across. Figure courtesy of E. Ben-Jacob
snowflakes. However, applications of DLA gave rise to a unification of this with the study of many other physical applications. Broad surveys are given in [6,7,10]. Here we will concentrate on a few illustrative examples. Rapid crystallization in a random environment is the most direct application of DLA scheme. One particularly accessible example is the growth of islands on surfaces in molecular beam epitaxy experiments. In the proper growth conditions it is easy to see patterns dominated by the Mullins–Sekerka instability. These are often referred to with the unlovely phrase “fractal-like”. An example is given in Fig. 7a. There are many examples of this type, e. g. [62]. A related example is the electrodeposition of metal ions from solution. For overpotential situations DLA patterns are often observed [44,63,64,65]. There are also examples of Laplacian growth. A case of this type was discovered by Matsushita and collaborators [66], and exploited in a series of very interesting studies by the group of Ben-Jacob [67] and others. This is the growth of colonies of bacteria on hard agar plates in conditions of low nutrient supply. In this case, bacteria movement is suppressed (by the hard agar) and the limiting step in colony growth is the diffusion of nutrients to the colony. Thus we have an almost literal realization of Laplacian growth, and, indeed, colonies do look like DLA clusters in these conditions; see Fig. 7b. The detailed study of this system has led to very interesting insights into the biophysics of bacteria: these are far from our subject here, and the reader should consult [67]. We remarked above that noisy viscous fingering patterns are similar to DLA clusters, but not the same in detail: in viscous fingering the surface tension boundary con-
dition is different from that of DLA which involves discrete particles. We should note that for viscous fingering patterns in a channel, the asymptotic state is not a disorderly cluster, but rather a single finger that fills half the channel [48] because surface tension smooths the finger. In radial growth this is not true: the Mullins–Sekerka instability gives rise to a disorderly pattern [51] which looks very much like a DLA cluster, see Fig. 8. Even for growth in a channel, if there is sufficient noise, patterns look rather like those in Fig. 4a. In fact, Tang [68] used a version of DLA which allowed him to reduce the noise (see below) and introduce surface tension to do simulations of viscous fingering.
Fractal Growth Processes, Figure 8 A radial viscous fingering pattern. Courtesy of M. Moore and E. Sharon
Fractal Growth Processes
We should ask if this resemblance is more than a mere coincidence. It is the case that the measured fractal dimension of patterns like the one in Fig. 8 is close to 1.71, as for DLA. On this basis there are several claims in the literature that the large-scale structure of the patterns is identical [9,51,69]. Many authors have disagreed and given ingenious arguments about how to verify this. For example, some have claimed that viscous patterns become two-dimensional at large scales [70], or that viscous fingering patterns are most like DBM patterns with 1:2 [71]. Most recently a measurement of the growth probability of a large viscous fingering pattern was found to agree with that of DLA [72]. On this basis, these authors claim that DLA and viscous fingering are in the same universality class. In our opinion, this subject is still open. Conformal Mapping For pattern formation in two dimensions the use of analytic function theory and conformal mapping methods allows a new look at growth processes. The idea is to think of a pattern in the z plane as the image of a simple reference shape, e. g. the unit circle in the w plane, under a time-dependent analytic function, z D F t (w). More precisely, we think of the region outside of the pattern as the image of the region outside of the reference shape. By the Riemann mapping theorem the map exists and is unique if we set a boundary condition such as F(w) ! r0 w as w ! 1. We will also use the inverse map, w D G(z) D F 1 (z). For Laplacian growth processes this idea is particularly interesting since the growth rules depend on solving the Laplace equation outside the cluster. We recall that the Laplace equation is conformally invariant; that is, it retains its form under a conformal transformation. Thus we can solve in the w plane, and transform the solution: if r 2 (w) D 0, and D 0 on the unit circle, and we take to be the real part of a complex potential ˚ in the w plane, then Re ˚ (G(z)) solves the Laplace equation in the z plane with Re ˚ D 0 on the cluster boundary. Thus we can solve for the potential outside the unit circle in w-space (which is easy): ˚ D ln(w). Then if we map to z space we have the solution outside of the cluster: ˚(z) D ln G(z) :
(11)
Note that the constant r0 has the interpretation of a mean radius since ˚ ! ln(z/r0 ) as z ! 1. In fact, r0 is the radius of the disk that gives the same potential as the cluster far away. This has another consequence: the electric field (i. e. the growth probability) is uniform around the unit circle in the w plane. This means that equal intervals on the unit
circle in w space map into equal regions of growth probability in the z plane. The map contains not only the shape of the cluster (the image of jwj D 1) but also information about the growth probability: using Eq. (11) we have: jr˚j D jG 0 j D
1 : jF 0 j
(12)
The problem remains to construct the maps G or F for a given cluster. Somfai, et al. gave a direct method [38]: for a given cluster release a large number, M, of random walkers and record where they hit, say at points zj . We know from the previous paragraph that the images of these points are spaced roughly equally on the unit circle in the w plane. That is, if we start somewhere on the cluster and number the landing positions sequentially around the cluster surface, the images in the w-plane, w j D r j e i j , are given by j D 2 j/M; r j D 1. Thus we have the boundary values of the map and by analytic continuation we can construct the entire map. In fact, if we represent F by a Laurent series: F(w) D r0 w C A0 C
1 X Aj ; wj
(13)
jD1
it is easy to see that the Fourier coefficients of the function F( j ) are the Aj . Unfortunately, for DLA this method only gives a map of the highly probable points on the surface of the cluster. The inner, frozen, regions are very badly sampled by random walkers, so the generated map will not represent these regions. Loewner Evolution and the Hastings–Levitov Scheme A more useful approach to finding F is to impose a dynamics on the map which gives rise to the growth of the cluster. This is very closely related to Loewner evolution, where a curve in the z plane is generated by a map that obeys an equation of motion: 2 dG t (z) D : dt G t (z) (t)
(14)
The map is to the upper half plane from the upper half plane minus the set of singularities, G D . (For an elementary discussion and references see [73].) If (t) is a stochastic process then many interesting statistical objects such as percolation clusters can be generated. For DLA a similar approach was presented by Hastings and Levitov [21,30]. In this case the evolution is discrete, and is designed to represent the addition of particles to a cluster. The process is iterative: suppose we know the
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Fractal Growth Processes, Figure 9 The Hastings–Levitov scheme for fractal growth. At each stage a “bump” of the proper size is added to the unit circle. The composition of the bump map, , with the map at stage N gives the map at stage N C 1
map for N particles, F N . Then we want to add a “bump” corresponding to a new particle. This is accomplished by adding the bump of area in the w-plane on the surface of the unit circle at angle . There are various explicit functions that generate bumps; for the most popular example see [21]. This is a function that depends on a parameter which gives the aspect ratio of the bump. Let us call the resulting transformation f; . If we use F N to transform the unit circle with a bump we get a cluster with an extra bump in the z-plane: that is, we have added a particle to the cluster and F NC1 D F N ı f . The scheme is represented in Fig. 9. There are two matters that need to be dealt with. First, we need to pick . Since the probability to grow is uniform on the circle in w-space, we take to be a random variable uniformly distributed between 0 and 2. Also, we need the bump in z-space to have a fixed area, 0 . That means that the bump in w-space needs to be adjusted because conformal transformations stretch lengths by jF 0 j. A first guess for the area is: NC1 D
0 0 jF N (e i NC1 ) j2
:
(15)
However, this is just a first guess. The stretching of lengths varies over the cluster, and there can be some regions where the approximation of Eq. (15) is not adequate. In this case an iterative procedure is necessary to get the area right [30,34]. The transformation itself is given by: F N D f1 ; 1 ı f2 ; 2 ı ı fN ; N :
(16)
Fractal Growth Processes, Figure 10 A DLA cluster made by iterated conformal maps. Courtesy of E. Somfai
All of the information needed to specify the mapping is contained in the list j ; j ; 1 j N. An example of a cluster made this way is shown in Fig. 10. If we choose uniformly in w-space, we have chosen points with the harmonic measure, jF 0 j1 and we make DLA clusters. To simulate DBM clusters with ¤ 1 we must choose the angles non-uniformly. Hastings [54] has shown how to do this: since a uniform distribution gives a distribution according to jF 0 j1 (see Eq. (12)) we have to pick angles with probability jF 0 j1 in order to grow with probability jrj . This can be done with a Metropolis algorithm with p( NC1 ) D jF N0 (e i )j1 playing the role of a Boltzmann factor. Applications of the Hastings–Levitov Method: Scaling and Crossovers The Hastings–Levitov method is a new numerical algorithm for DLA, but not a very efficient one. Constructing the composed map takes of order N steps so that the algorithm is of order N 2 . One may wonder what has been gained. The essential point is that new understanding arises from considering the objects that are produced in the course of the computation. For example, Davidovitch and collaborators [74] showed that averages of over the cluster boundary are related to the generalized dimensions of the growth probability measure, cf. Eq. (4). Further, the Laurent coefficients of the map, Eq. (13), are meaningful in themselves and have interesting scaling properties. We
Fractal Growth Processes
have seen above that r0 is the radius of a disk with the same capacitance as the cluster. It scales as N 1/D , as we expect. The constant, A0 gives the wandering of the center of the cluster from its original position. Its scaling can be worked out in terms of D and the generalized dimension, D2 . The other coefficients, Aj , are related to the moments of the charge density on the cluster surface. We expect them to scale in the same way as r0 for the following reason: there is an elementary theorem in complex analysis [75] for any univalent map that says, in our notation: r02 D S N C
X
kjA k j2 :
(17)
kD1
Here SN is the area of the cluster. However, this is just the area of N particles, and is linear in N. Therefore, in leading order, the sum must cancel the N 2/D dependence of r02 . The simplest way this can occur is if every term in the sum goes as N 2/D . This seemed to be incorrect according to the data in [74]: they found that for for the first few k the scaling exponents of the hjA k j2 i were smaller than 2/D. However, later work [38] showed that the asymptotic scaling of all of the coefficients is the same. The apparent difference in the exponents is due to a slow crossover. This effect also seems to resolve a long-standing controversy about the asymptotic scaling behavior of DLA clusters, namely the anomalous behavior of the penetration depth of random walkers as the clusters grow. The anomaly was pointed out numerically by Plischke and Racz [76] soon after the DLA algorithm was introduced. These authors showed numerically that the width of the region where deposition occurred, , (a measure of penetration of random walkers into the cluster) seemed to grow more slowly with N than the mean radius of deposition, Rdep . However for a simple fractal all of the characteristic lengths must scale in the same way, R / N 1/D . Mandelbrot and collaborators [36,37,77] used this and other numerical evidence to suggest that DLA would not be a simple fractal for large N. However, Meakin and Sander [78] gave numerical evidence that the anomaly in the scaling of is not due to a different exponent, but is a crossover. The controversy is resolved [38] by noting that the penetration depth can be estimated from the Laurent coefficients of f : 1X jA k j2 : (18) 2 D 2 kD1
It is easy to see [38] that this version of the penetration depth can be interpreted as the rms deviation of the position of the end of a field line from those for a disk of radius r0 . Thus an anomaly in the scaling of the Ak is related
to the effect discovered in [76]. Further, a slow crossover in Ak for small k is reasonable geometrically since this is a slow crossover in low moments of the charge. These low moments might be expected to have intrinsically slow dynamics. In [38,79,80] strong numerical evidence was given for the crossover and for universal asymptotic scaling of the Ak . Indeed, many quantities with the interpretation of a length fit an expression of the form: C 1/D N 1C ; (19) N where the subdominant correction to scaling is characterized by an exponent D 0:33˙0:06. This observation verifies the crossover of . The asymptotic value of the penetration depth is measured to be /Rdep ! 0:12. The same value is found in three dimensions [81]. In fact, the crossover probably accounts for the anomalies reported in [36,37,77]. Further, a careful analysis of numerical data shows that the reported multiscaling [82] (an apparent dependence of D on the distance from the center of the cluster) is also a crossover effect. These insights lead to the view that DLA clusters are simple fractals up to a slow crossover. The origin of the crossover exponent, 1/3, is not understood. In three dimensions a similar crossover exponent was found by different techniques [81] with a value of 0:22 ˙ 0:03. Conformal Mapping for Other Discrete Growth Processes The Hastings–Levitov scheme depends on the conformal invariance of the Laplace equation. Bazant and collaborators [83,84] pointed out that the same techniques can be used for more general growth processes. For example, consider growth of an aggregate in a flowing fluid. Now there are two relevant fields, the particle concentration, c, and the velocity of the fluid, v D r (for potential flow). The current associated with c can be written j D cv rc, where is a mobility and a diffusion coefficient. For steady incompressible flow we have, after rescaling: Per
rc D r 2 c ;
r2
D0:
(20)
Here Pe is the Péclet number, U L/, where U is the value of the velocity far from the aggregate and L its initial size. These equations are conformally invariant, and the problem of flow past a disk in the w plane is easily solved. In [83] growth was accomplished by choosing bumps with probability distribution @c/@n using the method described above. This scheme solves the flow equation past the complex growing aggregate and adds particles according to
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Fractal Growth Processes, Figure 11 An aggregate grown by the advection-diffusion mechanism, Eq. (20) for Pe=10. Courtesy of M. Bazant
their flux at the surface. An example of a pattern of this type is given in Fig. 11. It has long been known that it is possible to treat quasi-static fracture as a growth process similar to DLA [85,86,87]. This is due to the fact that the Lamé equation of elasticity is similar in form to the Laplace equation and the condition for breaking a region of the order of a process zone is related to the stress at the current crack, i. e., boundary values of derivatives of the displacement field. Recent work has exploited this similarity to give yet another application of conformal mapping. For example, for Mode III fracture the quasi-static elastic equation reduces to the Laplace equation, and it is only necessary to replace the growth probability and the boundary conditions in order to use the method of iterated conformal maps [88]. For Mode I and Mode II fracture it is necessary to solve for two analytic functions, but this can also be done [89]. Harmonic Measure The distribution of the boundary values of the normal derivatives of a potential on a electrode of complex shape is called the problem of the harmonic measure. For DLA it is equivalent to the distribution of growth probabilities on the surface of the cluster, or, in other terms, the penetration of random walkers into the cluster. For other complex shapes the problem is still interesting. Its practical significance is that of the study of electrodes [90] or catalytic surfaces. In some cases the harmonic measure has deep relationships to conformal field theory. For the case of the harmonic measure we can interpret the variable ˛ of Eq. (6) as the singularity strength of the electric field near a sharp tip. This is seen as follows: it is well known [91] that near the apex of a wedge-shaped conductor the electric field diverges as r/ˇ 1 where r is the
distance from the tip and the exterior angle of the wedge is ˇ. For example, near a square corner with ˇ D 3/2 there is a r1/3 divergence. Now the quantity pi is the integral over a box of size l. Thus a sequence of boxes centered on the tip will give a power-law l /ˇ D l ˛ . Smaller ˛ means stronger divergence. For fractals that can be treated with the methods of conformal field theory, a good deal is known about the harmonic measure. For example, for a percolation cluster at pc the harmonic measure is completely understood [92]. The Dq are given by a formula: Dq D
1 5 ; Cp 2 24q C 1 C 5
q
1 : 24
(21)
The f (˛) spectrum is easy to compute from this. This formula in good accord with the numerical results of Meakin and collaborators [93] who sampled the measure by firing many random walkers at a cluster. There is an interesting feature of this formula: D0 D 4/3 is the dimension of the support of the measure. This is less than the dimension of a percolation hull, 7/4. There is a large part of the surface of a percolation cluster which is inaccessible to random walkers – the interior surfaces are cut off from the exterior by narrow necks whose widths vanish in the scaling regime. For DLA the harmonic measure is much less well understood. Some results are known. For example, Makarov [94] proved that D1 D 1 under very general circumstances for a two-dimensional harmonic measure. Halsey [95] used Green’s functions for electrostatics to prove the following for DBM models with parameter : ( C 2) () D D ;
(22)
Here D is the fractal dimension of the cluster. For DLA D 1, and (1) D 0. Thus (3) D D D 1:71. We introduced the function f (˛) as the Legendre transform of D(q). There is an argument due to Turkevich and Sher [96] which allows us to see a feature of f (˛) directly by giving an estimate of the singularity associated with the most active tip of the growing cluster. Note that the growth rate of the extremal radius of the cluster is related to the fractal dimension because Rext / N 1/D . Suppose we imagine adding one particle per unit time and recall that ptip / (l/Rext )˛tip . Then: dRext dRext dN ˛ D / Rext tip dt dN dt D D 1 C ˛tip 1 C ˛min :
(23)
Since the singularity at the tip is close to being the most active one, we have an estimate of the minimum value of ˛.
Fractal Growth Processes
Fractal Growth Processes, Figure 12 A sketch of the f(˛) curve. Some authors find a maximum slope at a position like that marked by the dot so that the curve ends there. The curve is extended to the real axis with a straight line. This is referred to as a phase transition
There have been a very large number of numerical investigations of D(q); f (˛) for two dimensional DLA; see [7] for a comprehensive list. They proceed either by launching many random walkers, e. g. [93] or by solving the Laplace equation [97], or, most recently, by using the Hastings–Levitov method [98]. The general features of the results are shown in Fig. 12. There is fairly good agreement about the left hand side of the curve which corresponds to large probabilities. The intercept for small ˛ is close to the Turkevich–Sher relation above. The maximum of the curve corresponds to d f /d˛ D q D 0; thus it is the dimension of the support of the measure. This seems to be quite close to D so that the whole surface is accessible to random walkers. There is very little agreement about the right hand side of the curve. It arises from regions with small probabilities which are very hard to estimate. The most reliable current method is that of [98] where conformal maps are manipulated to get at probabilities as small as 1070 , quite beyond the reach of other techniques. Unfortunately, these computations are for rather small clusters (N 50; 000) which are the largest ones that can be made by the Hastings–Levitov method. A particularly active controversy relates to the value of ˛max , if it exists at all; that is, the question is whether there is a maximum value of the slope d/dq. The authors of [98] find ˛max 20. Scaling Theories Our understanding of non-equilibrium fractal growth processes is not very satisfactory compared to that of equi-
librium processes. A long-term goal of many groups has been to find a “theory of DLA” which has the nice features of the renormalization theory of critical phenomena. The result of a good deal of work is that we have theories that give the general features of DLA, but they do not explain things in satisfactory detail. We should note that a mere estimate of the fractal dimension, D, is not what we have in mind. There are several ad hoc estimates in the literature that give reasonable values of D [41,99]. We seek, rather, a theory that allows a good understanding of the fixed point of fractal growth with a description of relevant and irrelevant operators, crossovers, etc. There have been an number of such attempts. In the first section below we describe a semi-numerical scheme which sheds light on the fixed point. Then we look at several attempts at ab initio theory. Scaling of Noise The first question one might ask about DLA growth is the role of noise. It might seem that for a very large cluster the noise of individual particle arrivals should average out. However, this is not the case. Clusters fluctuate on all scales for large N. Further, the noise-free case seems to be unstable. We can see this by asking about the related question of the growth of viscous fingers in two dimensions. As was remarked long ago [51] this sort of growth is always unstable. Numerical simulations of viscous fingering [69] show that any asymmetric initial condition develops into a pattern with a fractal dimension close to that of DLA. DLA organizes its noise to a fixed point exactly as turbulence does. Several authors [79,81,100] have looked at the idea that the noise (measured in some way) flows to a fixed value as clusters grow large. For example, we could characterize the shape fluctuations by measuring the scaled variance of the number of particles necessary to grow to a fixed extremal radius: ˇ p ıN ˇˇ D A : (24) ˇ N Rext This is easy to measure for large clusters. In two dimensions A D :0036 [79]. In [100] it was argued that DLA will be at its fixed point if if one unit of growth acts as a coarse graining of DLA on finer length scales. This amounts to a kind of real-space renormalization. For the original DLA model the scaled noise to grow one unit of length is of order unity. We can reduce it to the fixed point value by noise reduction. The slow approach to the fixed point governed by the exponent can be in-
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Fractal Growth Processes, Figure 13 A small noise-reduced cluster and a larger one with no noise reduction. Noise reduction of the proper size accelerates the approach to the fixed point. From [79]
terpreted as the drift of the noise to its proper value as N grows. Noise reduction was introduced into lattice models by Tang [68]. He kept a counter on each site and only grew there if a certain number of random walkers, m, hit that point. For off-lattice clusters a similar reduction is obtained if shallow bumps of height A are added to the cluster by letting the particles overlap as they stick. We must add m D 1/A particles to advance the growth by one particle diameter [79]. For the Hastings–Levitov method it is equivalent to use a bump map with a small aspect ratio [30]. In either case if we examine a number of sites p whose mean advance is one unit, we will find ıN/N D A. We should expect that if we tune the input value of A to A we will get to asymptotic behavior quickly. This is what is observed in two dimensions [79] and in three dimensions [81]. The amplitude of the crossover, the parameter C in Eq. (19), is smallest for A near the fixed point. In Fig. 13 we show two clusters, a small one grown with noise reduction, and a much larger one with A D 1. They both have the same value of /Rdep , near the asymptotic value of 0.12. Attempts at Theoretical Description The last section described a semi-empirical approach to DLA. The reader may wonder why the techniques of phase transition theory are not simply applicable here. One problem is that one of the favorite methods in equilibrium theory, the "-expansion, cannot be used because DLA has no upper critical dimension [5,101]. To be more precise, in other cluster theories such as percolation there is a dimension, dc such that if d > d c the fractal dimension of the cluster does not change. For example, percolation clusters are 4 dimensional for all di-
mensions above 6. For DLA this is not true because if d is much bigger than D then random walkers will penetrate and fill up the cluster so that its dimension would increase. This results [5] from the fact that for a random walker the number of sites visited in radius R is Nw / R2 . In the same region there are RD sites of the cluster, so the density of cluster sites goes as R Dd . The mean number of intersections is R Dd R2 . Therefore if D C 2 < d there are a vanishing number of intersections, and the cluster will fill up. Thus we must have D d 2. A related argument [101] sharpens the bound to D d 1: the fractal dimension increases without limit as d increases. Halsey and collaborators have given a theoretical description based on branch competition [102,103]. This method has been used to give an estimate of Dq for positive q [104]. The idea is to think of two branches that are born from a single site. Then the probability to stick to the first or second is called p1 ; p2 where p1 C p2 D p b is the total probability to stick to that branch. Similarly there are numbers of particles, n1 C n2 D nb . Define x D p1 /p b ; y D n1 /nb . The two variables x; y regulate the branch competition. On the average p1 D dn1 /dnb so we have: dy nb Dxy: (25) dnb The equation of motion for x is assumed to be of similar form: dx nb D g(x; y) : (26) dnb Thus the competition is reduced to a two-dimensional dynamical system with an unstable fixed point at x D y D 1/2 corresponding to the point when the two branches start with equal numbers. The fixed point is unstable because one branch will eventually screen the other. If g is known, then the unstable manifold of this fixed point describes the growth of the dominant branch, and it turns out, by counting the number of particles that attach to the dominant branch, that the eigenvalue of the unstable manifold is 1/D. The starting conditions for the growth are taken to result from microscopic processes that distribute points randomly near the fixed point. The problem remains to find g. This was done several ways: numerically, by doing a large number of simulations of branches that start equally, or in terms of a complicated self-consistent equation [103]. The result is a fractal dimension of 1.66, and a multifractal spectrum that agrees pretty well with direct simulations [104]. Another approach is due to Pietronero and collaborators [105]. It is called the method of fixed scale transformations. It is a real-space method where a small system
Fractal Growth Processes
at one scale is solved essentially exactly, and the behavior at the next coarse-grained scale estimated by assuming that there is a scale-invariant dynamics and estimating the parameters from the fixed-scale solution. The method is much more general than a theory of DLA: in [105] it is applied to directed percolation, the Eden model, sandpile models, and DBM. For DLA the fractal dimension calculated is about 1.6. The rescaled noise (cf. the previous section) comes out to be of order unity rather than the small value, 0.0036, quoted above [106] . The most recent attempt at a fundamental theory is due to Ball and Somfai [71,107]. The idea depends on a mapping from DLA to an instance of the DBM which has different boundary conditions on the growing tip. The scaling of the noise and the multifractal spectrum (for small ˛) are successfully predicted.
Future Directions The DLA model is 27 years old as of this writing. Every year (including last year) there have been about 100 references to the paper. Needless to say, this author has only read a small fraction of them. Space and time prevented presenting here even the interesting ones that I am familiar with. For example, there is a remarkable literature associated with the viscous-fingering problem without surface tension which seems, on the one hand, to describe some facets of experiments [108] and on the other to have deep relationships with the theory of 2d quantum gravity [109,110]. Where this line of work will lead is a fascinating question. There are other examples: I hope that those whose work I have not covered will not feel slighted. There is simply too much going on. A direction that should be pursued is to use the ingenious techniques that have been developed for the DLA problem for problems in different areas; [59,83] are examples of this. It is clear that this field is as lively as ever after 27 years, and will certainly hold more surprises.
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50. Shraiman B, Bensimon D (1984) Singularities in nonlocal interface dynamics. Phys Rev A 30:2840–2842 51. Paterson L (1984) Diffusion-limited aggregation and 2-fluid displacements in porous-media. Phys Rev Lett 52(18): 1621–1624 52. Niemeyer L, Pietronero L, Wiesmann HJ (1984) Fractal dimension of dielectric breakdown. Phys Rev Lett 52:1033–1036 53. Sanchez A, Guinea F, Sander LM, Hakim V, Louis E (1993) Growth and forms of Laplacian aggregates. Phys Rev E 48:1296–1304 54. Hastings MB (2001) Fractal to nonfractal phase transition in the dielectric breakdown model. Phys Rev Lett 87:175502 55. Hastings MB (2001) Growth exponents with 3.99 walkers. Phys Rev E 64:046104 56. Meakin P (1983) Effects of particle drift on diffusion-limited aggregation. Phys Rev B 28(9):5221–5224 57. Nauenberg M, Richter R, Sander LM (1983) Crossover in diffusion-limited aggregation. Phys Rev B 28(3):1649–1651 58. Sander E, Sander LM, Ziff R (1994) Fractals and fractal correlations. Comput Phys 8:420 59. DeVita JP, Sander LM, Smereka P (2005) Multiscale kinetic monte carlo algorithm for simulating epitaxial growth. Phys Rev B 72(20):205421 60. Hou TY, Lowengrub JS, Shelley MJ (1994) Removing the stiffness from interfacial flow with surface-tension. J Comput Phys 114(2):312–338 61. Somfai E, Goold NR, Ball RC, DeVita JP, Sander LM (2004) Growth by random walker sampling and scaling of the dielectric breakdown model. Phys Rev E 70:051403 62. Radnoczi G, Vicsek T, Sander LM, Grier D (1987) Growth of fractal crystals in amorphous GeSe2 films. Phys Rev A 35(9):4012–4015 63. Brady RM, Ball RC (1984) Fractal growth of copper electrodeposits. Nature 309(5965):225–229 64. Grier D, Ben-Jacob E, Clarke R, Sander LM (1986) Morphology and microstructure in electrochemical deposition of zinc. Phys Rev Lett 56(12):1264–1267 65. Sawada Y, Dougherty A, Gollub JP (1986) Dendritic and fractal patterns in electrolytic metal deposits. Phys Rev Lett 56(12):1260–1263 66. Fujikawa H, Matsushita M (1989) Fractal growth of bacillussubtilis on agar plates. Phys J Soc Jpn 58(11):3875–3878 67. Ben-Jacob E, Cohen I, Levine H (2000) Cooperative self-organization of microorganisms. Adv Phys 49(4):395–554 68. Tang C (1985) Diffusion-limited aggregation and the Saffman–Taylor problem. Phys Rev A 31(3):1977–1979 69. Sander LM, Ramanlal P, Ben-Jacob E (1985) Diffusion-limited aggregation as a deterministic growth process. Phys Rev A 32:3160–3163 70. Barra F, Davidovitch B, Levermann A, Procaccia I (2001) Laplacian growth and diffusion limited aggregation: Different universality classes. Phys Rev Lett 87:134501 71. Ball RC, Somfai E (2002) Theory of diffusion controlled growth. Phys Rev Lett 89:135503 72. Mathiesen J, Procaccia I, Swinney HL, Thrasher M (2006) The universality class of diffusion-limited aggregation and viscous-limited aggregation. Europhys Lett 76(2):257–263 73. Gruzberg IA, Kadanoff LP (2004) The Loewner equation: Maps and shapes. J Stat Phys 114(5–6):1183–1198
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74. Davidovitch B, Hentschel HGE, Olami Z, Procaccia I, Sander LM, Somfai E (1999) Diffusion limited aggregation and iterated conformal maps. Phys Rev E 59:1368–1378 75. Duren PL (1983) Univalent Functions. Springer, New York 76. Plischke M, Rácz Z (1984) Active zone of growing clusters: Diffusion-limited aggregation and the Eden model. Phys Rev Lett 53:415–418 77. Mandelbrot BB, Kol B, Aharony A (2002) Angular gaps in radial diffusion-limited aggregation: Two fractal dimensions and nontransient deviations from linear self-similarity. Phys Rev Lett 88:055501 78. Meakin P, Sander LM (1985) Comment on “Active zone of growing clusters: Diffusion-limited aggregation and the Eden model”. Phys Rev Lett 54:2053–2053 79. Ball RC, Bowler NE, Sander LM, Somfai E (2002) Off-lattice noise reduction and the ultimate scaling of diffusion-limited aggregation in two dimensions. Phys Rev E 66:026109 80. Somfai E, Ball RC, Bowler NE, Sander LM (2003) Correction to scaling analysis of diffusion-limited aggregation. Physica A 325(1–2):19–25 81. Bowler NE, Ball RC (2005) Off-lattice noise reduced diffusion-limited aggregation in three dimensions. Phys Rev E 71(1):011403 82. Amitrano C, Coniglio A, Meakin P, Zannetti A (1991) Multiscaling in diffusion-limited aggregation. Phys Rev B 44:4974–4977 83. Bazant MZ, Choi J, Davidovitch B (2003) Dynamics of conformal maps for a class of non-laplacian growth phenomena. Phys Rev Lett 91(4):045503 84. Bazant MZ (2004) Conformal mapping of some non-harmonic functions in transport theory. Proceedings of the Royal Society of London Series A – Mathematical Physical and Engineering Sciences 460(2045):1433–1452 85. Louis E, Guinea F (1987) The fractal nature of fracture. Europhys Lett 3(8):871–877 86. Pla O, Guinea F, Louis E, Li G, Sander LM, Yan H, Meakin P (1990) Crossover between different growth regimes in crack formation. Phys Rev A 42(6):3670–3673 87. Yan H, Li G, Sander LM (1989) Fracture growth in 2d elastic networks with Born model. Europhys Lett 10(1):7–13 88. Barra F, Hentschel HGE, Levermann A, Procaccia I (2002) Quasistatic fractures in brittle media and iterated conformal maps. Phys Rev E 65(4) 89. Barra F, Levermann A, Procaccia I (2002) Quasistatic brittle fracture in inhomogeneous media and iterated conformal maps: Modes I, II, and III. Phys Rev E 66(6):066122 90. Halsey TC, Leibig M (1992) The double-layer impedance at a rough-surface – theoretical results. Ann Phys 219(1):109– 147 91. Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley, New York 92. Duplantier B (1999) Harmonic measure exponents for two-dimensional percolation. Phys Rev Lett 82(20):3940–3943 93. Meakin P, Coniglio A, Stanley HE, Witten TA (1986) Scaling properties for the surfaces of fractal and nonfractal objects – an infinite hierarchy of critical exponents. Phys Rev A 34(4):3325–3340
94. Makarov NG (1985) On the distortion of boundary sets under conformal-mappings. P Lond Math Soc 51:369–384 95. Halsey TC (1987) Some consequences of an equation of motion for diffusive growth. Phys Rev Lett 59:2067–2070 96. Turkevich LA, Scher H (1985) Occupancy-probability scaling in diffusion-limited aggregation. Phys Rev Lett 55(9):1026–1029 97. Ball RC, Spivack OR (1990) The interpretation and measurement of the f(alpha) spectrum of a multifractal measure. J Phys A 23:5295–5307 98. Jensen MH, Levermann A, Mathiesen J, Procaccia I (2002) Multifractal structure of the harmonic measure of diffusion-limited aggregates. Phys Rev E 65:046109 99. Ball RC (1986) Diffusion limited aggregation and its response to anisotropy. Physica A: Stat Theor Phys 140(1–2):62–69 100. Barker PW, Ball RC (1990) Real-space renormalization of diffusion-limited aggregation. Phys Rev A 42(10):6289–6292 101. Ball RC, Witten TA (1984) Causality bound on the density of aggregates. Phys Rev A 29(5):2966–2967 102. Halsey TC, Leibig M (1992) Theory of branched growth. Phys Rev A 46:7793–7809 103. Halsey TC (1994) Diffusion-limited aggregation as branched growth. Phys Rev Lett 72(8):1228–1231 104. Halsey TC, Duplantier B, Honda K (1997) Multifractal dimensions and their fluctuations in diffusion-limited aggregation. Phys Rev Lett 78(9):1719–1722 105. Erzan A, Pietronero L, Vespignani A (1995) The fixed-scale transformation approach to fractal growth. Rev Mod Phys 67(3):545–604 106. Cafiero R, Pietronero L, Vespignani A (1993) Persistence of screening and self-criticality in the scale-invariant dynamics of diffusion-limited aggregation. Phys Rev Lett 70(25):3939–3942 107. Ball RC, Somfai E (2003) Diffusion-controlled growth: Theory and closure approximations. Phys Rev E 67(2):021401, part 1 108. Ristroph L, Thrasher M, Mineev-Weinstein MB, Swinney HL (2006) Fjords in viscous fingering: Selection of width and opening angle. Phys Rev E 74(1):015201, part 2 109. Mineev-Weinstein MB, Wiegmann PB, Zabrodin A (2000) Integrable structure of interface dynamics. Phys Rev Lett 84(22):5106–5109 110. Abanov A, Mineev-Weinstein MB, Zabrodin A (2007) Self-similarity in Laplacian growth. Physica D – Nonlinear Phenomena 235(1–2):62–71
Books and Reviews Vicsek T (1992) Fractal Growth Phenomena, 2nd edn. World Scientific, Singapore Meakin P (1998) Fractals, scaling, and growth far from equilibrium. Cambridge University Press, Cambridge Godreche G (1991) Solids far from equilibrium. Cambridge, Cambridge, New York Sander LM (2000) Diffusion limited aggregation, a kinetic critical phenomenon? Contemporary Physics 41:203–218 Halsey TC (2000) Diffusion-limited aggregation: A model for pattern formation. Physics Today 53(4)
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Fractal Structures in Condensed Matter Physics TSUNEYOSHI N AKAYAMA Toyota Physical and Chemical Research Institute, Nagakute, Japan Article Outline Glossary Definition of the Subject Introduction Determining Fractal Dimensions Polymer Chains in Solvents Aggregates and Flocs Aerogels Dynamical Properties of Fractal Structures Spectral Density of States and Spectral Dimensions Future Directions Bibliography Glossary Anomalous diffusion It is well known that the meansquare displacement hr2 (t)i of a diffusing particle on a uniform system is proportional to the time t such as hr2 (t)i t. This is called normal diffusion. Particles on fractal networks diffuse more slowly compared with the case of normal diffusion. This slow diffusion called anomalous diffusion follows the relation given by hr2 (t)i t a , where the condition 0 < a < 1 always holds. Brownian motion Einstein published the important paper in 1905 opening the way to investigate the movement of small particles suspended in a stationary liquid, the so-called Brownian motion, which stimulated J. Perrin in 1909 to pursue his experimental work confirming the atomic nature of matter. The trail of a random walker provides an instructive example for understanding the meaning of random fractal structures. Fractons Fractons, excitations on fractal elastic-networks, were named by S. Alexander and R. Orbach in 1982. Fractons manifest not only static properties of fractal structures but also their dynamic properties. These modes show unique characteristics such as strongly localized nature with the localization length of the order of wavelength. Spectral density of states The spectral density of states of ordinary elastic networks are expressed by the Debye spectral density of states given by D(!) ! d1 , where d is the Euclidean dimensionality. The spec-
tral density of states of fractal networks is given by D(!) ! d s 1 , where ds is called the spectral or fracton dimension of the system. Spectral dimension This exponent characterizes the spectral density of states for vibrational modes excited on fractal networks. The spectral dimension constitutes the dynamic exponent of fractal networks together with the conductivity exponent and the exponent of anomalous diffusion. Definition of the Subject The idea of fractals is based on self-similarity, which is a symmetry property of a system characterized by invariance under an isotropic scale-transformation on certain length scales. The term scale-invariance has the implication that objects look the same on different scales of observations. While the underlying concept of fractals is quite simple, the concept is used for an extremely broad range of topics, providing a simple description of highly complex structures found in nature. The term fractal was first introduced by Benoit B. Mandelbrot in 1975, who gave a definition on fractals in a simple manner “A fractal is a shape made of parts similar to the whole in some way”. Thus far, the concept of fractals has been extensively used to understand the behaviors of many complex systems or has been applied from physics, chemistry, and biology for applied sciences and technological purposes. Examples of fractal structures in condensed matter physics are numerous such as polymers, colloidal aggregations, porous media, rough surfaces, crystal growth, spin configurations of diluted magnets, and others. The critical phenomena of phase transitions are another example where self-similarity plays a crucial role. Several books have been published on fractals and reviews concerned with special topics on fractals have appeared. Length, area, and volume are special cases of ordinary Euclidean measures. For example, length is the measure of a one-dimensional (1d) object, area the measure of a two-dimensional (2d) object, and volume the measure of a three-dimensional (3d) object. Let us employ a physical quantity (observable) as the measure to define dimensions for Euclidean systems, for example, a total mass M(r) of a fractal object of the size r. For this, the following relation should hold r / M(r)1/d ;
(1)
where d is the Euclidean dimensionality. Note that Euclidean spaces are the simplest scale-invariant systems. We extend this idea to introduce dimensions for selfsimilar fractal structures. Consider a set of particles with
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unit mass m randomly distributed on a d-dimensional Euclidean space called the embedding space of the system. Draw a sphere of radius r and denote the total mass of particles included in the sphere by M(r). Provided that the following relation holds in the meaning of statistical average such as r / hM(r)i1/D f ;
(2)
where h: : :i denotes the ensemble-average over different spheres of radius r, we call Df the similarity dimension. It is necessary, of course, that Df is smaller than the embedding Euclidean dimension d. The definition of dimension as a statistical quantity is quite useful to specify the characteristic of a self-similar object if we could choose a suitable measure. There are many definitions to allocate dimensions. Sometimes these take the same value as each other and sometimes not. The capacity dimension is based on the coverage procedure. As an example, the length of a curved line L is given by the product of the number N of straightline segment of length r needed to step along the curve from one end to the other such as L(r) D N(r)r. While, the area S(r) or the volume V(r) of arbitrary objects can be measured by covering it with squares or cubes of linear size r. The identical relation, M(r) / N(r)r d
(3)
should hold for the total mass M(r) as measure, for example. If this relation does not change as r ! 0, we have the relation N(r) / rd . We can extend the idea to define the dimensions of fractal structures such as N(r) / rD f ;
(4)
from which the capacity dimension Df is given by Df :D lim
r!0
ln N(r) : ln(1/r)
(5)
The definition of Df can be rendered in the following implicit form lim N(r)r D f D const :
r!0
(6)
Equation (5) brings out a key property of the Hausdorff dimension [10], where the product N(r)r D f remains finite as r ! 0. If Df is altered even by an infinitesimal amount, this product will diverge either to zero or to infinity. The Hausdorff dimension coincides with the capacity dimension for many fractal structures, although the Hausdorff dimension is defined less than or equal to the capacity dimension. Hereafter, we refer to the capacity dimension or the Hausdorff dimension mentioned above as the fractal dimension. Introduction Fractal structures are classified into two categories; deterministic fractals and random fractals. In condensed matter physics, we encounter many examples of random fractals. The most important characteristic of random fractals is the spatial and/or sample-to-sample fluctuations in their properties. We must discuss their characteristics by averaging over a large ensemble. The nature of deterministic fractals can be easily understood from some examples. An instructive example is the Mandelbrot–Given fractal [12], which can be constructed by starting with a structure with eight line segments as shown in Fig. 1a (the first stage of the Mandelbrot–Given fractal). In the second stage, each line segment of the initial structure is replaced by the initial structure itself (Fig. 1b). This process is repeated indef-
Fractal Structures in Condensed Matter Physics, Figure 1 Mandelbrot–Given fractal. a The initial structure with eight line segments, b the object obtained by replacing each line segment of the initial structure by the initial structure itself (the second stage), and c the third stage of the Mandelbrot–Given fractal obtained by replacing each line segment of the second-sage structure by the initial structure
Fractal Structures in Condensed Matter Physics
Fractal Structures in Condensed Matter Physics, Figure 2 a Homogeneous random structure in which particles are randomly but homogeneously distributed, and b the distribution functions of local densities , where (l) is the average mass density independent of l
initely. The Mandelbrot–Given fractal possesses an obvious dilatational symmetry, as seen from Fig. 1c, i. e., when we magnify a part of the structure, the enlarged portion looks just like the original one. Let us apply (5) to determine Df of the Mandelbrot–Given fractal. The Mandelbrot–Given fractal is composed of 8 parts of size 1/3, hence, N(1/3) D 8, N((1/3)2 ) D 82 , and so on. We thus have a relation of the form N(r) / r ln3 8 , which gives the fractal dimension Df D ln3 8 D 1:89278 : : :. The Mandelbrot–Given fractal has many analogous features with percolation networks (see Sect. “Dynamical Properties of Fractal Structures”), a typical random fractal, such that the fractal dimension of a 2d percolation network is Df D 91/48 D 1:895833: : :, which is very close to that of the Mandelbrot–Given fractal. The geometric characteristics of random fractals can be understood by considering two extreme cases of random structures. Figure 2a represents the case in which particles are randomly but homogeneously distributed in a d-dimensional box of size L, where d represents ordinary Euclidean dimensionality of the embedding space. If we divide this box into smaller boxes of size l, the mass density of the ith box is
i (l) D
M i (l) ; ld
(7)
where M i (l) represents the total mass (measure) inside box i. Since this quantity depends on the box i, we plot the distribution function P( ), from which curves like those in Fig. 2b may be obtained for two box sizes l1 and (l2 < l2 ). We see that the central peak position of the distribution function P( ) is the same for each case. This means that the average mass density yields
(l) D
hM i (l)i i ld
becomes constant, indicating that hM i (l)i i / l d . The above is equivalent to
D
m ad
;
(8)
where a is the average distance (characteristic lengthscale) between particles and the mass of a single particle. This indicates that there exists a single length scale a characterizing the random system given in Fig. 2a. The other type of random structure is shown in Fig. 3a, where particle positions are correlated with each other and i (l) greatly fluctuates from box to box, as shown in Fig. 3b. The relation hM i (l)i i / l d may not hold at all for this type of structure. Assuming the fractality for this system, namely, if the power law hM i (l)i i / lfD holds, the average mass density becomes ¯ D
(l)
hM i (l)i i / l D f d ; ld
(9)
¯ dewhere i (l) D 0 is excluded. In the case Df < d, (l) pends on l and decreases with increasing l. Thus, there is no characteristic length scale for the type of random structure shown in Fig. 3a. If (9) holds with Df < d, so that hM i (l)i i is proportional to lfD , the structure is said to be fractal. It is important to note that there is no characteristic length scale for the type of random fractal structure shown in Fig. 3b. Thus, we can extend the idea of self-similarity not only for deterministic self-similar structures, but also for random and disordered structures, the so-called random fractals, in the meaning of statistical average. The percolation network made by putting particles or bonds on a lattice with the probability p is a typical example of random fractals. The theory of percolation was initiated in 1957 by S.R. Broadbent and J.M. Hammersley [5] in connection with the diffusion of gases through porous
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Fractal Structures in Condensed Matter Physics, Figure 3 a Correlated random fractal structure in which particles are randomly distributed, but correlated with each other, and b the distribution functions of local densities with finite values, where the average mass densities depend on l
media. Since their work, it has been widely accepted that the percolation theory describes a large number of physical and chemical phenomena such as gelation processes, transport in amorphous materials, hopping conduction in doped semiconductors, the quantum Hall effect, and many other applications. In addition, it forms the basis for studies of the flow of liquids or gases through porous media. Percolating networks thus serve as a model which helps us to understand physical properties of complex fractal structures. For both deterministic and random fractals, it is remarkable that no characteristic length scale exists, and this is a key feature of fractal structures. In other words, fractals are defined to be objects invariant under isotropic scale transformations, i. e., uniform dilatation of the system in every spatial direction. In contrast, there exist systems which are invariant under anisotropic transformations. These are called self-affine fractals. Determining Fractal Dimensions There are several methods to determine fractal dimensions Df of complex structures encountered in condensed matter physics. The following methods for obtaining the fractal dimension Df are known to be quite efficient. Coverage Method The idea of coverage in the definition of the capacity dimension (see (5)) can be applied to obtain the fractal dimension Df of material surfaces. An example is the fractality of rough surfaces or inner surfaces of porous media. The fractal nature is probed by changing the sizes of adsorbed molecules on solid surfaces. Power laws are verified by plotting the total number of adsorbed molecules versus
their size r. The area of a surface can be estimated with the aid of molecules weakly adsorbed by van der Waals forces. Gas molecules are adsorbed on empty sites until the surface is uniformly covered with a layer one molecule thick. Provided that the radius r of one adsorbed molecule and the number of adsorbed molecules N(r) are known, the surface area S obtained by molecules is given by S(r) / N(r)r2 :
(10)
If the surface of the adsorbate is perfectly smooth, we expect the measured area to be independent of the radius r of the probe molecules, which indicates the power law N(r) / r2 :
(11)
However, if the surface of the adsorbate is rough or contains pores that are small compared with r, less of the surface area S is accessible with increasing size r. For a fractal surface with fractal dimension Df , (11) gives the relation N(r) / rD f :
(12)
Box-Counting Method Consider as an example a set of particles distributed in a space. First, we divide the space into small boxes of size r and count the number of boxes containing more than one particle, which we denote by N(r). From the definition of the capacity dimension (4), the number of particle N(r) / rD f :
(13)
For homogeneous objects distributed in a d-dimensional space, the number of boxes of size r becomes, of course N(r) / rd :
Fractal Structures in Condensed Matter Physics
Fractal Structures in Condensed Matter Physics, Figure 4 a 2d site-percolation network and circles with different radii. b The power law relation holds between r and the number of particles in the sphere of radius r, indicating the fractal dimension of the 2d network is Df D 1:89 : : : D 91/48
Correlation Function The fractal dimension Df can be obtained via the correlation function, which is the fundamental statistical quantity observed by means of X-ray, light, and neutron scattering experiments. These techniques are available to bulk materials (not surface), and is widely used in condensed matter physics. Let (r) be the number density of atoms at position r. The density-density correlation function G(r; r 0 ) is defined by G(r; r 0 ) D h (r) (r 0 )i ;
(14)
where h: : :i denotes an ensemble average. This gives the correlation of the number-density fluctuation. Provided that the distribution is isotropic, the correlation function becomes a function of only one variable, the radial distance r D jr r 0 j, which is defined in spherical coordinates. Because of the translational invariance of the system on average, r 0 can be fixed at the coordinate origin r 0 D 0. We can write the correlation function as G(r) D h (r) (0)i :
(15)
The quantity h (r) (0)i is proportional to the probability that a particle exists at a distance r from another particle. This probability is proportional to the particle density (r) within a sphere of radius r. Since (r) / r D f d for a fractal distribution, the correlation function becomes G(r) / r D f d ;
(16)
where Df and d are the fractal and the embedded Euclidean dimensions, respectively. This relation is often used directly to determine Df for random fractal structures. The scattering intensity in an actual experiment is proportional to the structure factor S(q), which is the Fourier transform of the correlation function G(r). The structure factor is calculated from (16) as Z 1 S(q) D G(r)e i qr dr / qD f (17) V V where V is the volume of the system. Here dr is the d-dimensional volume element. Using this relation, we can determine the fractal dimension Df from the data obtained by scattering experiments. When applying these methods to obtain the fractal dimension Df , we need to take care over the following point. Any fractal structures found in nature must have upper and lower length-limits for their fractality. There usually exists a crossover from homogeneous to fractal. Fractal properties should be observed only between these limits. We describe in the succeeding Sections several examples of fractal structures encountered in condensed matter physics. Polymer Chains in Solvents Since the concept of fractal was coined by B.B. Mandelbrot in 1975, scientists reinterpreted random complex structures found in condensed matter physics in terms of fractals. They found that a lot of objects are classified as fractal
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structures. We show at first from polymer physics an instructive example exhibiting the fractal structure. That is an early work by P.J. Flory in 1949 on the relationship between the mean-square end-to-end distance of a polymer chain hr2 i and the degree of polymerization N. Consider a dilute solution of separate coils in a solvent, where the total length of a flexible polymer chain with a monomer length a is Na. The simplest idealization views the polymer chain in analogy with a Brownian motion of a random walker. The walk is made by a succession of N steps from the origin r D 0 to the end point r. According to the central limit theorem of the probability theory, the probability to find a walker at r after N steps (N 1) follows the diffusion equation and we have the expression for the probability to find a particle after N steps at r PN (r) D (2 Na 2/3)3/2 exp(3r2 /2Na2) ;
(18)
where the prefactor arises from the normalization of PN (r). The mean squared distance calculated from PN (r) becomes Z hr2 i D r2 PN (r)d3 r D Na2 : (19) Then, the mean-average end-to-end distance of a polymer chain yields R D hr2 i1/2 D N 1/2 a. Since the number of polymerization N corresponds to the total mass M of a polymer chain, the use of (19) leads to the relation such as M(R) R2 . The mass M(R) can be considered as a measure of a polymer chain, the fractal dimension of this ideal chain as well as the trace of Brown motion becomes Df D 2 for any d-dimensional embedding space. The entropy of the idealized chain of the length L D Na is obtained from (18) as S(r) D S(0)
3r2 ; 2R2
(20)
from which the free energy Fel D U T S is obtained as Fel (r) D Fel(0) C
3k B Tr2 : 2R2
(21)
Here U is assumed to be independent of distinct configurations of polymer chains. This is an elastic energy of an ideal chain due to entropy where Fel decreases as N ! large. P.J. Flory added the repulsive energy term due to monomer-monomer interactions, the so-called excluded volume effect. This has an analogy with self-avoiding random walk. The contribution to the free energy is obtained by the virial expansion into the power series on the concentration cint D N/r d . According to the mean field
2 , we have the total theory on the repulsive term Fint / cint free-energy F such as
F v(T)N 2 3r2 C ; D kB T 2Na2 rd
(22)
where v(T) is the excluded volume parameter. We can obtain a minimum of F(r) at r D R by differentiating F(r) with respect to r such that M(R) / R
dC2 3
:
(23)
Here the number of polymerization N corresponds to the total mass M(R) of a polymer chain. Thus, we have the fractal dimension Df D (d C 2)/3, in particular, Df D 5/3 D 1:666 : : : for a polymer chain in a solvent. Aggregates and Flocs The structures of a wide variety of flocculated colloids in suspension (called aggregates or flocs) can be described in terms of fractals. A colloidal suspension is a fluid containing small charged particles that are kept apart by Coulomb repulsion and kept afloat by Brownian motion. A change in the particle-particle interaction can be induced by varying the chemical composition of the solution and in this manner an aggregation process can be initiated. Aggregation processes are classified into two simple types: diffusion-limited aggregation (DLA) and diffusion-limited cluster-cluster aggregation (DLCA), where a DLA is due to the cluster-particle coalescence and a DLCA to the cluster-cluster flocculation. In most cases, actual aggregates involve a complex interplay between a variety of flocculation processes. The pioneering work was done by M.V. Smoluchowski in 1906, who formulated a kinetic theory for the irreversible aggregation of particles into clusters and further clusters combining with clusters. The inclusion of cluster-cluster aggregation makes this process distinct from the DLA process due to particle-cluster interaction. There are two distinct limiting regimes of the irreversible colloidal aggregation process: the diffusion-limited CCA (DLCA) in dilute solutions and the reaction-limited CCA (RLCA) in dense solutions. The DLCA is due to the fast process determined by the time for the clusters to encounter each other by diffusion, and the RLCA is due to the slow process since the cluster-cluster repulsion has to dominate thermal activation. Much of our understanding on the mechanism forming aggregates or flocs has been mainly due to computer simulations. The first simulation was carried out by Vold in 1963 [23], who used the ballistic aggregation model and found that the number of particles N(r) within a distance r measured from the first seed particle is given by
Fractal Structures in Condensed Matter Physics
N(r) r2:3 . Though this relation surely exhibits the scaling form of (2), the applicability of this model for real systems was doubted in later years. The researches on fractal aggregates has been developed from a simulation model on DLA introduced by T.A. Witten and L.M. Sander in 1981 [26] and on the DLCA model proposed by P. Meakin in 1983 [14] and M. Kolb et al. in 1983 [11], independently. The DLA has been used to describe diverse phenomena forming fractal patterns such as electro-depositions, surface corrosions and dielectric breakdowns. In the simplest version of the DLA model for irreversible colloidal aggregation, a particle is located at an initial site r D 0 as a seed for cluster formation. Another particle starts a random walk from a randomly chosen site in the spherical shell of radius r with width dr( r) and center r D 0. As a first step, a random walk is continued until the particle contacts the seed. The cluster composed of two particles is then formed. Note that the finite-size of particles is the very reason of dendrite structures of DLA. This procedure is repeated many times, in each of which the radius r of the starting spherical shell should be much larger than the gyration radius of the cluster. If the number of particles contained in the DLA cluster is huge (typically 104 108 ), the cluster generated by this process is highly branched, and forms fractal structures in the meaning of statistical average. The fractality arises from the fact that the faster growing parts of the cluster shield the other parts, which therefore become less accessible to incoming particles. An arriving random walker is far more likely to attach to one of the tips of the cluster. Thus, the essence of the fractal-pattern formation arises surely from nonlinear process. Figure 5 illustrates a simulated result for a 2d DLA cluster obtained by the procedure mentioned above. The number of particles N inside a sphere of radius L ( the gyration radius of the cluster) follows the scaling law given by N / L Df ;
(24)
where the fractal dimension takes a value of Df 1:71 for the 2d DLA cluster and Df 2:5 for the 3d DLA cluster without an underlying lattice. Note that these fractal dimensions are sensitive to the embedding lattice structure. The reason for this open structure is that a wandering molecule will settle preferentially near one of the tips of the fractal, rather than inside a cluster. Thus, different sites have different growth probabilities, which are high near the tips and decrease with increasing depth inside a cluster. One of the most extensively studied DLA processes is the growth of metallic forms by electrochemical deposition. The scaling properties of electrodeposited metals were pointed out by R.M. Brady and R.C. Ball in 1984 for
Fractal Structures in Condensed Matter Physics, Figure 5 Simulated results of a 2d diffusion-limited aggregation (DLA). The number of particles contained in this DLA cluster is 104
copper electrodepositions. The confirmation of the fractality for zinc metal leaves was made by M. Matsushita et al. in 1984. In their experiments [13], zinc metal leaves are grown two-dimensionally by electrodeposition. The structures clearly recover the pattern obtained by computer simulations for the DLA model proposed by T.A. Witten and L.M. Sander in 1981. Figure 6 shows a typical zinc dendrite that was deposited on the cathode in one of these experiments. The fractal dimensionality Df D 1:66 ˙ 0:33 was obtained by computing the density-density correlation function G(r) for patterns grown at applied voltages of less than 8V. The fractality of uniformly sized gold-colloid aggregates according to the DLCA was experimentally demonstrated by D.A. Weitz in 1984 [25]. They used transmission-electron micrographs to determine the fractal dimension of this systems to be Df D 1:75. They also performed quasi-elastic light-scattering experiments to investigate the dynamic characteristics of DLCA of aqueous gold colloids. They confirmed the scaling behaviors for the dependence of the mean cluster size on both time and initial concentration. These works were performed consciously to examine the fractality of aggregates. There had been earlier works exhibiting the mass-size scaling relationship for actual aggregates. J.M. Beeckmans [2] pointed out in 1963 the power law behaviors by analyzing the data for aerosol and
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Fractal Structures in Condensed Matter Physics, Figure 6 The fractal structures of zinc metal leaves grown by electrodeposition. Photographs a–d were taken 3,5,9, and 15 min after initiating the electrolysis, respectively. After [13]
precipitated smokes in the literature (1922–1961). He used in his paper the term “aggregates-within-aggregates”, implying the fractality of aggregates. However, the data available at that stage were not adequate and scattered. Therefore, this work did not provide decisive results on the fractal dimensions of aggregates. There were smarter experiments by N. Tambo and Y. Watanabe in 1967 [20], which precisely determined fractal dimensions of flocs formed in an aqueous solution. These were performed without being aware of the concept of fractals. Original works were published in Japanese. The English versions of these works were published in 1979 [21]. We discuss these works below. Flocs generated in aqueous solutions have been the subject of numerous studies ranging from basic to applied sciences. In particular, the settling process of flocs formed in water incorporating kaolin colloids is relevant to water and wastewater treatment. The papers by N. Tambo and Y. Watanabe pioneered the discussion on the socalled fractal approach to floc structures; they performed their own settling experiments to clarifying the size dependences of mass densities for clay-aluminum flocs by using Stokes’ law ur / (r)r2 where is the difference between the densities of water and flocs taking so-small values 0:01–0:001 g/cm3 . Thus, the settling velocities ur are very slow of the order of 0:001 m/sec for flocs of sizes r 0:1 mm, which enabled them to perform precise
measurements. Since flocs are very fragile aggregates, they made the settling experiments with special cautions on convection and turbulence, and by careful and intensive experiments of flocculation conditions. They confirmed from thousands of pieces of data the scaling relationship between settling velocities ur and sizes of aggregates such as ur / r b . From the analysis of these data, they found the scaling relation between effective mass densities and sizes of flocs such as (r) / rc , where the exponents c were found to take values from 1.25 to 1.00 depending on the aluminum-ion concentration, showing that the fractal dimensions become Df D 1:75 to 2.00 with increasing aluminum-ion concentration. This is because the repulsive force between charged clay-particles is screened, and van der Waals attractive force dominates between the pair of particles. It is remarkable that these fractal dimensions Df show excellent agreement with those determined for actual DLCA and RLCA clusters in the 1980s by using various experimental and computer simulation methods. Thus, they had found that the size dependences of mass densities of flocs are controlled by the aluminum-ion concentration dosed/suspended particle concentration, which they named the ALT ratio. These correspond to the transition from DLCA (established now taking the value of Df 1:78 from computer simulations) process to the RLCA one (established at present from computer simulations as Df 2:11). The ALT ratio has since the publica-
Fractal Structures in Condensed Matter Physics
of the network is preserved and decimeter-size monolithic blocks with a range of densities from 50 to 500 kg/m3 can be obtained. As a consequence, aerogels exhibit unusual physical properties, making them suitable for a number of practical applications, such as Cerenkov radiation detectors, supports for catalysis, or thermal insulators. Silica aerogels possess two different length scales. One is the radius r of primary particles. The other length is the correlation length of the gel. At intermediate length scales, lying between these two length scales, the clusters possess a fractal structure and at larger length scales the gel is a homogeneous porous glass. Aerogels have a very low thermal conductivity, solid-like elasticity, and very large internal surfaces. In elastic neutron scattering experiments, the scattering differential cross-section measures the Fourier components of spatial fluctuations in the mass density. For aerogels, the differential cross-section is the product of three factors, and is expressed by d D Af 2 (q)S(q)C(q) C B : d˝
Fractal Structures in Condensed Matter Physics, Figure 7 Observed scaling relations between floc densities and their diameters where aluminum chloride is used as coagulants. After [21]
tion of the paper been used in practice as a criterion for the coagulation to produce flocs with better settling properties and less sludge volume. We show their experimental data in Fig. 7, which demonstrate clearly that flocs (aggregates) are fractal. Aerogels Silica aerogels are extremely light materials with porosities as high as 98% and take fractal structures. The initial step in the preparation is the hydrolysis of an alkoxysilane Si(OR)4 , where R is CH3 or C2 H5 . The hydrolysis produces silicon hydroxide Si(OH)4 groups which polycondense into siloxane bonds –Si–O–Si–, and small particles start to grow in the solution. These particles bind to each other by diffusion-limited cluster-cluster aggregation (DLCA) (see Sect. “Aggregates and Flocs”) until eventually they produce a disordered network filling the reaction volume. After suitable aging, if the solvent is extracted above the critical point, the open porous structure
(25)
Here A is a coefficient proportional to the particle concentration and f (q) is the primary-particle form factor. The structure factor S(q) describes the correlation between particles in a cluster and C(q) accounts for cluster-cluster correlations. The incoherent background is expressed by B. The structure factor S(q) is proportional to the spatial Fourier transform of the density-density correlation function defined by (16), and is given by (17). Since the structure of the aerogel is fractal up to the correlation length of the system and homogeneous for larger scales, the correlation function G(r) is expressed by (25) for r and G(r) D Const. for r . Corresponding to this, the structure factor S(q) is given by (17) for q 1, while S(q) is independent of q for q 1. The wavenumber regime for which S(q) becomes a constant is called the Guinier regime. The value of Df can be deduced from the slope of the observed intensity versus momentum transfer (q 1) in a double logarithmic plot. For very large q, there exists a regime called the Porod regime in which the scattering intensity is proportional to q4 . The results in Fig. 8 by R. Vacher et al. [22] are from small-angle neutron scattering experiments on silica aerogels. The various curves are labeled by the macroscopic density of the corresponding sample in Fig. 8. For example, 95 refers to a neutrally reacted sample with D 95 kg/m3 . Solid lines represent best fits. They are presented even in the particle regime q > 0:15 Å-1 to emphasize that the fits do not apply in the region, particularly for the denser samples. Remarkably, Df is independent of sample
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and the mean-square displacement follows the power law hr2 (t)i / t 2/d w ;
(26)
where dw is termed the exponent of anomalous diffusion. The exponents is expressed as dw D 2 C with a positive > 0 (see (31)), implying that the diffusion becomes slower compared with the case of normal diffusion. This is because the inequality 2/dw < 1 always holds. This slow diffusions on fractal supports are called anomalous diffusion. The scaling relation between the length-scale and the time-scale can be easily extended to the problem of atomic vibrations of elastic fractal-networks. This is because various types of equations governing dynamics can be mapped onto the diffusion equation. This implies that both equations are governed by the same eigenvalue problem, namely, the replacement of eigenvalues ! ! ! 2 between the diffusion equation and the equation of atomic vibrations is justified. Thus, the basic properties of vibrations of fractal networks, such as the density of states, the dispersion relation and the localization/delocalization property, can be derived from the same arguments for diffusion on fractal networks. The dispersion relation between the frequency ! and the wavelength (!) is obtained from (26) by using the reciprocal relation t ! ! 2 (here the diffusion problem is mapped onto the vibrational one) and hr2 (t)i ! (!)2 . Thus we obtain the dispersion relation for vibrational excitations on fractal networks such as ! / (!)d w /2 : Fractal Structures in Condensed Matter Physics, Figure 8 Scattered intensities for eight neutrally reacted samples. Curves are labeled with in kg/m3 . After [22]
density to within experimental accuracy: Df D 2:40 ˙ 0:03 for samples 95 to 360. The departure of S(q) from the qD f dependence at large q indicates the presence of particles with gyration radii of a few Å. Dynamical Properties of Fractal Structures The dynamics of fractal objects is deeply related to the time-scale problems such as diffusion, vibration and transport on fractal support. For the diffusion of a particle on any d-dimensional ordinary Euclidean space, it is well known that the mean-square displacement hr2 (t)i is proportional to the time such as hr2 (t)i / t for any Euclidean dimension d (see also (19)). This is called normal diffusion. While, on fractal supports, a particle more slowly diffuses,
(27)
If dw D 2, we have the ordinary dispersion relation ! / (!) for elastic waves excited on homogeneous systems. Consider the diffusion of a random walker on a percolating fractal network. How does hr2 (t)i behave in the case of fractal percolating networks? For this, P.G. de Gennes in 1976 [7] posed the problem called an ant in the labyrinth. Y. Gefen et al. in 1983 [9] gave a fundamental description of this problem in terms of a scaling argument. D. BenAvraham and S. Havlin in 1982 [3] investigated this problem in terms of Monte Carlo simulations. The work by Y. Gefen [9] triggered further developments in the dynamics of fractal systems, where the spectral (or fracton) dimension ds is a key dimension for describing the dynamics of fractal networks, in addition to the fractal dimension Df . The fractal dimension Df characterizes how the geometrical distribution of a static structure depends on its length scale, whereas the spectral dimension ds plays a central role in characterizing dynamic quantities on fractal networks. These dynamical properties are described in a unified way by introducing a new dynamic exponent called
Fractal Structures in Condensed Matter Physics
the spectral or fracton dimension defined by ds D
2Df : dw
(28)
The term fracton, coined by S. Alexander and R. Orbach in 1982 [1], denotes vibrational modes peculiar to fractal structures. The characteristics of fracton modes cover a rich variety of physical implications. These modes are strongly localized in space and their localization length is of the order of their wavelengths. We give below the explicit form of the exponent of anomalous diffusion dw by illustrating percolation fractal networks. The mean-square displacement hr2 (t)i after a sufficiently long time t should follow the anomalous diffusion described by (26). For a finite network with a size , the mean-square distance at sufficiently large time becomes hr2 (t)i 2 , so we have the diffusion coefficient for anomalous diffusion from (26) such as D / 2d w :
(29)
For percolating networks, the diffusion constant D in the vicinity of the critical percolation density pc behaves D / (p pc ) tˇ / (tˇ )/ ;
(30)
where t is called the conductivity exponent defined by dc (p pc ) t , ˇ the exponent for the percolation order parameter defined by S(p) / (p pc )ˇ , and the exponent for the correlation length defined by / jp pc j , respectively. Comparing (29) and (30), we have the relation between exponents such as dw D 2 C
tˇ D2C :
(31)
Due to the condition t > ˇ, and hence > 0, implying that the diffusion becomes slow compared with the case of normal diffusion. This slow diffusion is called anomalous diffusion. Spectral Density of States and Spectral Dimensions The spectral density of states of atomic vibrations is the most fundamental quantity describing the dynamic properties of homogeneous or fractal systems such as specific heats, heat transport, scattering of waves and others. The simplest derivation of the spectral density of states (abbreviated, SDOS) of a homogeneous elastic system is given below. The density of states at ! is defined as the number of modes per particle, which is expressed by D(!) D
1 ; !Ld
(32)
where ! is the frequency interval between adjacent eigenfrequencies close to ! and L is the linear size of the system. In the lowest frequency region, ! is the lowest eigenfrequency which depends on the size L. The relation between the frequency ! and L is obtained from the well-known linear dispersion relationship ! D ( k, where ( is the velocity of phonons (quantized elastic waves) such that ! D
2( 1 / : L
(33)
The substitution of (33) into (32) yields D(!) / ! d1 :
(34)
Since this relation holds for any length scale L due to the scale-invariance property of homogeneous systems, we can replace the frequency ! by an arbitrary !. Therefore, we obtain the conventional Debye density of states as D(!) / ! d1 :
(35)
It should be noted that this derivation is based on the scale invariance of the system, suggesting that we can derive the SDOS for fractal networks in the same line with this treatment. Consider the SDOS of a fractal structure of size L with fractal dimension Df . The density of states per particle at the lowest frequency ! for this system is, as in the case of (32), written as D(!) /
1 : LfD !
(36)
Assuming that the dispersion relation for ! corresponding to (33) is ! / Lz ;
(37)
we can eliminate L from (36) and obtain D(!) / ! D f /z1 :
(38)
The exponent z of the dispersion relation (37) is evaluated from the exponent of anomalous diffusion dw . Considering the mapping correspondence between diffusion and atomic vibrations, we can replace hr2 (t)i and t by L2 and 1/! 2 , respectively. Equation (26) can then be read as L / ! 2/d w :
(39)
The comparison of (28),(37) and (39) leads to zD
dw Df : D 2 ds
(40)
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Fractal Structures in Condensed Matter Physics, Figure 9 a Spectral densities of states (SDOS) per atom for 2d, 3d, and 4d BP networks at p D pc . The angular frequency ! is defined with mass units m D 1 and force constant Kij D 1. The networks are formed on 1100 1100 (2d), 100 100 100 (3d), and 30 30 30 30 (4d) lattices with periodic boundary conditions, respectively. b Integrated densities of states for the same
Since the system has a scale-invariant fractal (self-similar) structure !, can be replaced by an arbitrary frequency !. Hence, from (38) and (40) the SDOS for fractal networks is found to be D(!) / ! d s 1 ;
(41)
and the dispersion relation (39) becomes ! / L(!)D f /d s :
(42)
For percolating networks, the spectral dimension is obtained from (40) ds D
2Df 2( Df D : 2C 2( C ˇ
(43)
This exponent ds is called the fracton dimension after S. Alexander and R. Orbach [1] or the spectral dimension after R. Rammal and G. Toulouse [17], hereafter we use the term spectral dimension for ds . S. Alexander and R. Orbach [1] estimated the values of ds for percolating networks on d-dimensional Euclidean lattices from the known values of the exponents Df ; (; and ˇ. They pointed out that, while these exponents depend largely on d, the spectral dimension (fracton) dimension ds does not. The spectral dimension ds can be obtained from the value of the conductivity exponent t or vice versa. In the case of percolating networks, the conductivity exponent t is related to ds through (43), which means that the con-
ductivity dc (p pc ) t is also characterized by the spectral dimension ds . In this sense, the spectral dimension ds is an intrinsic exponent related to the dynamics of fractal systems. We can determine the precise values of ds from the numerical calculations of the spectral density of states of percolation fractal networks. The fracton SDOS for 2d, 3d, and 4d bond percolation networks at the percolation threshold p D pc are given in Fig. 9a and b, which were calculated by K. Yakubo and T. Nakayama in 1989. These were obtained by large-scale computer simulations [27]. At p D pc , the correlation length diverges as / jp pc j and the network has a fractal structure at any length scale. Therefore, fracton SDOS should be recovered in the wide frequency range !L ! !D , where !D is the Debye cutoff frequency and !L is the lower cutoff determined by the system size. The SDOSs and the integrated SDOSs per atom are shown by the filled squares for a 2d bond percolation (abbreviated, BP) network at pc D 0:5. The lowest frequency !L is quite small (! 105 for the 2d systems) as seen from the results in Fig. 9 because of the large sizes of the systems. The spectral dimension ds is obtained as ds D 1:33 ˙ 0:11 from Fig. 9a, whereas data in Fig. 9b give the more precise value ds D 1:325 ˙ 0:002. The SDOS and the integrated SDOS for 3d BP networks at pc D 0:249 are given in Fig. 9a and b by the filled triangles (middle). The spectral dimension ds is obtained as ds D 1:31 ˙ 0:02 from Fig. 9a and ds D 1:317 ˙ 0:003 from Fig. 9b. The SDOS and the integrated SDOS of 4d BP networks at pc D 0:160.
Fractal Structures in Condensed Matter Physics
Fractal Structures in Condensed Matter Physics, Figure 10 a Typical fracton mode (! D 0:04997) on a 2d network. Bright region represents the large amplitude portion of the mode. b Crosssection of the fracton mode shown in a along the white line. The four figures are snapshots at different times. After [28]
A typical mode pattern of a fracton on a 2d percolation network is shown in Fig. 10a, where the eigenmode belongs to the angular frequency ! D 0:04997. To bring out the details more clearly, Fig. 10b by K. Yakubo and T. Nakayama [28] shows cross-sections of this fracton mode along the line drawn in Fig. 10a. Filled and open circles represent occupied and vacant sites in the percolation network, respectively. We see that the fracton core (the largest amplitude) possesses very clear boundaries for the edges of the excitation, with an almost step-like character and a long tail in the direction of the weak segments. It should be noted that displacements of atoms in dead ends (weakly connected portions in the percolation network) move in phase, and fall off sharply at their edges. The spectral dimension can be obtained exactly for deterministic fractals. In the case of the d-dimensional Sierpinski gasket, the spectral dimension is given by [17] ds D
2 log(d C 1) : log(d C 3)
We see from this that the upper bound for a Sierpinski gasket is ds D 2 as d ! 1. The spectral dimension for the Mandelbrot–Given fractal depicted is also calculated ana-
lytically as ds D
2 log 8 D 1:345 : : : : log 22
This value is close to those for percolating networks mentioned above, in addition to the fact that the fractal dimension ds D log 8/ log 3 of the Mandelbrot–Given fractal is close to Df D 91/48 for 2d percolating networks and that the Mandelbrot–Given fractal has a structure with nodes, links, and blobs as in the case of percolating networks. For real systems, E. Courtens et al. in 1988 [6] observed fracton excitations in aerogels by means of inelastic light scattering. Future Directions The significance of fractal researches in sciences is that the very idea of fractals opposes reductionism. Modern physics has developed by making efforts to elucidate the physical mechanisms of smaller and smaller structures such as molecules, atoms, and elementary particles. An example in condensed matter physics is the band theory of electrons in solids. Energy spectra of electrons can be obtained by incorporating group theory based on the translational and
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rotational symmetry of the systems. The use of this mathematical tool greatly simplifies the treatment of systems composed of 1022 atoms. If the energy spectrum of a unit cell molecule is solved, the whole energy spectrum of the solid can be computed by applying the group theory. In this context, the problem of an ordered solid is reduced to that of a unit cell. Weakly disordered systems can be handled by regarding impurities as a small perturbation to the corresponding ordered systems. However, a different approach is required for elucidating the physical properties of strongly disordered/complex systems with correlations, or of medium-scale objects, for which it is difficult to find an easily identifiable small parameter that would allow a perturbative analysis. For such systems, the concept of fractals plays an important role in building pictures of the realm of nature. Our established knowledge on fractals is mainly due to experimental observations or computer simulations. The researches are at the phenomenological level stage, not at the intrinsic level, except for a few examples. Concerning future directions of the researches on fractals in condensed matter physics apart from such a question as “What kinds of fractal structures are involved in condensed matter?”, we should consider two directions: one is the very basic aspect such as the problem “Why are there numerous examples showing fractal structures in nature/condensed matter?” However, this type of question is hard. The kinetic growth-mechanisms of fractal systems have a rich variety of applications from the basic to applied sciences and attract much attention as one of the important subjects in non-equilibrium statistical physics and nonlinear physics. Network formation in society is one example where the kinetic growth is relevant. However, many aspects related to the mechanisms of network formations remain puzzling because arguments are at the phenomenological stage. If we compare with researches on Brownian motion as an example, the DLA researches need to advance to the stage of Einstein’s intrinsic theory [8], or that of Smoluchowski [18] and H. Nyquist [15]. It is notable that the DLA is a stochastic version of the Hele–Shaw problem, the flow in composite fluids with high and low viscosities: the particles diffuse in the DLA, while the fluid pressure diffuses in Hele–Shaw flow [19]. These are deeply related to each other and involve many open questions for basic physics and mathematical physics. Concerning the opposite direction, one of the important issues in fractal research is to explore practical uses of fractal structures. In fact, the characteristics of fractals are applied to many cases such as the formation of tailormade nano-scale fractal structures, fractal-shaped antennae with much reduced sizes compared with those of ordi-
nary antennae, and fractal molecules sensitive to frequencies in the infrared region of light. Deep insights into fractal physics in condensed matter will open the door to new sciences and its application to technologies in the near future.
Bibliography Primary Literature 1. Alexander S, Orbach R (1982) Density of states: Fractons. J Phys Lett 43:L625–631 2. Beeckmans JM (1963) The density of aggregated solid aerosol particles. Ann Occup Hyg 7:299–305 3. Ben-Avraham D, Havlin S (1982) Diffusion on percolation at criticality. J Phys A 15:L691–697 4. Brady RM, Ball RC (1984) Fractal growth of copper electro-deposits. Nature 309:225–229 5. Broadbent SR, Hammersley JM (1957) Percolation processes I: Crystals and mazes. Proc Cambridge Philos Soc 53:629–641 6. Courtens E, Vacher R, Pelous J, Woignier T (1988) Observation of fractons in silica aerogels. Europhys Lett 6:L691–697 7. de Gennes PG (1976) La percolation: un concept unificateur. Recherche 7:919–927 8. Einstein A (1905) Über die von der molekularkinetischen Theorie der Waerme geforderte Bewegung von in Ruhenden Fluessigkeiten Suspendierten Teilchen. Ann Phys 17:549–560 9. Gefen Y, Aharony A, Alexander S (1983) Anomalous diffusion on percolating clusters. Phys Rev Lett 50:70–73 10. Hausdorff F (1919) Dimension und Aeusseres Mass. Math Ann 79:157–179 11. Kolb M, Botel R, Jullien R (1983) Scaling of kinetically growing clusters. Phys Rev Lett 51:1123–1126 12. Mandelbrot BB, Given JA (1984) Physical properties of a new fractal model of percolation clusters. Phys Rev Lett 52: 1853–1856 13. Matsushita M, et al (1984) Fractal structures of zinc metal leaves grown by electro-deposition. Phys Rev Lett 53:286–289 14. Meakin P (1983) Formation of fractal clusters and networks by irreversible diffusion-limited aggregation. Phys Rev Lett 51:1119–1122 15. Nyquist H (1928) Termal agitation nof electric charge in conductors. Phys Rev 32:110–113 16. Perrin J (1909) Movement brownien et realite moleculaire. Ann Chim Phys 19:5–104 17. Rammal R, Toulouze G (1983) Random walks on fractal structures and percolation clusters. J Phys Lett 44:L13–L22 18. Smoluchowski MV (1906) Zur Kinematischen Theorie der Brownschen Molekular Bewegung und der Suspensionen. Ann Phys 21:756–780 19. Saffman PG, Taylor GI (1959) The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous fluid. Proc Roy Soc Lond Ser A 245:312–329 20. Tambo N, Watanabe Y (1967) Study on the density profiles of aluminium flocs I (in japanese). Suidou Kyoukai Zasshi 397:2– 10; ibid (1968) Study on the density profiles of aluminium flocs II (in japanese) 410:14–17 21. Tambo N, Watanabe Y (1979) Physical characteristics of flocs
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I: The floc density function and aluminium floc. Water Res 13:409–419 Vacher R, Woignier T, Pelous J, Courtens E (1988) Structure and self-similarity of silica aerogels. Phys Rev B 37:6500– 6503 Vold MJ (1963) Computer simulation of floc formation in a colloidal suspension. J Colloid Sci 18:684–695 Weitz DA, Oliveria M (1984) Fractal structures formed by kinetic aggregation of aqueous cold colloids. Phys Rev Lett 52: 1433–1436 Weitz DA, Huang JS, Lin MY, Sung J (1984) Dynamics of diffusion-limited kinetics aggregation. Phys Rev Lett 53:1657–1660 Witten TA, Sander LM (1981) Diffusion-limited aggregation, a kinetic critical phenomenon. Phys Rev Lett 47:1400–1403 Yakubo K, Nakayama T (1989) Direct observation of localized fractons excited on percolating nets. J Phys Soc Jpn 58: 1504–1507 Yakubo K, Nakayama T (1989) Fracton dynamics of percolating elastic networks: energy spectrum and localized nature. Phys Rev B 40:517–523
Books and Reviews Barabasi AL, Stanley HE (1995) Fractal concepts in surface growth. Cambridge University Press, Cambridge Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge Bunde A, Havlin S (1996) Fractals and disordered systems. Springer, New York de Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Itahca
Falconer KJ (1989) Fractal geometry: Mathematical foundations and applications. Wiley, New York Feder J (1988) Fractals. Plenum, New York Flory PJ (1969) Statistical mechanics of chain molecules. Inter Science, New York Halsey TC (2000) Diffusion-limited aggregation: A model for pattern formation. Phys Today 11:36–41 Kadanoff LP (1976) Domb C, Green MS (eds) Phase transitions and critical phenomena 5A. Academic Press, New York Kirkpatrick S (1973) Percolation and conduction. Rev Mod Phys 45:574–588 Mandelbrot BB (1979) Fractals: Form, chance and dimension. Freeman, San Francisco Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco Meakin P (1988) Fractal aggregates. Adv Colloid Interface Sci 28:249–331 Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge Nakayama T, Yakubo K, Orbach R (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 Nakayama T, Yakubo K (2003) Fractal concepts in condensed matter. Springer, Heidelberg Sahimi M (1994) Applications of percolation theory. Taylor and Francis, London Schroeder M (1991) Fractals, chaos, power laws. W.H. Freeman, New York Stauffer D, Aharony A (1992) Introduction to percolation theory, 2nd edn. Taylor and Francis, London Vicsek T (1992) Fractal growth phenomena, 2nd edn. World Scientific, Singapore
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Fractals and Economics MISAKO TAKAYASU1 , HIDEKI TAKAYASU2 1 Tokyo Institute of Technology, Tokyo, Japan 2 Sony Computer Science Laboratories Inc, Tokyo, Japan Article Outline Glossary Definition of the Subject Introduction Examples in Economics Basic Models of Power Laws Market Models Income Distribution Models Future Directions Bibliography Glossary Fractal An adjective or a noun representing complex configurations having scale-free characteristics or selfsimilar properties. Mathematically, any fractal can be characterized by a power law distribution. Power law distribution For this distribution the probability density is given by a power law, p(r) D c r˛1 , where c and ˛ are positive constants. Foreign exchange market A free market of currencies, exchanging money in one currency for other, such as purchasing a United States dollar (USD) with Japanese yen (JPY). The major banks of the world are trading 24 hours and it is the largest market in the world. Definition of the Subject Market price fluctuation was the very first example of fractals, and since then many examples of fractals have been found in the field of Economics. Fractals are everywhere in economics. In this article the main attention is focused on real world examples of fractals in the field of economics, especially market properties, income distributions, money flow, sales data and network structures. Basic mathematics and physics models of power law distributions are reviewed so that readers can start reading without any special knowledge. Introduction Fractal is the scientific word coined by B.B. Mandelbrot in 1975 from the Latin word fractus, meaning “fractured” [25]. However, fractal does not directly mean fracture itself. As an image of a fractal Fig. 1 shows a photo of
Fractals and Economics, Figure 1 Fractured pieces of plaster fallen on a hard floor (provided by H. Inaoka)
fractured pieces of plaster fallen on a hard floor. There are several large pieces, many middle size pieces and countless fine pieces. If you have a microscope and observe a part of floor carefully then you will find in your vision several large pieces, many small pieces and countless fine pieces, again in the microscopic world. Such scale-invariant nature is the heart of the fractal. There is no explicit definition on the word fractal, it generally means a complicated scale-invariant configuration. Scale-invariance can be defined mathematically [42]. Let P( r) denote the probability that the diameter of a randomly chosen fractured piece is larger than r, then this distribution is called scale-invariant if this function satisfies the following proportional relation for any positive scale factor in a considering scale range: P( r) / P( r) :
(1)
The proportional factor should be a function of , so we can re-write Eq. (1) as P( r) D C()P( r) :
(2)
Assuming that P( r) is a differentiable function, and differentiate Eq. (2) by , and then let D 1. rP 0 ( r) D C 0 (1)P( r)
(3)
As C 0 (1) is a constant this differential equation is readily integrated as P( r) D c0 r C
0 (1)
:
(4)
P( r) is a cumulative distribution and it is a non-increasing function in general, the exponent C 0 (1) can be replaced by ˛ where ˛ is a positive constant. Namely, from
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the scale-invariance with the assumption of differentiability we have the following power law: P( r) D c0 r˛ :
(5)
The reversed logic also holds, namely for any power law distribution there is a fractal configuration or a scale-invariant state. In the case of real impact fracture, the size distribution of pieces is experimentally obtained by repeating sieves of various sizes, and it is empirically well-known that a fractured piece’s diameter follows a power law with the exponent about ˛ D 2 independent of the details about the material or the way of impact [14]. This law is one of the most stubborn physical laws in nature as it is known to hold from 106 m to 105 m, from glass pieces around us to asteroids. From theoretical viewpoint this phenomenon is known to be described by a scale-free dynamics of crack propagation and the universal properties of the exponent value are well understood [19]. Usually fractal is considered geometric concept introducing the quantity fractal dimension or the concept of self-similarity. However, in economics there are very few geometric objects, so, the concept of fractals in economics are mostly used in the sense of power law distributions. It should be noted that any geometrical fractal object accompanies a power law distribution even a deterministic fractal such as Sierpinski gasket. Figure 2 shows Sierpinski gasket which is usually characterized by the fractal dimension D given by DD
log 3 : log 2
(6)
Paying attention to the distribution of length r of white triangles in this figure, it is easy to show that the probability that a randomly chosen white triangle’s side is larger than r, P( r), follows the power law, P( r) / r ˛ ;
˛DDD
Fractals and Economics, Figure 2 Sierpinski gasket
log 3 : log 2
(7)
Here, the power law exponent of distribution equals to the fractal dimension; however, such coincidence occurs only when the considering distribution is for a length distribution. For example, in Sierpinski gasket the area s of white triangles follow the power law, P( s) / s ˛ ;
˛D
log 3 : log 4
(8)
The fractal dimension is applicable only for geometric fractals, however, power law distributions are applicable for any fractal phenomena including shapeless quantities. In such cases the power law exponent is the most important quantity for quantitative characterization of fractals. According to Mandelbrot’s own review on his life the concept of fractal was inspired when he was studying economics data [26]. At that time he found two basic properties in the time series data of daily prices of New York cotton market [24]: (A) Geometrical similarity between large scale chart and an expanded chart. (B) Power law distribution of price changes in a unit time interval, which is independent of the time scale of the unit. He thought such scale invariance in both shape and distribution is a quite general property, not only in price charts but also in nature at large. His inspiration was correct and the concept of fractals spread over physics first and then over almost all fields of science. In the history of science it is a rare event that a concept originally born in economics has been spread widely to all area of sciences. Basic mathematical properties of cumulative distribution can be summarized as follows (here we consider distribution of non-negative quantity for simplicity): 1. P( 0) D 1 , P( 1) D 0. 2. P( r) is a non-increasing function of r. 3. The probability density is given as p(r) drd P( r). As for power law distributions there are three peculiar characteristics: 4. Difficulty in normalization. Assuming that P( r) D c0 r˛ for all in the range 0 r < 1, then the normalization factor c0 must be 0 considering the limit of r ! 0. To avoid this difficulty it is generally assumed that the power law does not hold in the vicinity of r D 0. In the case of observing distribution from real data there are naturally lower and upper bounds, so this difficulty should be necessary only for theoretical treatment. 5. Divergence R 1of moments. As for moments defined by hr n i 0 r n p(r)dr, hr n i D 1 for n ˛. In the
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special case of 2 ˛ > 0 the basic statistical quantity, the variance, diverges, 2 hr2 i hri2 D 1. In the case of 1 ˛ > 0 even the average can not be defined as hri D 1. 6. Stationary or non-stationary? In view of the data analysis, the above characteristics of diverging moments is likely to cause a wrong conclusion that the phenomenon is non-stationary by observing its averaged value. For example, assume that we observe k samples fr1 ; r2 ; : : : ; r k g independently from the power law distribution with the exponent, 1 ˛ > 0. Then, the sample average, hri k 1k fr1 C r2 C C r k g, is shown to diverge as, hri k / k 1/˛ . Such tendency of monotonic increase of averaged quantity might be regarded as a result of non-stationarity, however, this is simply a general property of a power law distribution. The best way to avoid such confusion is to observe the distribution directly from the data. Other than the power law distribution there is another important statistical quantity in the study of fractals, that is, the autocorrelation. For given time series, fx(t)g, the autocorrelation is defined as, C(T)
hx(t C T)x(t)i hx(t)i2 ; hx(t)2 i hx(t)i2
(9)
where h i denotes an average over realizations. The autocorrelation can be defined only for stationary time series with finite variance, in which any statistical quantities do not depend on the location of the origin of time axis. For any case, the autocorrelation satisfies the following basic properties, 1. 2. 3.
C(0) D 1 and C(1) D 0 jC(T)j 1 for any T 0. The R 1 Wiener–Khinchin theorem holds, C(T) D 0 S( f ) cos 2 f d f , where S(f ) is the power spectrum defined by S( f R) hb x( f )b x( f )i, with the Fourier transform, b x( f ) x(t)e2 i f t dt.
In the case that the autocorrelation function is characterized by a power law, C(T) / T ˇ , ˇ > 0, then the time series fx(t)g is said to have a fractal property, in the sense that the autocorrelation function is scale-independent for any scale-factor, > 0, C(T) / C(T). In the case 1 > ˇ > 0 the corresponding power spectrum is given as S( f ) / f 1Cˇ . The power spectrum can be applied to any time series including non-stationary situations. A simple way of telling non-stationary situation is to check the power law exponent of S( f ) / f 1Cˇ in the vicinity of f D 0, for 0 > ˇ the time series is non-stationary.
Three basic examples of fractal time series are the followings: 1. White noise. In the case that fx(t)g is a stationary independent noise, the autocorrelation is given by the Kronecker’s function, C(T) D ıT , where ( 1; T D 0 ıT D 0; T ¤ 0: The corresponding power spectrum is S( f ) / f 0 . This case is called white noise from an analogy that superposition of all frequency lights with the same amplitude make a colorless white light. White noise is a plausible model of random phenomena in general including economic activities. 2. Random walk. This is defined by summation of a white Pt x(s), and the power specnoise, X(t) D X(0) C sD0 trum is given by S( f ) / f 2 . In this case the autocorrelation function can not be defined because the data is non-stationary. Random walks are quite generic models widely used from Brownian motions of colloid to market prices. The graph of a random walk has a fractal property such that an expansion of any part of the graph looks similar to the whole graph. 3. The 1/f noise. The boundary of stationary and nonstationary states is given by the so-called 1/f noise, S( f ) / f 1 . This type of power spectrum is also widely observed in various fields of sciences from electrical circuit noise [16] to information traffics in the Internet [53]. The graph of this 1/f noise also has the fractal property. Examples in Economics In this chapter fractals observed in real economic activities are reviewed. Mathematical models derived from these empirical findings will be summarized in the next chapter. As mentioned in the previous chapter the very first example of a fractal was the price fluctuation of the New York cotton market analyzed by Mandelbrot with the daily data for a period of more than a hundred years [24]. This research attracted much attention at that time, however, there was no other good market data available for scientific analysis, and no intensive follow-up research was done until the 1990s. Instead of earnest scientific data analysis artificial mathematical models of market prices based on random walk theory became popular by the name of Financial Technology during the years 1960–1980. Fractal properties of market prices are confirmed with huge amount of high resolution market data since the 1990s [26,43,44]. This is due to informationization of fi-
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nancial markets in which transaction orders are processed by computers and detail information is recorded automatically, while until the 1980s many people gathered at a market and prices are determined by shouting and screaming which could not be recorded. Now there are more than 100 financial market providers in the world and the number of transacted items exceeds one million. Namely, millions of prices in financial markets are changing with time scale in seconds, and you can access any market price at real time if you have a financial provider’s terminal on your desk via the Internet. Among these millions of items one of the most representative financial markets is the US Dollar-Japanese Yen (USD-JPY) market. In this market Dollar and Yen are exchanged among dealers of major international banks. Unlike the case of stock markets there is no physical trading place, but major international banks are linked by computer networks and orders are emitted from each dealer’s terminal and transactions are done at an electronic broking system. Such a broking system and the computer networks are provided by financial provider companies like Reuters. The foreign exchange markets are open 24 hours and deals are done whenever buy- and sell-orders meet. The minimum unit of a deal is one million USD (called a bar), and about three million bars are traded everyday in the whole foreign exchange markets in which more than 100 kinds of currencies are exchanged continuously. The total amount of money flow is about 100 times bigger than the total amount of daily world trade, so it is believed that most of deals are done not for the real world’s needs, but they are based on speculative strategy or risk hedge, that is, to get profit by buying at a low price and selling at a high price, or to avoid financial loss by selling decreasing currency. In Fig. 3 the price of one US Dollar paid by Japanese Yen in the foreign exchange markets is shown for 13 years [30]. The total number of data points is about 20 million, that is, about 10 thousand per day or the averaged transaction interval is seven seconds. A magnified part of the top figure for one year is shown in the second figure. The third figure is the enlargement of one month in the second figure. The bottom figure is again a part of the third figure, here the width is one day. It seems that at least the top three figures look quite similar. This is one of the fractal properties of market price (A) introduced in the previous chapter. This geometrical fractal property can be found in any market, so that this is a very universal market property. However, it should be noted that this geometrical fractal property breaks down for very short time scale as typi-
Fractals and Economics, Figure 3 Dollar-Yen rate for 13 years (Top). Dark areas are enlarged in the following figure [30]
Fractals and Economics, Figure 4 Market price changes in 10 minutes
cally shown in Fig. 4. In this figure the abscissa is 10 minutes range and we can observe each transaction separately. Obviously the price up down is more zigzag and more discrete than the large scale continuous market fluctuations
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Fractals and Economics, Figure 5 Log–log plot of cumulative distribution of rate change [30]
shown in Fig. 3. In the case of USD-JPY market the time scale that this breakdown of scale invariance occurs typically at time scale of several hours. The distribution of rate change in a unit time (one minute) is shown in Fig. 5. Here, there are two plots of cumulative distributions, P(> x) for positive rate changes and P(> jxj) for negative rate changes, which are almost identical meaning that the up-down symmetry of rate changes is nearly perfect. In this log–log plot the estimated power law distribution’s exponent is 2.5. In the original finding of Mandelbrot, (B) in the previous chapter, the reported exponent value is about 1.7 for cotton prices. In the case of stock markets power laws are confirmed universally for all items, however, the power exponents are not universal, taking value from near one to near five, typically around three [15]. Also the exponent values change in time year by year. In order to demonstrate the importance of large fluctuations, Fig. 6 shows a comparison of three market prices. The top figure is the original rate changes for a week. The middle figure is produced from the same data, but it is consisted of rate changes of which absolute values are larger than 2, that is, about 5 % of all the data. In the bottom curve such large rate changes are omitted and the residue of 95 % of small changes makes the fluctuations. As known from these figures the middle figure is much closer to the original market price changes. Namely, the contribution from the power law tails of price change distribution is very large for macro-scale market prices. Power law distribution of market price changes is also a quite general property which can be confirmed for any market. Up-down symmetry also holds universally in short time scale in general, however, for larger unit time the distribution of price changes gradually deforms and for very large unit time the distribution becomes closer to
Fractals and Economics, Figure 6 USD-JPY exchange rate for a week (top) Rate changes smaller than 2 are neglected (middle) Rate changes larger than 2 are neglected (bottom)
a Gaussian distribution. It should be noted that in special cases of market crashes or bubbles or hyper-inflations the up-down symmetry breaks down and the power law distribution is also likely to be deformed. The autocorrelation of the time sequence of price changes generally decays quickly to zero, sometimes accompanied by a negative correlation in a very short time. This result implies that the market price changes are apparently approximated by white noise, and market prices are known to follow nearly a random walk as a result. However, market price is not a simple random walk. In Fig. 7 the autocorrelation of volatility, which is defined by the square of price change, is shown in log–log scale. In the case of a simple random walk this autocorrelation should also decay quickly. The actual volatility autocorrelation nearly satisfies a power law implying that the volatility time series has a fractal clustering property. (See also Fig. 31 representing an example of price change clustering.) Another fractal nature of markets can be found in the intervals of transactions. As shown in Fig. 8 the transaction intervals fluctuate a lot in very short time scale. It is known that the intervals make clusters, namely, shorter intervals tend to gather. To characterize such clustering effect we can make a time sequence consisted of 0 and 1, where 0 denotes no deal was done at that time, and 1 denotes a deal was done. The corresponding power spectrum follows a 1/f power spectrum as shown in Fig. 9 [50].
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Fractals and Economics, Figure 7 Autocorrelation of volatility [30]
Fractals and Economics, Figure 10 Income distribution of companies in Japan
Fractals and Economics, Figure 8 Clustering of transaction intervals
Fractal properties are found not only in financial markets. Company’s income distribution is known to follow also a power law [35]. A company’s income is roughly given by subtraction of incoming money flow minus outgoing money flow, which can take both positive and negative values. There are about six million companies in Japan and Fig. 10 shows the cumulative distribution of annual income of these companies. Clearly we have a power law distribution of income I with the exponent very close to 1 in the middle size range, so-called the Zipf’s law, P(> I) / I ˇ ;
Fractals and Economics, Figure 9 Power spectrum of transaction intervals [50]
ˇ D 1:
(10)
Although in each year every company’s income fluctuates, and some percentage of companies disappear or are newly born, this power law is known to hold for more than 30 years. Similar power laws are confirmed in various countries, the case of France is plotted in Fig. 11 [13]. Observing more details by categorizing the companies, it is found that the income distribution in each job category follows nearly a power law with the exponent depending on the job category as shown in Fig. 12 [29]. The implication of this phenomenon will be discussed in Sect. “Income Distribution Models”. A company’s size can also be viewed by the amount of whole sale or the number of employee. In Figs. 13 and 14 distributions of these quantities are plotted [34]. In both cases clear power laws are confirmed. The size distribution of debts of bankrupted companies is also known to follow a power law as shown Fig. 15 [12].
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Fractals and Economics, Figure 11 Income distribution of companies in France [13]
Fractals and Economics, Figure 14 The distribution of employee numbers [34]
Fractals and Economics, Figure 12 Income distribution of companies in each category [29]
Fractals and Economics, Figure 15 The size distribution of debts of bankrupted companies [12]
Fractals and Economics, Figure 13 The distribution of whole sales [34]
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Fractals and Economics, Figure 16 Personal income distribution in Japan [1]
Fractals and Economics, Figure 17 The distribution of the amount of transferred money [21]
A power law distribution can also be found in personal income. Figure 16 shows the personal income distribution in Japan in a log–log plot [1]. The distribution is clearly separated into two parts. The majority of people’s incomes are well approximated by a log-normal distribution (the left top part of the graph), and the top few percent of people’s income distribution is nicely characterized by a power law (the linear line in the left part of the graph). The majority of people are getting salaries from companies. This type of composite of two distributions is well-known from the pioneering study by Pareto about 100 years ago and it holds in various countries [8,22]. A typical value of the power exponent is about two, significantly larger than the income distribution of companies. However, the exponent of the power law seems to be not universal and the value changes county by country or year by year. There is a tendency that the exponent is smaller, meaning more rich people, when the economy is improving [40]. Another fractal in economics can be found in a network of economic agents such as banks’ money transfer network. As a daily activity banks transfer money to other banks for various reasons. In Japan all of these interbank money transfers are done via a special computer network provided by the Bank of Japan. Detailed data of actual money transfer among banks are recorded and analyzed for the basic study. The total amount of money flow among banks in a day is about 30 1012 yen with the number of transactions about 10 000. Figure 17 shows the distribution of the amount of money at a transaction. The range is not wide enough but we can find a power law with an exponent about 1.3 [20].
The number of banks is about 600, so the daily transaction number is only a few percent of the theoretically possible combinations. It is confirmed that there are many pairs of banks which never transact directly. We can define active links between banks for pairs with the averaged number of transaction larger than one per day. By this criterion the number of links becomes about 2000, that is, about 0.5 percent of all possible link numbers. Compared with the complete network, the actual network topologies are much more sparse. In Fig. 18 the number distribution of active links per site are plotted in log–log plot [21]. As is known from this graph, there is an intermediate range in which the link number distribution follows a power law. In the terminology of recent complex network study, this property is called the scale-free network [5]. The scale-free network structure among these intermediate banks is shown in Fig. 19. There are about 10 banks with large link numbers which deviate from the power law, also small link number banks with link number less than four are out of the power law. Such small banks are known to make a satellite structure that many banks linked to one large link number banks. It is yet to clarify why intermediate banks make fractal network, and also to clarify the role of large banks and small banks which are out of the fractal configuration. In relation with the banks, there are fractal properties other than cash flow and the transaction network. The distribution of the whole amount of deposit of Japanese bank is approximated by a power law as shown in Fig. 20 [57]. In recent years large banks merged making a few mega banks and the distribution is a little deformed. Historically there were more than 6000 banks in Japan, however,
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Fractals and Economics, Figure 20 Distribution of total deposit for Japanese banks [57] Power law breaks down from 1999 Fractals and Economics, Figure 18 The number distribution of active links per site [20]
Fractals and Economics, Figure 19 Scale-free network of intermediate banks [20]
now we have about 600 as mentioned. It is very rare that a bank disappears, instead banks are merged or absorbed. The number distribution of banks which are historically behind a present bank is plotted in Fig. 21, again a power law can be confirmed. Other than the example of the bank network, network structures are very important generally in economics. In production process from materials, through various parts to final products the network structure is recently studied in view of complex network analysis [18]. Trade networks among companies can also be described by network terminology. Recently, network characterization quantities such as link numbers (Fig. 22), degrees of authority, and Pageranks are found to follow power laws from real trade data for nearly a million of companies in Japan [34]. Still more power laws in economics can be found in sales data. A recent study on the distribution of expen-
Fractals and Economics, Figure 21 Distribution of bank numbers historically behind a present bank [57]
diture at convenience stores in one shopping trip shows a clear power law distribution with the exponent close to two as shown in Fig. 23 [33]. Also, book sales, movie hits, news paper sales are known to be approximated by power laws [39]. Viewing all these data in economics, we may say that fractals are everywhere in economics. In order to understand why fractals appear so frequently, we firstly need to make simple toy models of fractals which can be analyzed completely, and then, based on such basic models we can make more realistic models which can be directly comparable with real data. At that level of study we will be able to predict or control the complex real world economy.
Fractals and Economics
a power law, P(> y) D y ˛ for y 1. This is a useful transformation in case of numerical simulation using random variable following power laws. 2. Let x be a stochastic variable following an exponential distribution, P(> x) D ex , for positive x, then, y e x/˛ satisfies a power law, P(> y) / y ˛ . As exponential distributions occur frequently in random process such as the Poisson process, or energy distribution in thermal equilibrium, this simple exponential variable transformation can make it a power law. Superposition of Basic Distributions
Fractals and Economics, Figure 22 Distribution of in-degrees and out-degrees in Japanese company network [34]
A power law distribution can also be easily produced by superposition of basic distributions. Let x be a Gaussian distribution with the probability density given by p R 2 R p R (x) D p e 2 x ; (11) 2 and R be a 2 distribution with degrees of freedom ˛, 1 ˛/2 w(R) D
2
˛ 2
˛
R
R 2 1 e 2 :
(12)
Then, the superposition of Gaussian distribution, Eq. (11), with the weight given by Eq. (12) becomes the T-distribution having power law tails: Z1 p(x) D
W(R)p R (x)dR 0
˛C1 1 2 / jxj˛1 ; Dp ˛ 2 (1 C x 2 ) ˛C1 2 Fractals and Economics, Figure 23 Distribution of expenditure in one shopping trip [33]
Basic Models of Power Laws In this chapter we introduce general mathematical and physical models which produce power law distributions. By solving these simple and basic cases we can deepen our understanding of the underlying mechanism of fractals or power law distributions in economics. Transformation of Basic Distributions A power law distribution can be easily produced by variable transformation from basic distributions. 1. Let x be a stochastic variable following a uniform distribution in the range (0, 1], then, y x 1/˛ satisfies
(13)
which is P(> jxj) / jxj˛ in cumulative distribution. In the special case that R, the inverse of variance of the normal distribution, distributes exponentially, the value of ˛ is 2. Similar super-position can be considered for any basic distributions and power law distributions can be produced by such superposition. Stable Distributions Assume that stochastic variables, x1 ; x2 ; : : : ; x n , are independent and follow the same distribution, p(x), then consider the following normalized summation; Xn
x1 C x2 C C x n n : n1/˛
(14)
If there exists ˛ > 0 and n , such that the distribution of X n is identical to p(x), then, the distribution belongs to
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one of the Levy stable distributions [10]. The parameter ˛ is called the characteristic exponent which takes a value in the range (0, 2]. The stable distribution is characterized by four continuous parameters, the characteristic exponent, an asymmetry parameter which takes a value in [1, 1], the scale factor which takes a positive value and the location parameter which takes any real number. Here, we introduce just a simple case of symmetric distribution around the origin with the unit scale factor. The probability density is then given as p(x; ˛) D
1 2
Z1
˛
ei x ej j d :
(15)
the power law is obtained, P( x) D
1 Mlog x
0
:
(18)
In other words, a power law distribution maximizes the entropy in the situation where products are conserved. To be more precise, consider two time dependent random variables interacting each other satisfying the relation, x1 (t) x2 (t) D x1 (t 0 ) x2 (t 0 ), then the equilibrium distribution follows a power law. Another entropy approach to the power laws is to generalize the entropy by the following form [56],
1
For large jxj the cumulative distribution follows the power law, P(> x; ˛) / jxj˛ except the case of ˛ D 2. The stable distribution with ˛ D 2 is the Gaussian distribution. The most important property of the stable distribution is the generalized central limit theorem: If the distribution of sum of any independent identically distributed random variables like X n in Eq. (14) converges in the limit of n ! 1 for some value of ˛, then the limit distribution is a stable distribution with the characteristic exponent ˛. For any distribution with finite variance, the ordinary central limit theory holds, that is, the special case of ˛ D 2. For any infinite variance distribution the limit distribution is ˛ ¤ 2 with a power law tail. Namely, a power law realizes simply by summing up infinitely many stochastic variables with diverging variance.
x x0
1 Sq
1 R
p(x)q dx
x0
q1
;
(19)
where q is a real number. This function is called the q-entropy and the ordinary entropy, Eq. (15), recovers in the limit of q ! 1. Maximizing the q-entropy keeping the variance constant, so-called a q-Gaussian distribution is obtained, which has the same functional form with the T-distribution, Eq. (12), with the exponent ˛ given by ˛D
q3 : 1q
(20)
This generalized entropy formulation is often applied to nonlinear systems having long correlations, in which power law distributions play the central role.
Entropy Approaches Let x0 be a positive constant and consider a probability density p(x) defined in the interval [x0 ; 1), the entropy of this distribution is given by Z1
Here, we find a distribution that maximizes the entropy with a constraint such that the expectation of logarithm of x is a constant, hlog xi D M. Then, applying the variational principle to the following function,
L x0
01 1 Z p(x) log p(x)dx 1 @ p(x)dx 1A 0 C 2 @
x0
Z1
x0
(21)
(16)
x0
Z1
Stochastic time evolution described by the following formulation is called the multiplicative process, x(t C 1) D b(t)x(t) C f (t) ;
p(x) log p(x)dx :
S
Random Multiplicative Process
P(> x) / jxj˛ ;
1
p(x) log xdx M A
where b(t) and f (t) are both independent random variables [17]. In the case that b(t) is a constant, the distribution of x(t) depends on the distribution of f (t), for example, if f (t) follows a Gaussian distribution, then the distribution of x(t) is also a Gaussian. However, in the case that b(t) fluctuates randomly, the resulting distribution of x(t) is known to follows a power law independent of f (t),
(17)
(22)
where the exponent ˛ is determined by solving the following equation [48], hjb(t)j˛ i D 1 :
(23)
Fractals and Economics
This steady distribution exists when hlog jb(t)ji < 0 and f (t) is not identically 0. As a special case that b(t) D 0 with a finite probability, then a steady state exists. It is proved rigorously that there exists only one steady state, and starting from any initial distribution the system converges to the power law steady state. In the case hlog jb(t)ji 0 there is no statistically steady state, intuitively the value of jb(t)j is so large that x(t) is likely to diverge. Also in the case f (t) is identically 0 there is no steady state as known from Eq. (21) that log jx(t)j follows a simple random walk with random noise term, log jb(t)j. The reason why this random multiplicative process produces a power law can be understood easily by considering a special case that b(t) D b > 1 with probability 0.5 and b(t) D 0 otherwise, with a constant value of f (t) D 1. In such a situation the value of x(t) is 1 C b C b2 C C b K with probability (0:5)K . From this we can directly evaluate the distribution of x(t), b KC1 1 D 2KC1 i: e: P b1 (24) log 2 P( x) D 4(1 C (b 1)x)˛ ; ˛D : log b As is known from this discussion, the mechanism of this power law is deeply related to the above mentioned transformation of exponential distribution in Sect. “Transformation of Basic Distributions”. The power law distribution of a random multiplicative process can also be confirmed experimentally by an electrical circuit in which resistivity fluctuates randomly [38]. In an ordinary electrical circuit the voltage fluctuations in thermal equilibrium is nearly Gaussian, however, for a circuit with random resistivity a power law distribution holds. Aggregation with Injection Assume the situation that many particles are moving randomly and when two particles collide they coalesce making a particle with mass conserved. Without any injection of particles the system converges to the trivial state that only one particle remains. In the presence of continuous injection of small mass particles there exists a non-trivial statistically steady state in which mass distribution follows a power law [41]. Actually, the mass distribution of aerosol in the atmosphere is known to follow a power law in general [11]. The above system of aggregation with injection can be described by the following model. Let j be the discrete space, and x j (t) be the mass on site j at time t, then choose
one site and let the particle move to another site and particles on the visited site merge, then add small mass particles to all sites, this process can be mathematically given as, 8 ˆ <x j (t) C x k (t) C f j (t) ; x j (tC1) D x j (t) C f j (t) ; ˆ : f j (t) ;
prob D 1/N prob D (N 2)/N prob D 1/N (25)
where N is the total number of sites and f j (t) is the injected mass to the site j. The characteristic function, Z( ; t) he x j (t) i, which is the Laplace transform of the probability density, satisfies the following equation by assuming uniformity, Z( ; t C 1)
N 2 1 1 D Z( ; t)2 C Z( ; t) C he f j (t) i : N N N (26) The steady state solution in the vicinity of D 0 is obtained as p (27) Z( ) D 1 h f i 1/2 C : From this behavior the following power law steady distribution is obtained. P( x) / x ˛ ;
˛D
1 : 2
(28)
By introducing a collision coefficient depending on the size of particles power laws with various values of exponents realized in the steady state of such aggregation with injection system [46]. Critical Point of a Branching Process Consider the situation that a branch grows and splits with probability q or stops growing with probability 1 q as shown in Fig. 24. What is the size distribution of the branch? This problem can be solved in the following way. Let p(r) be the probability of finding a branch of size r, then the next relation holds. p(r C 1) D q
r1 X
p(s)p(r s) :
(29)
sD1
Multiplying y rC1 and summing up by r from 0 to 1, a closed equation of the generating function, M(y), is obtained, M(y)1Cq D q y M(y)2 ;
M(y)
1 X rD0
y r p(r) : (30)
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tributed in N-sites. At each time step choose one site randomly, and transport a finite portion, x j (t), to another randomly chosen site, where is a parameter in the range [0, 1]. x j (t C 1) D (1 )x j (t) ; x k (t C 1) D x k (t) C x j (t) :
Fractals and Economics, Figure 24 Branching process (from left to right)
Solving this quadratic equation and expanding in terms of y, we have the probability density, p(r) / r3/2 eQ(q)r ;
Q(q) log 4q(1 q) :
(31)
For q < 0:5 the probability decays exponentially for large r, in this case all branches has a finite size. At q D 0:5 the branch size follows the power law, P( r) / r1/2 , and the average size of branch becomes infinity. For q > 0:5 there is a finite probability that a branch grows infinitely. The probability of having an infinite branch, p(1) D 1M(1), is given as, p 2q 1 C 1 4q(1 q) p(1) D ; (32) 2q which grows monotonically from zero to one in the range q D [0:5; 1]. It should be noted that the power law distribution realizes at the critical point between the finite-size phase and the infinite-size phase [42]. Compared with the preceding model of aggregation with injection, Eq. (28), the mass distribution is the same as the branch size distribution at the critical point in Eq. (31). This coincidence is not an accident, but it is known that aggregation with injection automatically chooses the critical point parameter. Aggregation and branching are reversed process and the steady occurrence of aggregation implies that branching numbers keep a constant value on average and this requires the critical point condition. This type of critical behaviors is called the self-organized criticality and examples are found in various fields [4].
It is known that for small positive the statistically steady distribution x is well approximated by a Gaussian like the case of thermal fluctuations. For close to 1 the fluctuation of x is very large and its distribution is close to a power law. In the limit goes to 1 and the distribution converges to Eq. (28), the aggregation with injection case. For intermediate values of the distribution accompanies a fat tail between Gaussian and a power law [49]. Fractal Tiling A fractal tiling is introduced as the final basic model. Figure 25 shows an example of fractal tiling of a plane by squares. Like this case Euclidean space is covered by various sizes of simple shapes like squares, triangles, circles etc. The area size distribution of squares in Fig. 25 follows the power law, P( x) / x ˛ ;
˛D 1/2 :
(34)
Generalizing this model in d-dimensional space, the distribution of d-dimensional volume x is characterized by a power law distribution with an exponent, ˛ D (d 1)/d, therefore, the Zipf’s law which is the case of ˛ D 1 realizes in the limit of d D 1. The fracture size distribution measured in mass introduced in the beginning of this article corresponds to the case of d D 3.
Finite Portion Transport Here, a kind of mixture of aggregation and branching is considered. Assume that conserved quantities are dis-
(33)
Fractals and Economics, Figure 25 An example of fractal tiling
Fractals and Economics
ence as, x(t) D (t) f (t) ;
(35)
where f (t) is a random variable following a Gaussian distribution with 0 mean and variance unity, the local variance (t) is given as (t)2 D c0 C
k X
c k (x(t k))2 ;
(36)
jD1
Fractals and Economics, Figure 26 Fractal tiling by river patterns [45]
A classical example of fractal tiling is the Apollonian gasket, that is, a plane is covered totally by infinite number of circles which are tangent each other. For a given river pattern like Fig. 26 the basin area distribution follows a power law with exponent about ˛ D 0:4 [45]. Although these are very simple geometric models, simple models may sometimes help our intuitive understanding of fractal phenomena in economics.
Market Models In this chapter market price models are reviewed in view of fractals. There are two approaches for construction of market models. One is modeling the time sequences directly by some stochastic model, and the other is modeling markets by agent models which are artificial markets in computer consisted of programmed dealers. The first market price model was proposed by Bachelier in 1900 written as his Ph.D thesis [3], that is, five years before the model of Einstein’s random walk model of colloid particles. His idea was forgotten for nearly 50 years. In 1950’s Markowitz developed the portfolio theory based on a random walk model of market prices [28]. The theory of option prices by Black and Scholes was introduced in the 1970s, which is also based on random walk model of market prices, or to be more precise a logarithm of market prices in continuum description [7]. In 1982 Engle introduced a modification of the simple random walk model, the ARCH model, which is the abbreviation of auto-regressive conditional heteroscedasticity [9]. This model is formulated for market price differ-
with adjustable positive parameters, fc0 ; c1 ; : : : ; c k g. By the effect of this modulation on variance, the distribution of price difference becomes superposition of Gaussian distribution with various values of variance, and the distribution becomes closer to a power law. Also, volatility clustering occurs automatically so that the volatility autocorrelation becomes longer. There are many variants of ARCH models, such as GARCH and IGARCH, but all of them are based on purely probabilistic modeling, and the probability of prices going up and that of going down are identical. Another type of market price model has been proposed from physics view point [53]. The model is called the PUCK model, an abbreviation of potentials of unbalanced complex kinetics, which assumes the existence of market’s time-dependent potential force, UM (x; t), and the time evolution of market price is given by the following set of equations; ˇ ˇ d C f (t) ; (37) x(tC1)x(t) D UM (x; t)ˇˇ dx xDx(t)x M (t) b(t) x 2 ; (38) UM (x; t) M1 2 where M is the number of moving average needed to define the center of potential force, xM (t)
M1 1 X x(t k) : M
(39)
kD0
In this model f (t) is the external noise and b(t) is the curvature of quadratic potential which changes with time. When b(t) D 0 the model is identical to the simple random walk model. When b(t) > 0 the market prices are attracted to the moving averaged price, xM (t), the market is stable, and when b(t) < 0 prices are repelled from xM (t) so that the price fluctuation is large and the market is unstable. For b(t) < 2 the price motion becomes an exponential function of time, which can describe singular behavior such as bubbles and crashes very nicely.
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In the simplest case of M D 2 the time evolution equation becomes, x(t C 1) D
b(t) x(t) C f (t) : 2
(40)
As is known from this functional form in the case b(t) fluctuates randomly, the distribution of price difference follows a power law as mentioned in the previous Sect. “Random Multiplicative Process”, Random multiplicative process. Especially the PUCK model derives the ARCH model by introducing a random nonlinear potential function [54]. The value of b(t) can be estimated from the data and most of known empirical statistical laws including fractal properties are fulfilled as a result [55]. The peculiar difference of this model compared with financial technology models is that directional prediction is possible in some sense. Actually, from the data it is known that b(t) changes slowly in time, and for non-zero b(t) the autocorrelation is not zero implying that the up-down statistics in the near future is not symmetric. Moreover in the case of b(t) < 2 the price motion show an exponential dynamical growth hence predictable. As introduced in Sect. “Examples in Economics” the tick interval fluctuations can be characterized by the 1/f power spectrum. This power law can be explained by a model called the self-modulation model [52]. Let t j be the jth tick interval, and we assume that the tick interval can be approximated by the following random process, t jC1 D j
K1 1 X t jk C g j ; K
Fractals and Economics, Figure 27 Tick intervals of Poisson process (top) and the self-modulation process (bottom) [52]
(41)
kD0
where j is a positive random number following an exponential distribution with the mean value 1, and K is an integer which means the number of moving average, g j is a positive random variable. Due to the moving average term in Eq. (41) the tick interval automatically make clusters as shown in Fig. 27, and the corresponding power spectrum is proved to be proportional to 1/ f as typically represented in Fig. 28. The market data of tick intervals are tested whether Eq. (41) really works or not. In Fig. 29 the cumulative probability of estimated value of j from market data is plotted where the moving average size is determined by the physical time of 150 seconds and 400 seconds. As known from this figure, the distribution fits very nicely with the exponential distribution when the moving average size is 150 seconds. This result implies that dealers in the market are mostly paying attention to the latest transaction for about a few minutes only. And the dealers’ clocks in their minds move quicker if the market becomes busier. By this
Fractals and Economics, Figure 28 The power spectrum of the self-modulation process [52]
Fractals and Economics, Figure 29 The distribution of normalized time interval [50]
Fractals and Economics
self-modulation effect transactions in markets automatically make a fractal configuration. Next, we introduce a dealer model approach to the market [47]. In any financial market dealers’ final goal is to gain profit from the market. To this end dealers try to buy at the lowest price and to sell at the highest price. Assume that there are N dealers at a market, and let the jth dealer’s buying and selling prices in their mind B j (t) and S j (t). For each dealer the inequality, B j (t) < S j (t), always holds. We pay attention to the maximum price of fB j (t)g called the best bid, and to the minimum price of fS j (t)g called the best ask. Transactions occur in the market if there exists a pair of dealers, j and k, who give the best bid and best ask respectively, and they fulfill the following condition, B j (t) S k (t) :
(42)
Fractals and Economics, Figure 30 Price evolution of a market with deterministic three dealers
In the model the market price is given by the mean value of these two prices. As a simple situation we consider a deterministic time evolution rule for these dealers. For all dealers the spread, S j (t) B j (t), is set to be a constant L. Each dealer has a position, either a seller or a buyer. When the jth dealer’s position is a seller the selling price in mind, S j (t), decreases every time step until he can actually sell. Similar dynamics is applied to a buyer with the opposite direction of motion. In addition we assume that all dealers shift their prices in mind proportional to a market price change. When this proportional coefficient is positive, the dealer is categorized as a trend-follower. If this coefficient is negative, the dealer is called a contrarian. These rules are summarized by the following time evolution equations. B j (t C 1) D B j (t) C a j S j C b j x(t) ;
(43)
where Sj takes either +1 or 1 meaning the buyer position or seller position, respectively, x(t) gives the latest market price change, fa j g are positive numbers given initially, fb j g are also parameters given initially. Figure 30 shows an example of market price evolution in the case of three dealers. It should be noted that although the system is deterministic, namely, the future price is determined uniquely by the set of initial values, resulting market price fluctuates almost randomly even in the minimum case of three dealers. The case of N D 2 gives only periodic time evolution as expected, while for N 3 the system can produce market price fluctuations similar to the real market price fluctuations, for example, the fractal properties of price chart and the power law distribution of price difference are realized. In the case that the value of fb j g are identical for all dealers, b, then the distribution of market price difference follows a power law where the exponent is controllable by
Fractals and Economics, Figure 31 Cumulative distribution of a dealer model for different values of b. For weaker trend-follow the slope is steeper [38]
this trend-follow parameter, b as shown in Fig. 31 [37]. The volatility clustering is also observed automatically for large dealer number case as shown in Fig. 32 (bottom) which looks quite similar to a real price difference time series Fig. 32 (top). By adding a few features to this basic dealer model it is now possible to reproduce almost all statistical characteristics of market, such as tick-interval fluctuations, abnormal diffusions etc. [58]. In this sense the study of market behaviors are now available by computer simulations based on the dealer model. Experiments on the market is either impossible or very difficult for a real market, however, in an artificial market we can repeat occurrence of bubbles and crashes any times, so that we might be able to find a way to avoid catastrophic market behaviors by numerical simulation.
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This mathematical answer implies that even X is one million dollars this lottery is generous enough and you should buy because expectation is infinity. But, would you dare to buy this lottery, in which you will win only one dollar with probability 0.5, and two dollars with probability 0.25, . . . ? Bernoulli’s answer to this paradox is to introduce the human feeling of value, or utility, which is proportional to the logarithm of price, for example. Based on this expected utility hypothesis the fair value of X is given as follows, XD
1 1 X X U(2n ) log(2n ) D D 1 C log 2 nC1 2 2nC1 nD0 nD0
1:69 ;
Fractals and Economics, Figure 32 Price difference time series for a real market (top) and a dealer model (bottom)
Income Distribution Models Let us start with a famous historical problem, the St. Petersburg Paradox, as a model of income. This paradox was named after Daniel Bernoulli’s paper written when he was staying in the Russian city, Saint Petersburg, in 1738 [6]. This paradox treats a simple lottery as described in the following, which is deeply related to the infinite expected value problem in probability theory and also it has been attracting a lot of economists’ interest in relation with the essential concept in economics, the utility [2]. Assume that you enjoy a game of chance, you pay a fixed fee, X dollars, to enter, and then you toss a fair coin repeatedly until a tail firstly appears. You win 2n dollars where n is the number of heads. What is the fair price of the entrance fee, X? Mathematically a fair price should be equal to the expectation value, therefore, it should be given as, 1 X 1 XD 2n nC1 D 1 : (44) 2 nD0
(45)
where the utility function, U(x) D 1Clog x, is normalized to satisfy U(1) D 1. This result implies that the appropriate entry fee X should be about two dollars. The idea of utility was highly developed in economics for description of human behavior, in the way that human preference is determined by maximal point of utility function, the physics concept of the variational principle applied to human action. Recently, in the field of behavioral finance which emerged from psychology the actual observation of human behaviors about money is the main task and the St. Petersburg paradox is attracting attention [36]. Although Bernoulli’s solution may explain the human behavior, the fee X D 2 is obviously so small that the bookmaker of this lottery will bankrupt immediately if the entrance fee is actually fixed as two dollars and if a lot of people actually buy it. The paradox is still a paradox. To clarify what is the problem we calculate the distribution of income of a gambler. As an income is 2 n with probability 2n1 , the cumulative distribution of income is readily obtained as, P( x) /
1 : x
(46)
This is the power law which we observed for income distribution of companies in Sect. “Examples in Economics”. The key of this lottery is the mechanism that the prize money doubles at each time a head appears and the coin toss stops when a tail appears. By denoting the number of coin toss by t, we can introduce a stochastic process or a new lottery which is very much related to the St. Petersburg lottery. x(t C 1) D b(t)x(t) C 1 ;
(47)
where b(t) is 2 with probability 0.5 and is 0 otherwise. As introduced in Sect. “Random Multiplicative Process”, this problem is solved easily and it is confirmed that the steady
Fractals and Economics
state cumulative distribution of x(t) also follows Eq. (46). The difference between the St. Petersburg lottery and the new lottery Eq. (47) is the way of payment of entrance fee. In the case of St. Petersburg lottery the entrance fee X is paid in advance, while in the case of new lottery you have to add one dollar each time you toss a coin. This new lottery is fair from both the gambler side and the bookmaker side because the expectation of income is given by hx(t)i D t and the amount of paid fee is also t. Now we introduce a company’s income model by generalizing this new fair lottery in the following way, I(t C 1) D b(t)I(t) C f (t) ;
(48)
where I(t) denotes the annual income of a company, b(t) represents the growth rate which is given randomly from a growth rate distribution g(b), and f (t) is a random noise. Readily from the results of Sect. “Random Multiplicative Process”, we have a condition to satisfy the empirical relation, Eq. (10), Z hb(t)i D bg(b) D 1 : (49)
An implication of this result is that if a job category is expanding, namely, hb(t)i > 1, then the power law exponent determined by Eq. (50) is smaller than 1. On the other hand if a job category is shrinking, we have an exponent that is larger than 1. This type of company’s income model can be generalized to take into account the effect of company’s size dependence on the distribution of growth rate. Also, the magnitude of the random force term can be estimated from the probability of occurrence of negative income. Then, assuming that the present growth rate distribution continues we can perform a numerical simulation of company’s income distribution starting from a uniform distribution as shown in Fig. 34 for Japan and in Fig. 35 for USA. It is shown that in the case of Japan, the company size distribution converges to the power law with exponent 1 in 20 years, while in the case of USA the steady power law’s slope is about 0.7 and it takes about 100 years to converge [31]. According to this result extremely large com-
This relation is confirmed to hold approximately in actual company data [32]. In order to explain the job category dependence of the company’s income distribution already shown in Fig. 12, we plot the comparison of exponents in Fig. 33. Empirically estimated exponents are plotted in the ordinate and the solutions of the following equation calculated in each job category are plotted in the abscissa, hb(t)ˇ i D 1 :
(50)
The data points are roughly on a straight line demonstrating that the simple growth model of Eq. (48) seems to be meaningful.
Fractals and Economics, Figure 33 Theoretical predicted exponent value vs. observed value [29]
Fractals and Economics, Figure 34 Numerical simulation of income distribution evolution of Japanese companies [32]
Fractals and Economics, Figure 35 Numerical simulation of income distribution evolution of USA companies [32]
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panies with size about 10 times bigger than the present biggest company will appear in USA in this century. Of course the growth rate distribution will change faster than this prediction, however, this model can tell the qualitative direction and the speed of change in very macroscopic economical conditions. Other than this simple random multiplicative model approach there are various approaches to explain empirical facts about company’s statistics assuming a hierarchical structure of organization, for example [23]. Future Directions Fractal properties generally appear in almost any huge data in economics. As for financial market models, empirical fractal laws are reproduced and the frontier of study is now at the level of practical applications. However, there are more than a million markets in the world and little is known about their interaction. More research on market interaction will be promising. Company data so far analyzed show various fractal properties as introduced in Sect. “Examples in Economics”, however, they are just a few cross-sections of global economics. Especially, companies’ interaction data are inevitable to analyze the underlying network structures. Not only money flow data it will be very important to observe material flow data in manufacturing and consumption processes. From the viewpoint of environmental study, such material flow network will be of special importance in the near future. Detail sales data analysis is a new topic and progress is expected. Bibliography Primary Literature 1. Aoyama H, Nagahara Y, Okazaki MP, Souma W, Takayasu H, Takayasu M (2000) Pareto’s law for income of individuals and debt of bankrupt companies. Fractals 8:293–300 2. Aumann RJ (1977) The St. Petersburg paradox: A discussion of some recent comments. J Econ Theory 14:443–445 http://en. wikipedia.org/wiki/Robert_Aumann 3. Bachelier L (1900) Theory of Speculation. In: Cootner PH (ed) The Random Character of Stock Market Prices. MIT Press, Cambridge (translated in English) 4. Bak P (1996) How Nature Works. In: The Science of Self-Organized Criticality. Springer, New York 5. Barabási AL, Réka A (1999) Emergence of scaling in random networks. Science 286:509–512; http://arxiv.org/abs/ cond-mat/9910332 6. Bernoulli D (1738) Exposition of a New Theory on the Measurement of Risk; Translation in: (1954) Econometrica 22:22–36 7. Black F, Scholes M (1973) The Pricing of Options and Corporate Liabilities. J Political Econ 81:637–654
8. Brenner YS, Kaelble H, Thomas M (1991) Income Distribution in Historical Perspective. Cambridge University Press, Cambridge 9. Engle RF (1982) Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of UK Inflation. Econometrica 50:987–1008 10. Feller W (1971) An Introduction to Probability Theory and Its Applications, 2nd edn, vol 2. Wiley, New York 11. Friedlander SK (1977) Smoke, dust and haze: Fundamentals of aerosol behavior. Wiley-Interscience, New York 12. Fujiwara Y (2004) Zipf Law in Firms Bankruptcy. Phys A 337:219–230 13. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Phys A 335:197–216 14. Gilvarry JJ, Bergstrom BH (1961) Fracture of Brittle Solids. II Distribution Function for Fragment Size in Single Fracture (Experimental). J Appl Phys 32:400–410 15. Gopikrishnan P, Meyer M, Amaral LAN, Stanley HE (1998) Inverse Cubic Law for the Distribution of Stock Price Variations. Eur Phys J B 3:139–143 16. Handel PH (1975) 1/f Noise-An “Infrared” Phenomenon. Phys Rev Lett 34:1492–1495 17. Havlin S, Selinger RB, Schwartz M, Stanley HE, Bunde A (1988) Random Multiplicative Processes and Transport in Structures with Correlated Spatial Disorder. Phys Rev Lett 61: 1438–1441 18. Hidalgo CA, Klinger RB, Barabasi AL, Hausmann R (2007) The Product Space Conditions the Development of Nations. Science 317:482–487 19. Inaoka H, Toyosawa E, Takayasu H (1997) Aspect Ratio Dependence of Impact Fragmentation. Phys Rev Lett 78:3455–3458 20. Inaoka H, Ninomiya T, Taniguchi K, Shimizu T, Takayasu H (2004) Fractal Network derived from banking transaction – An analysis of network structures formed by financial institutions. Bank of Japan Working Paper. http://www.boj.or.jp/en/ ronbun/04/data/wp04e04.pdf 21. Inaoka H, Takayasu H, Shimizu T, Ninomiya T, Taniguchi K (2004) Self-similarity of bank banking network. Phys A 339: 621–634 22. Klass OS, Biham O, Levy M, Malcai O, Solomon S (2006) The Forbes 400 and the Pareto wealth distribution. Econ Lett 90:290–295 23. Lee Y, Amaral LAN, Canning D, Meyer M, Stanley HE (1998) Universal Features in the Growth Dynamics of Complex Organizations. Phys Rev Lett 81:3275–3278 24. Mandelbrot BB (1963) The variation of certain speculative prices. J Bus 36:394–419 25. Mandelbrot BB (1982) The Fractal Geometry of Nature. W.H. Freeman, New York 26. Mandelbrot BB (2004) The (mis)behavior of markets. Basic Books, New York 27. Mantegna RN, Stanley HE (2000) An Introduction to Economics: Correlations and Complexity in Finance. Cambridge Univ Press, Cambridge 28. Markowitz HM (1952) Portfolio Selection. J Finance 7:77–91 29. Mizuno T, Katori M, Takayasu H, Takayasu M (2001) Statistical laws in the income of Japanese companies. In: Empirical Science of Financial Fluctuations. Springer, Tokyo, pp 321–330 30. Mizuno T, Kurihara S, Takayasu M, Takayasu H (2003) Analysis of high-resolution foreign exchange data of USD-JPY for 13 years. Phys A 324:296–302
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31. Mizuno T, Kurihara S, Takayasu M, Takayasu H (2003) Investment strategy based on a company growth model. In: Takayasu H (ed) Application of Econophysics. Springer, Tokyo, pp 256–261 32. Mizuno T, Takayasu M, Takayasu H (2004) The mean-field approximation model of company’s income growth. Phys A 332:403–411 33. Mizuno T, Toriyama M, Terano T, Takayasu M (2008) Pareto law of the expenditure of a person in convenience stores. Phys A 387:3931–3935 34. Ohnishi T, Takayasu H, Takayasu M (in preparation) 35. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Phys A 269:125– 131. http://www.ingentaconnect.com/content/els/03784371; jsessionid=5e5wq937wfsqu.victoria 36. Rieger MO, Wang M (2006) Cumulative prospect theory and the St. Petersburg paradox. Econ Theory 28:665–679 37. Sato AH, Takayasu H (1998) Dynamic numerical models of stock market price: from microscopic determinism to macroscopic randomness. Phys A 250:231–252 38. Sato AH, Takayasu H, Sawada Y (2000) Power law fluctuation generator based on analog electrical circuit. Fractals 8: 219–225 39. Sinha S, Pan RK (2008) How a “Hit” is Born: The Emergence of Popularity from the Dynamics of Collective Choice. http:// arxiv.org/PS_cache/arxiv/pdf/0704/0704.2955v1.pdf 40. Souma W (2001) Universal structure of the personal income distribution. Fractals 9:463–470; http://www.nslij-genetics. org/j/fractals.html 41. Takayasu H (1989) Steady-state distribution of generalized aggregation system with injection. Phys Rev Lett 63: 2563–2566 42. Takayasu H (1990) Fractals in the physical sciences. Manchester University Press, Manchester 43. Takayasu H (ed) (2002) Empirical Science of Financial Fluctuations–The Advent of Econophysics. Springer, Tokyo 44. Takayasu H (ed) (2003) Application of Econophysics. Springer, Tokyo 45. Takayasu H, Inaoka H (1992) New type of self-organized criticality in a model of erosion. Phys Rev Lett 68:966–969
46. Takayasu H, Takayasu M, Provata A, Huber G (1991) Statistical properties of aggregation with injection. J Stat Phys 65: 725–745 47. Takayasu H, Miura H, Hirabayashi T, Hamada K (1992) Statistical properties of deterministic threshold elements–The case of market price. Phys A 184:127–134 48. Takayasu H, Sato AH, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys Rev Lett 79:966–969 49. Takayasu M, Taguchi Y, Takayasu H (1994) Non-Gaussian distribution in random transport dynamics. Inter J Mod Phys B 8:3887–3961 50. Takayasu M (2003) Self-modulation processes in financial market. In: Takayasu H (ed) Application of Econophysics. Springer, Tokyo, pp 155–160 51. Takayasu M (2005) Dynamics Complexity in Internet Traffic. In: Kocarev K, Vatty G (eds) Complex Dynamics in Communication Networks. Springer, New York, pp 329–359 52. Takayasu M, Takayasu H (2003) Self-modulation processes and resulting generic 1/f fluctuations. Phys A 324:101–107 53. Takayasu M, Mizuno T, Takayasu H (2006) Potentials force observed in market dynamics. Phys A 370:91–97 54. Takayasu M, Mizuno T, Takayasu H (2007) Theoretical analysis of potential forces in markets. Phys A 383:115–119 55. Takayasu M, Mizuno T, Watanabe K, Takayasu H (preprint) 56. Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52:479–487; http://en.wikipedia.org/wiki/ Boltzmann_entropy 57. Ueno H, Mizuno T, Takayasu M (2007) Analysis of Japanese bank’s historical network. Phys A 383:164–168 58. Yamada K, Takayasu H, Takayasu M (2007) Characterization of foreign exchange market using the threshold-dealer-model. Phys A 382:340–346
Books and Reviews Takayasu H (2006) Practical Fruits of Econophysics. Springer, Tokyo Chatterjee A, Chakrabarti BK (2007) Econophysics of Markets and Business Networks (New Economic Windows). Springer, New York
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Fractals and Multifractals, Introduction to DANIEL BEN-AVRAHAM1 , SHLOMO HAVLIN2 1 2
Clarkson University, Potsdam, USA Bar-Ilan University, Ramat Gan, Israel
Fractals generalize Euclidean geometrical objects to noninteger dimensions and allow us, for the first time, to delve into the study of complex systems, disorder, and chaos. In the words of B. Mandelbrot: “Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, nor does lightning travel in a straight line,” [1]. Indeed, much has changed in our perception of nature, and today it is hard to conceive of natural phenomena that are not fractal, in the same way that it is hard to conceive of everyday life dynamical systems that are not non-linear. The discovery of fractals over three decades ago signaled a profound shift in the way we understand and analyze the physical world around us. Not only does fractal geometry model complex disordered objects such as clouds and mountains, coastlines and lightning, but it also finds a beautiful new symmetry in the midst of all this complexity – invariance under dilation of space – and it is this self-similarity symmetry that lends fractals their tremendous power and analytical appeal ( Fractals and Multifractals, Introduction to). The same symmetry plays a significant role in critical phase transitions, and one of the earliest applications of fractals has been to the study of the paradigmatic model of percolation ( Fractals and Percolation). Soon after their advent, researchers systematically used fractals to explain a host of anomalous phenomena (anomalous with respect to the expectations from regular Euclidean objects) in condensed matter and solid state physics ( Fractal Structures in Condensed Matter Physics). One of the most influential models of nonequilibrium growth, Diffusion Limited Aggregation (DLA), produces a fractal object that is still studied today for its fascinating connections to the Laplace equation and electrical discharge, conformal mapping, patterns of bacterial growth, and viscous fingering ( Fractal Growth Processes). Fractals and fractal scaling also arise in regular systems with superposed disorder, for example when resistances from a random distribution are assigned to the bonds of a regular lattice ( Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks). More recently, interest has grown in the study of large, stochas-
tic complex nets, which seem infinite-dimensional at the outset, but even so can be found to have fractal geometry ( Fractal and Transfractal Scale-Free Networks). Having realized that the geometry of everyday life objects around us is more likely fractal than not, it was only natural to ask about the physics of phase transitions in disordered media, for example, for the magnetization of spins in the Ising model, placed on the nodes of fractals ( Phase Transitions on Fractals and Networks). Early on, interest developed in the physics of transport in fractal substrates. It was quickly discovered that fractal media diffusion, the most elementary mode of transport, does not obey Fick’s law but is, rather, anomalous ( Anomalous Diffusion on Fractal Networks), leading to entirely new ways of viewing and analyzing transport, most notably among them Lévy flights, continuum random walks, and fractional diffusion equations. Because of the anomalous diffusion of reactants in fractal disordered media, so are the kinetics of reactions among them anomalous ( Reaction Kinetics in Fractals). The graph Laplacian in fractals has an anomalous spectrum that displays a peculiar scaling, having a profound effect on dynamics in general ( Dynamics on Fractals). Fractals refer not only to geometrical objects but, more broadly, to any kind of phenomena possessing scaling that exhibits dilation symmetry, or scale invariance, often characterized by the appearance of a power-law distribution. A whole suite of techniques has evolved to analyze this type of fractal scaling ( Fractal and Multifractal Time Series). Fractal phenomena of this type (and occasionally fractal objects) find applications in several diverse fields of interest, such as finances and economics ( Fractals and Economics), geology ( Fractals in Geology and Geophysics), the analysis of DNA sequences ( Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences?), biology ( Fractals in Biology), and even in path integrals in the quantum theory of spacetime ( Fractals in the Quantum Theory of Spacetime). Fractals and the modern study of nonlinear systems, strange attractors, and chaos have infused and enriched one another from their very incipience ( Fractals Meet Chaos). Together they provide us with a fundamentally new way of understanding the everyday world around us. Despite our best intentions, this Section on Fractals and Multifractals remains incomplete, as it fails to include numerous important subjects such as the practical use of fractals for image compression, their ubiquity in medicine, astronomy, and their prominence in the study of real systems of porous materials, electrodeposition, molecular
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surfaces, colloids, and polymers, to name just a few. We hope that some, if not all of these subjects will be included in future editions.
Bibliography 1. Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco
Fractals and Percolation
Fractals and Percolation YAKOV M. STRELNIKER1 , SHLOMO HAVLIN1 , ARMIN BUNDE2 1 Department of Physics, Bar–Ilan University, Ramat–Gan, Israel 2 Institut für Theoretische Physik III, Justus-Liebig-Universität, Giessen, Germany
Definition of the Subject Percolation theory is useful for characterizing many disordered systems. Percolation is a pure random process of choosing sites to be randomly occupied or empty with certain probabilities. However, the topology obtained in such processes has a rich structure related to fractals. The structural properties of percolation clusters have become clearer thanks to the development of fractal geometry since the 1980s.
Article Outline Glossary Definition of the Subject Introduction Percolation Percolation Clusters as Fractals Anomalous Transport on Percolation Clusters: Diffusion and Conductivity Networks Summary and Future Directions Bibliography Glossary Percolation In the traditional meaning, percolation concerns the movement and filtering of fluids through porous materials. In this chapter, percolation is the subject of physical and mathematical models of porous media that describe the formation of a long-range connectivity in random systems and phase transitions. The most common percolation model is a lattice, where each site is occupied randomly with a probability p or empty with probability 1 p. At low p values, there is no connectivity between the edges of the lattice. Above some concentration pc , the percolation threshold, connectivity appears between the edges. Percolation represents a geometric critical phenomena where p is the analogue of temperature in thermal phase transitions. Fractal A fractal is a structure which can be subdivided into parts, where the shape of each part is similar to that of the original structure. This property of fractals is called self-similarity, and it was first recognized by G.C. Lichtenberg more than 200 years ago. Random fractals represent models for a large variety of structures in nature, among them porous media, colloids, aggregates, flashes, etc. The concepts of self-similarity and fractal dimensions are used to characterize percolation clusters. Self-similarity is strongly related to renormalization properties used in critical phenomena, in general, and in percolation phase transition properties.
Introduction Percolation represents the simplest model of a phase transition [1,8,13,14,26,27,30,48,49,61,64,65,68]. Assume a regular lattice (grid) where each site (or bond) is occupied with probability p or empty with probability 1 p. At a critical threshold, pc , a long-range connectivity first appears: pc is called the percolation threshold (see Fig. 1). Occupied and empty sites (or bonds) may stand for very different physical properties. For example, occupied sites may represent electrical conductors, empty sites may represent insulators, and electrical current may flow only through nearest-neighbor conducting sites. Below pc , the grid represents an isolator since there is no conducting path between two adjacent bars of the lattice, while above pc , conducting paths start to occur and the grid becomes a conductor. One can also consider percolation as a model for liquid filtration (i. e., invasion percolation (see Fig. 2), which is the source of this terminology) through porous media. A possible application of bond percolation in chemistry is the polymerization process [25,31,44], where small branching molecules can form large molecules by activating more and more bonds between them. If the activation probability p is above the critical concentration, a network of chemical bonds spanning the whole system can be formed, while below pc only macromolecules of finite size can be generated. This process is called a sol-gel transition. An example of this gelation process is the boiling of an egg, which at room temperature is liquid but, upon heating, becomes a solid-like gel. An example from biology concerns the spreading of an epidemic [35]. In its simplest form, an epidemic starts with one sick individual which can infect its nearest neighbors with probability p in one time step. After one time step, it dies, and the infected neighbors in turn can infect their (so far) uninfected neighbors, and the process is continued. Here the critical concentration separates a phase at low p where the epidemic always dies out after a finite number of time steps, from a phase where the epidemic can continue forever. The same process can be used as a model for
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Fractals and Percolation, Figure 1 Square lattice of size 20 20. Sites have been randomly occupied with probability p (p D 0:20, 0.59, 0.80). Sites belonging to finite clusters are marked by full circles, while sites on the infinite cluster are marked by open circles
Fractals and Percolation, Figure 2 Invasion percolation through porous media
forest fires [52,60,64,71], with the infection probability replaced by the probability that a burning tree can ignite its nearest-neighbor trees in the next time step. In addition to these simple examples, percolation concepts have been found useful for describing a large number of disordered systems in physics and chemistry. The first study introducing the concept of percolation is attributable to Flory and Stockmayer about 65 years ago, when studying the gelation process [32]. The name percolation was proposed by Broadbent and Hammersley in 1957 when they were studying the spreading of fluids in random media [15]. They also introduced relevant geometrical and probabilistic concepts. The developments of phase transition theory in the following years, in particular the series expansion method by Domb [27] and renormalization group theory by Wilson, Fisher and Kadanoff [51,65], very much stimulated research activities into the geometric percolation transition. At the percolation threshold, the conducting (as well as insulating) clusters are self-similar (see Fig. 3) and, therefore, can be described by fractal geometry [53], where various fractal dimensions are introduced to quantify the clusters and their physical properties.
Fractals and Percolation, Figure 3 Self-similarity of the random percolation cluster at the critical concentration; courtesy of M. Meyer
Percolation As above (see Sect. “Introduction”), consider a square lattice, where each site is occupied randomly with probability p (see Fig. 1). For simplicity, let us assume that the occupied sites are electrical conductors and the empty sites represent insulators. At low concentration p, the occupied sites either are isolated or form small clusters (Fig. 1a). Two occupied sites belong to the same cluster if they are connected by a path of nearest-neighbor occupied sites and a current can flow between them. When p is increased, the average size of the clusters increases. At a critical concentration pc (also called the percolation threshold), a large cluster appears which connects opposite edges of the lattice (Fig. 1). This cluster is called the infinite cluster, since its size diverges when the size of the lattice is increased to infinity. When p is increased further, the density of the infinite cluster increases, since more and more sites become
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part of the infinite cluster, and the average size of the finite clusters decreases (Fig. 1c). The percolation threshold separates two different phases and, therefore, the percolation transition is a geometrical phase transition, which is characterized by the geometric features of large clusters in the neighborhood of pc . At low values of p, only small clusters of occupied sites exist. When the concentration p is increased, the average size of the clusters increases. At the critical concentration pc , a large cluster appears which connects opposite edges of the lattice. Accordingly, the average size of the finite clusters which do not belong to the infinite cluster decreases. At p D 1, trivially, all sites belong to the infinite cluster. Similar to site percolation, it is possible to consider bond percolation when the bonds between sites are randomly occupied. An example of bond percolation in physics is a random resistor network, where the metallic wires in a regular network are cut at random. If sites are occupied with probability p and bonds are occupied with probability q, we speak of site-bond percolation. Two occupied sites belong to the same cluster if they are connected by a path of nearest-neighbor occupied sites with occupied bonds in between. The definitions of site and bond percolation on a square lattice can easily be generalized to any lattice in d-dimensions. In general, in a given lattice, a bond has more nearest neighbors than a site. Thus, large clusters of bonds can be formed more effectively than large clusters of sites, and therefore, on a given lattice, the percolation threshold for bonds is smaller than the percolation threshold for sites (see Table 1). A natural example of percolation, is continuum percolation, where the positions of the two components of Fractals and Percolation, Table 1 Percolation thresholds for the Cayley tree and several two- and three-dimensional lattices (see Refs. [8,14,41,64,75] and references therein)
Lattice Triangular Square Honeycomb Face Centered Cubic Body Centered Cubic Simple Cubic (1st nn) Simple Cubic (2nd nn) Simple Cubic (3rd nn) Cayley Tree
Percolation of Sites Bonds 1/2 2 sin(/18) 0.5927460 1/2 0.6962 1 2 sin(/18) 0.198 0.119 0.245 0.1803 0.31161 0.248814 0.137 – 0.097 – 1/(z 1) 1/(z 1)
Fractals and Percolation, Figure 4 Continuum percolation: Swiss cheese model
a random mixture are not restricted to the discrete sites of a regular lattice [9,73]. As a simple example, consider a sheet of conductive material, with circular holes punched randomly in it (Swiss cheese model, see Fig. 4). The relevant quantity now is the fraction p of remaining conductive material. Hopping Percolation Above, we have discussed traditional percolation with only two values of local conductivities, 0 and 1 (insulator– conductor) or 1 and 1 (superconductor–normal conductor). However, quantum systems should be treated by hopping conductivity, which can be described by an exponential function representing the local conductivities (between ith and jth sites): i j exp(x i j ). Here can be interpreted as the dimensionless mean hopping distance or as the degree of disorder (the smaller the density of the deposited grains, the larger becomes), and xij is a random number taken from a uniform distribution in the range (0,1) [70]. In contrast to the traditional bond (or site) percolation model, in which the system is either a metal or an insulator, in the hopping percolation model the system always conducts some current. However, there are two regimes of such percolation [70]. A regime with many conducting paths which is not sensitive to the removal of a single bond (weak disorder L/ > 1, where L is size of the system and is percolation critical exponent) and a regime with a single or only a few dominating conducting paths which is very sensitive to the removal of a specific single bond with the highest current (strong disorder L/ 1). In the strong disorder regime, the trajectories along which
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P1 can be identified as the order parameter similar to magnetization, m(T) (Tc T)ˇ , in magnetic materials. With decreasing temperature, T, more elementary magnetic moments (spins) become aligned in the same direction, and the system becomes more ordered. The linear size of the finite clusters, below and above pc , is characterized by correlation length . Correlation length is defined as the mean distance between two sites on the same finite cluster. When p approaches pc , increases as jp pc j ;
(2)
with the same exponent below and above the threshold. The mean number of sites (mass) of a finite cluster also diverges, S jp pc j ;
Fractals and Percolation, Figure 5 A color density plot of the current distribution in a bond-percolating lattice for which voltage is applied in the vertical direction for strong disorder with D 10. The current between the sites is shown by the different colors (orange corresponds to the highest value, green to the lowest). a The location of the resistor, on which the value of the local current is maximal, is shown by a circle. b The current distribution after removing the above resistor. This removal results in a significant change of the current trajectories
(3)
again with the same exponent above and below pc . Analogous to S in magnetic systems is the susceptibility (see Fig. 6 and Table 2). The exponents ˇ, , and describe the critical behavior of typical quantities associated with the percolation transition, and are called critical exponents. The exponents are universal and do not depend on the structural details of the lattice (e. g., square or triangular) nor on the type of percolation (site, bond, or continuum), but depend only on the dimension d of the lattice.
the highest current flows (analogous to the spanning cluster at criticality in the traditional percolation network, see Fig. 5) can be distinguished and a single bond can determine the transport properties of the entire macroscopic system. Percolation as a Critical Phenomenon In percolation, the concentration p of occupied sites plays the same role as temperature in thermal phase transitions. The percolation transition is a geometrical phase transition where the critical concentration pc separates a phase of finite clusters (p < pc ) from a phase where an infinite cluster is present (p > pc ). An important quantity is the probability P1 that a site (or a bond) belongs to the infinite cluster. For p < pc , only finite clusters exist, and P1 D 0. For p > pc , P1 increases with p by a power law P1 (p pc )ˇ :
(1)
Fractals and Percolation, Figure 6 P1 and S compared with magnetization M and susceptibility Fractals and Percolation, Table 2 Exact and best estimate values for the critical exponents for percolation (see Refs. [8,14,41,64] and references therein) Percolation Order parameter P1 : ˇ Correlation length : Mean cluster size S:
dD2 5/36 4/3 43/18
dD3 0:417 ˙ 0:003 0:875 ˙ 0:008 1:795 ˙ 0:005
d6 1 1/2 1
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This universality property is a general feature of phase transitions, where the order parameter vanishes continuously at the critical point (second order phase transition). In Table 2, the values of the critical exponents ˇ, , and for percolation in two, three, and six dimensions. The exponents considered here describe the geometrical properties of the percolation transition. The physical properties associated with this transition also show power-law behavior near pc and are characterized by critical exponents. Examples include the conductivity in a random resistor or random superconducting network and the spreading velocity of an epidemic disease near the critical infection probability. It is believed that the “dynamical” exponents cannot be generally related to the geometric exponents discussed above. Note that all quantities described above are defined in the thermodynamic limit of large systems. In a finite system, for example, P1 , is not strictly zero below pc .
Fractals and Percolation, Figure 7 Examples of regular systems with dimensions d D 1, d D 2, and dD3
fractals. Thus, to include fractal structures, (4) we can generalize M(bL) D b d f M(L) ; and M(L) D ALd f ;
Percolation Clusters as Fractals As first noticed by Stanley [66], the structure of percolation clusters (when the length scale is smaller than ) can be well described by the fractal concept [53]. Fractal geometry is a mathematical tool for dealing with complex structures that have no characteristic length scale. Scaleinvariant systems are usually characterized by noninteger (“fractal”) dimensions. This terminology is associated with B. Mandelbrot [53] (though some notion of noninteger dimensions and several basic properties of fractal objects were studied earlier by G. Cantor, G. Peano, D. Hilbert, H. von Koch, W. Sierpinski, G. Julia, F. Hausdorff, C. F. Gauss, and A. Dürer).
Sierpinski Gasket This fractal is generated by dividing a full triangle into four smaller triangles and removing the central triangle (see Fig. 8). In subsequent iterations, this procedure is repeated by dividing each of the remaining triangles into four smaller triangles and removing the central triangles. To obtain the fractal dimension, we consider
In regular systems (with uniform density) such as long wires, large thin plates, or large filled cubes, the dimension d characterizes how the mass M(L) changes with the linear size L of the system. If we consider a smaller part of a system of linear size bL (b < 1), then M(bL) is decreased by a factor of bd , i. e., (4)
The solution of the functional Eq. (4) is simply M(L) D ALd . For a long wire, mass changes linearly with b, i. e., d D 1. For the thin plates, we obtain d D 2, and for cubes d D 3; see Fig. 7. Mandelbrot coined the name “fractal dimension”, and those objects described by a fractal dimension are called
(6)
where df is the fractal dimension and can be a noninteger. Below, we present two examples of dealing with df : (i) the deterministic Sierpinski Gasket and (ii) random percolation clusters and criticality.
Fractal Dimension df
M(bL) D b d M(L) :
(5)
Fractals and Percolation, Figure 8 2D Sierpinski gasket. Generation and self-similarity
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Fractals and Percolation
the mass of the gasket within a linear size L and compare it with the mass within 12 L. Since M( 21 L) D 13 M(L), we have d f D log 3/ log 2 Š 1:585. Percolation Fractal We assume that at pc ( D 1) the clusters are fractals. Thus for p > pc , we expect length scales smaller than to have critical properties and therefore a fractal structure. For length scales larger than , one expects a homogeneous system which is composed of many unit cells of size : ( rd f ; r ; M(r) d (7) r ; r:
Shortest Path Dimensions, dmin and d` The fractal dimension, however, is not sufficient to fully characterize a percolation cluster, since two clusters with very different topologies may have the same fractal dimension df . As an additional characterization of a fractal, one can consider, e. g., the shortest path between two arbitrary sites A and B on the cluster (see Figs. 10, 11) [3,16,35, 42,55,58]. The structure formed by the sites of this path is also self-similar and is described by a fractal dimension dmin [46,67]. Accordingly, the length ` of the path, which is often called the “chemical distance”, scales with the “Eu-
For a demonstration of this feature see Fig. 9. One can relate the fractal dimension df of percolation clusters to the exponents ˇ and . The probability that an arbitrary site within a circle of radius r smaller than belongs to the infinite cluster, is the ratio between the number of sites on the infinite cluster and the total number of sites, P1 r d f /r d ;
r4 N 2/3 ; As shown above, for < 3, there is no percolation threshold, and therefore no spanning percolation cluster. Note that SF networks can be regarded as a generalization of ER networks, since for > 4 one obtains the ER network results. Summary and Future Directions The percolation problem and its numerous modifications can be useful in describing several physical, chemical, and biological processes, such as the spreading of epidemics or forest fires, gelation processes, and the invasion of water into oil in porous media, which is relevant for the process of recovering oil from porous rocks. In some cases, modi-
Fractals and Percolation
fication changes the universality class of a percolation. We begin with an example in which the universality class does not change. We showed that a random process such as percolation can lead naturally to fractal structures. This may be one of the reasons why fractals occur frequently in nature. Bibliography Primary Literature 1. Aharony A (1986) In: Grinstein G, Mazenko G (eds) Directions in condensed matter physics. World Scientific, Singapore 2. Albert R, Barabasi A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47 3. Alexandrowicz Z (1980) Phys Lett A 80:284 4. Alexander S, Orbach R (1982) J Phys Lett 43:L625 5. Alexander S, Bernasconi J, Schneider WR, Orbach R (1981) Rev Mod Phys 53:175; Alexander S (1983) In: Deutscher G, Zallen R, Adler J (eds) Percolation Structures and Processes. Adam Hilger, Bristol, p 149 6. Avnir D (ed) (1989) The fractal approach to heterogeneos chemistry. Wiley, Chichester 7. Ben-Avraham D, Havlin S (1982) J Phys A 15:L691; Havlin S, BenAvraham D, Sompolinsky H (1983) Phys Rev A 27: 1730 8. Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge 9. Benguigui L (1984) Phys Rev Lett 53:2028 10. Blumen A, Klafter J, Zumofen G (1986) In: Zschokke I (ed) Optical spectroscopy of glasses. Reidel, Dordrecht, pp 199–265 11. Bollobás B (1985) Random graphs. Academic Press, London 12. Bouchaud JP, Georges A (1990) Phys Rep 195:127 13. Bunde A (1986) Adv Solid State Phys 26:113 14. Bunde A, Havlin S (eds) (1996) Fractals and disordered systems, 2nd edn. Springer, Berlin; Bunde A, Havlin S (eds) (1995) Fractals in Science, 2nd edn. Springer, Berlin 15. Broadbent SR, Hammersley JM (1957) Proc Camb Phil Soc 53:629 16. Cardey JL, Grassberger P (1985) J Phys A 18:L267 17. Cardy J (1998) J Phys A 31:L105 18. Clerc JP, Giraud G, Laugier JM, Luck JM (1990) Adv Phys 39:191 19. Cohen R, Havlin S (2003) Scale-free networks are ultrasmall. Phys Rev Lett 90:058701 20. Cohen R, Havlin S (2008) Complex networks: Structure, stability and function. Cambridge University Press, Cambridge 21. Cohen R, Erez K, Ben-Avraham D, Havlin S (2000) Resilience of the internet to random breakdowns. Phys Rev Lett 85:4626 22. Coniglio A (1982) J Phys A 15:3829 23. Coniglio A (1982) Phys Rev Lett 46:250 24. de Gennes PG (1976) La Recherche 7:919 25. de Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Ithaca 26. Deutscher G, Zallen R, Adler J (eds) (1983) A collection of review articles: percolation structures and processes. Adam Hilger, Bristol 27. Domb C (1983) In: Deutscher G, Zallen R, Adler J (eds) Percolation structures and processes. Adam Hilger, Bristol; Domb C, Stoll E, Schneider T (1980) Contemp Phys 21: 577 28. Elam WT, Kerstein AR, Rehr JJ (1984) Phys Rev Lett 52:1515
˝ P, Rényi A (1959) On random graphs. Publicationes 29. Erdos Mathematicae 6:290; (1960) Publ Math Inst Hung Acad Sci 5:17 30. Essam JW (1980) Rep Prog Phys 43:843 31. Family F, Landau D (eds) (1984) Kinetics of aggregation and gelation. North Holland, Amsterdam; For a review on gelation see: Kolb M, Axelos MAV (1990) In: Stanley HE, Ostrowsky N (eds) Correlatios and Connectivity: Geometric Aspects of Physics, Chemistry and Biology. Kluwer, Dordrecht, p 225 32. Flory PJ (1971) Principles of polymer chemistry. Cornell University, New York; Flory PJ (1941) J Am Chem Soc 63:3083–3091– 3096; Stockmayer WH (1943) J Chem Phys 11:45 33. Gefen Y, Aharony A, Alexander S (1983) Phys Rev Lett 50:77 34. Gouyet JF (1992) Phys A 191:301 35. Grassberger P (1986) Math Biosci 62:157; (1985) J Phys A 18:L215; (1986) J Phys A 19:1681 36. Grassberger P (1992) J Phys A 25:5867 37. Grassberger P (1999) J Phys A 32:6233 38. Grossman T, Aharony A (1987) J Phys A 20:L1193 39. Haus JW, Kehr KW (1987) Phys Rep 150:263 40. Havlin S, Ben-Avraham D (1987) Adv Phys 36:695 41. Havlin S, Ben-Avraham D (1987) Diffusion in random media. Adv Phys 36:659 42. Havlin S, Nossal R (1984) Topological properties of percolation clusters. J Phys A 17:L427 43. Havlin S, Nossal R, Trus B, Weiss GH (1984) J Stat Phys A 17:L957 44. Herrmann HJ (1986) Phys Rep 136:153 45. Herrmann HJ, Stanley HE (1984) Phys Rev Lett 53:1121; Hong DC, Stanley HE (1984) J Phys A 16:L475 46. Herrmann HJ, Stanley HE (1988) J Phys A 21:L829 47. Herrmann HJ, Hong DC, Stanley HE (1984) J Phys A 17:L261 48. Kesten H (1982) Percolation theory for mathematicians. Birkhauser, Boston (A mathematical approach); Grimmet GR (1989) Percolation. Springer, New York 49. Kirkpatrick S (1979) In: Maynard R, Toulouse G (eds) Le Houches Summer School on Ill Condensed Matter. North Holland, Amsterdam 50. Kopelman R (1976) In: Fong FK (ed) Topics in applied physics, vol 15. Springer, Heidelberg 51. Ma SK (1976) Modern theory of critical phenomena. Benjamin, Reading 52. Mackay G, Jan N (1984) J Phys A 17:L757 53. Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco; Mandelbrot BB (1977) Fractals: Form, Chance and Dimension. Freeman, San Francisco 54. Meakin P, Majid I, Havlin S, Stanley HE (1984) J Phys A 17:L975 55. Middlemiss KM, Whittington SG, Gaunt DC (1980) J Phys A 13:1835 56. Montroll EW, Shlesinger MF (1984) In: Lebowitz JL, Montroll EW (eds) Nonequilibrium phenomena II: from stochastics to hydrodynamics. Studies in Statistical Mechanics, vol 2. NorthHolland, Amsterdam 57. Paul G, Ziff RM, Stanley HE (2001) Phys Rev E 64:26115 58. Pike R, Stanley HE (1981) J Phys A 14:L169 59. Porto M, Bunde A, Havlin S, Roman HE (1997) Phys Rev E 56:1667 60. Ritzenberg AL, Cohen RI (1984) Phys Rev B 30:4036 61. Sahimi M (1993) Application of percolation theory. Taylor Francis, London 62. Saleur H, Duplantier B (1987) Phys Rev Lett 58:2325 63. Sapoval B, Rosso M, Gouyet JF (1985) J Phys Lett 46:L149
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64. Stauffer D, Aharony A (1994) Introduction to percolation theory, 2nd edn. Taylor Francis, London 65. Stanley HE (1971) Introduction to phase transition and critical phrenomena. Oxford University, Oxford 66. Stanley HE (1977) J Phys A 10:L211 67. Stanley HE (1984) J Stat Phys 36:843 68. Turcotte DL (1992) Fractals and chaos. In: Geology and geophysics. Cambridge University Press, Cambridge 69. Toulouse G (1974) Nuovo Cimento B 23:234 70. Tyc S, Halperin BI (1989) Phys Rev B 39:R877; Strelniker YM, Berkovits R, Frydman A, Havlin S (2004) Phys Rev E 69:R065105; Strelniker YM, Havlin S, Berkovits R, Frydman A (2005) Phys Rev E 72:016121; Strelniker YM (2006) Phys Rev B 73:153407 71. von Niessen W, Blumen A (1988) Canadian J For Res 18:805 72. Webman I (1991) Phys Rev Lett 47:1496 73. Webman I, Jortner J, Cohen MH (1976) Phys Rev B 14:4737 74. Weiss GH, Rubin RJ (1983) Adv Chem Phys 52:363; Weiss GH (1994) Aspects and applications of the random walk. North Holland, Amsterdam 75. Ziff RM (1992) Phys Rev Lett 69:2670 76. Ziff RM (1999) J Phys A 32:L457
Books and Reviews Bak P (1996) How nature works: The science of self organized criticality. Copernicus, New York Barabási A-L (2003) Linked: How everything is connected to every-
thing else and what it means for business, science and everyday life. Plume Bergman DJ, Stroud D (1992) Solid State Phys 46:147–269 Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: From biological nets to the internet and www (physics). Oxford University Press, Oxford Eglash R (1999) African fractals: Modern computing and indigenous design. Rutgers University Press, New Brunswick, NJ Feder J (1988) Fractals. Plenum, New York Gleick J (1997) Chaos. Penguin Books, New York Gould H, Tobochnik J (1988) An introduction to computer simulation methods. In: Application to physical systems. AddisonWesley, Reading, MA Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge Pastor-Satorras R, Vespignani A (2004) Evolution and structure of the internet: A statistical physics approach. Cambridge University Press, Cambridge Peitgen HO, Jurgens H, Saupe D (1992) Chaos and fractals. Springer, New York Peng G, Decheng T (1990) The fractal nature of a fracture surface. J Physics A 14:3257–3261 Pikovsky A, Rosenblum M, Kurths J, Chirikov B, Cvitanovic P, Moss F, Swinney H (2003) Synchronization: A universal concept in nonlinear sciences. Cambridge University Press, Cambridge Vicsek T (1992) Fractal growth phenomena. World Scientific, Singapore
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Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences? ALAIN ARNEODO1 , BENJAMIN AUDIT1 , EDWARD-BENEDICT BRODIE OF BRODIE1 , SAMUEL N ICOLAY2 , MARIE TOUCHON3,5 , YVES D’AUBENTON-CARAFA 4 , MAXIME HUVET4 , CLAUDE THERMES4 1 Laboratoire Joliot–Curie and Laboratoire de Physique, ENS-Lyon CNRS, Lyon Cedex, France 2 Institut de Mathématique, Université de Liège, Liège, Belgium 3 Génétique des Génomes Bactériens, Institut Pasteur, CNRS, Paris, France 4 Centre de Génétique Moléculaire, CNRS, Gif-sur-Yvette, France 5 Atelier de Bioinformatique, Université Pierre et Marie Curie, Paris, France Article Outline Glossary Definition of the Subject Introduction A Wavelet-Based Multifractal Formalism: The Wavelet Transform Modulus Maxima Method Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome Future Directions Acknowledgments Bibliography Glossary Fractal Fractals are complex mathematical objects that are invariant with respect to dilations (self-similarity) and therefore do not possess a characteristic length scale. Fractal objects display scale-invariance properties that can either fluctuate from point to point (multifractal) or be homogeneous (monofractal). Mathematically, these properties should hold over all scales.
However, in the real world, there are necessarily lower and upper bounds over which self-similarity applies. Wavelet transform The continuous wavelet transform (WT) is a mathematical technique introduced in the early 1980s to perform time-frequency analysis. The WT has been early recognized as a mathematical microscope that is well adapted to characterize the scaleinvariance properties of fractal objects and to reveal the hierarchy that governs the spatial distribution of the singularities of multifractal measures and functions. More specifically, the WT is a space-scale analysis which consists in expanding signals in terms of wavelets that are constructed from a single function, the analyzing wavelet, by means of translations and dilations. Wavelet transform modulus maxima method The WTMM method provides a unified statistical (thermodynamic) description of multifractal distributions including measures and functions. This method relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform modulus maxima (WTMM). This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the D(h) singularity spectrum as the equivalent of a thermodynamic potential (entropy). With some appropriate choice of the analyzing wavelet, one can show that the WTMM method provides a natural generalization of the classical box-counting and structure function techniques. Compositional strand asymmetry The DNA double helix is made of two strands that are maintained together by hydrogen bonds involved in the base-pairing between Adenine (resp. Guanine) on one strand and Thymine (resp. Cytosine) on the other strand. Under no-strand bias conditions, i. e. when mutation rates are identical on the two strands, in other words when the two strands are strictly equivalent, one expects equimolarities of adenine and thymine and of guanine and cytosine on each DNA strand, a property named Chargaff’s second parity rule. Compositional strand asymmetry refers to deviations from this rule which can be assessed by measuring departure from intrastrand equimolarities. Note that two major biological processes, transcription and replication, both requiring the opening of the double helix, actually break the symmetry between the two DNA strands and can thus be at the origin of compositional strand asymmetries. Eukaryote Organisms whose cells contain a nucleus, the structure containing the genetic material arranged into
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
chromosomes. Eukaryotes constitute one of the three domains of life, the two others, called prokaryotes (without nucleus), being the eubacteria and the archaebacteria. Transcription Transcription is the process whereby the DNA sequence of a gene is enzymatically copied into a complementary messenger RNA. In a following step, translation takes place where each messenger RNA serves as a template to the biosynthesis of a specific protein. Replication DNA replication is the process of making an identical copy of a double-stranded DNA molecule. DNA replication is an essential cellular function responsible for the accurate transmission of genetic information though successive cell generations. This process starts with the binding of initiating proteins to a DNA locus called origin of replication. The recruitment of additional factors initiates the bi-directional progression of two replication forks along the chromosome. In eukaryotic cells, this binding event happens at a multitude of replication origins along each chromosome from which replication propagates until two converging forks collide at a terminus of replication. Chromatin Chromatin is the compound of DNA and proteins that forms the chromosomes in living cells. In eukaryotic cells, chromatin is located in the nucleus. Histones Histones are a major family of proteins found in eukaryotic chromatin. The wrapping of DNA around a core of 8 histones forms a nucleosome, the first step of eukaryotic DNA compaction. Definition of the Subject The continuous wavelet transform (WT) is a mathematical technique introduced in signal analysis in the early 1980s [1,2]. Since then, it has been the subject of considerable theoretical developments and practical applications in a wide variety of fields. The WT has been early recognized as a mathematical microscope that is well adapted to reveal the hierarchy that governs the spatial distribution of singularities of multifractal measures [3,4,5]. What makes the WT of fundamental use in the present study is that its singularity scanning ability equally applies to singular functions than to singular measures [3,4,5,6,7, 8,9,10,11]. This has led Alain Arneodo and his collaborators [12,13,14,15,16] to elaborate a unified thermodynamic description of multifractal distributions including measures and functions, the so-called Wavelet Transform Modulus Maxima (WTMM) method. By using wavelets instead of boxes, one can take advantage of the freedom in the choice of these “generalized oscillating boxes” to get
rid of possible (smooth) polynomial behavior that might either mask singularities or perturb the estimation of their strength h (Hölder exponent), remedying in this way for one of the main failures of the classical multifractal methods (e. g. the box-counting algorithms in the case of measures and the structure function method in the case of functions [12,13,15,16]). The other fundamental advantage of using wavelets is that the skeleton defined by the WTMM [10,11], provides an adaptative space-scale partitioning from which one can extract the D(h) singularity spectrum via the Legendre transform of the scaling exponents (q) (q real, positive as well as negative) of some partition functions defined from the WT skeleton. We refer the reader to Bacry et al. [13], Jaffard [17,18] for rigorous mathematical results and to Hentschel [19] for the theoretical treatment of random multifractal functions. Applications of the WTMM method to 1D signals have already provided insights into a wide variety of problems [20], e. g., the validation of the log-normal cascade phenomenology of fully developed turbulence [21,22,23, 24] and of high-resolution temporal rainfall [25,26], the characterization and the understanding of long-range correlations in DNA sequences [27,28,29,30], the demonstration of the existence of causal cascade of information from large to small scales in financial time series [31,32], the use of the multifractal formalism to discriminate between healthy and sick heartbeat dynamics [33,34], the discovery of a Fibonacci structural ordering in 1D cuts of diffusion limited aggregates (DLA) [35,36,37,38]. The canonical WTMM method has been further generalized from 1D to 2D with the specific goal to achieve multifractal analysis of rough surfaces with fractal dimensions D F anywhere between 2 and 3 [39,40,41]. The 2D WTMM method has been successfully applied to characterize the intermittent nature of satellite images of the cloud structure [42,43], to perform a morphological analysis of the anisotropic structure of atomic hydrogen (H I ) density in Galactic spiral arms [44] and to assist in the diagnosis in digitized mammograms [45]. We refer the reader to Arneodo et al. [46] for a review of the 2D WTMM methodology, from the theoretical concepts to experimental applications. In a recent work, Kestener and Arneodo [47] have further extended the WTMM method to 3D analysis. After some convincing test applications to synthetic 3D monofractal Brownian fields and to 3D multifractal realizations of singular cascade measures as well as their random function counterpart obtained by fractional integration, the 3D WTMM method has been applied to dissipation and enstrophy 3D numerical data issued from direct numerical simulations (DNS) of isotropic turbulence. The results so-obtained have revealed that the multifractal spatial structure of both
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dissipation and enstrophy fields are likely to be well described by a multiplicative cascade process clearly nonconservative. This contrasts with the conclusions of previous box-counting analysis [48] that failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master non-conservative singular cascade measures [47]. For many years, the multifractal description has been mainly devoted to scalar measures and functions. However, in physics as well as in other fundamental and applied sciences, fractals appear not only as deterministic or random scalar fields but also as vector-valued deterministic or random fields. Very recently, Kestener and Arneodo [49,50] have combined singular value decomposition techniques and WT analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method has been applied to turbulent velocity and vorticity fields generated in (256)3 DNS of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship Dv (h C 1) D D! (h) between the singularity spectra of these two vector fields that are found significantly more intermittent that previously estimated from longitudinal and transverse velocity increment statistics. Furthermore, thanks to the singular value decomposition, the TWTMM method looks very promising for future simultaneous multifractal and structural (vorticity sheets, vorticity filaments) analysis of turbulent flows [49,50]. Introduction The possible relevance of scale invariance and fractal concepts to the structural complexity of genomic sequences has been the subject of considerable increasing interest [20,51,52]. During the past fifteen years or so, there has been intense discussion about the existence, the nature and the origin of the long-range correlations (LRC) observed in DNA sequences. Different techniques including mutual information functions [53,54], auto-correlation functions [55,56], power-spectra [54,57,58], “DNA walk” representation [52,59], Zipf analysis [60,61] and entropies [62,63], were used for the statistical analysis of DNA sequences. For years there has been some permanent debate on rather struggling questions like the fact that the reported LRC might be just an artifact of the compositional heterogeneity of the genome organization [20,27,52, 55,56,64,65,66,67]. Another controversial issue is whether or not LRC properties are different for protein-coding (exonic) and non-coding (intronic, intergenic) sequences [20, 27,52,54,55,56,57,58,59,61,68]. Actually, there were many
objective reasons for this somehow controversial situation. Most of the pioneering investigations of LRC in DNA sequences were performed using different techniques that all consisted in measuring power-law behavior of some characteristic quantity, e. g., the fractal dimension of the DNA walk, the scaling exponent of the correlation function or the power-law exponent of the power spectrum. Therefore, in practice, they all faced the same difficulties, namely finite-size effects due to the finiteness of the sequence [69, 70,71] and statistical convergence issue that required some precautions when averaging over many sequences [52, 65]. But beyond these practical problems, there was also a more fundamental restriction since the measurement of a unique exponent characterizing the global scaling properties of a sequence failed to resolve multifractality [27], and thus provided very poor information upon the nature of the underlying LRC (if they were any). Actually, it can be shown that for a homogeneous (monofractal) DNA sequence, the scaling exponents estimated with the techniques previously mentioned, can all be expressed as a function of the so-called Hurst or roughness exponent H of the corresponding DNA walk landscape [20, 27,52]. H D 1/2 corresponds to classical Brownian, i. e. uncorrelated random walk. For any other value of H, the steps (increments) are either positively correlated (H > 1/2: Persistent random walk) or anti-correlated (H < 1/2: Anti-persistent random walk). One of the main obstacles to LRC analysis in DNA sequences is the genuine mosaic structure of these sequences which are well known to be formed of “patches” of different underlying composition [72,73,74]. When using the “DNA walk” representation, these patches appear as trends in the DNA walk landscapes that are likely to break scale-invariance [20,52,59,64,65,66,67,75,76]. Most of the techniques, e. g. the variance method, used for characterizing the presence of LRC are not well adapted to study non-stationary sequences. There have been some phenomenological attempts to differentiate local patchiness from LRC using ad hoc methods such as the so-called “min-max method” [59] and the “detrended fluctuation analysis” [77]. In previous works [27,28], the WT has been emphasized as a well suited technique to overcome this difficulty. By considering analyzing wavelets that make the WT microscope blind to low-frequency trends, any bias in the DNA walk can be removed and the existence of powerlaw correlations with specific scale invariance properties can be revealed accurately. In [78], from a systematic WT analysis of human exons, CDSs and introns, LRC were found in non-coding sequences as well as in coding regions somehow hidden in their inner codon structure. These results made rather questionable the model based
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
on genome plasticity proposed at that time to account for the reported absence of LRC in coding sequences [27,28, 52,54,59,68]. More recently, some structural interpretation of these LRC has emerged from a comparative multifractal analysis of DNA sequences using structural coding tables based on nucleosome positioning data [29,30]. The application of the WTMM method has revealed that the corresponding DNA chain bending profiles are monofractal (homogeneous) and that there exists two LRC regimes. In the 10–200 bp range, LRC are observed for eukaryotic sequences as quantified by a Hurst exponent value H ' 0:6 (but not for eubacterial sequences for which H D 0:5) as the signature of the nucleosomal structure. These LRC were shown to favor the autonomous formation of small (a few hundred bps) 2D DNA loops and in turn the propensity of eukaryotic DNA to interact with histones to form nucleosomes [79,80]. In addition, these LRC might induce some local hyperdiffusion of these loops which would be a very attractive interpretation of the nucleosomal repositioning dynamics. Over larger distances (& 200 bp), stronger LRC with H ' 0:8 seem to exist in any sequence [29,30]. These LRC are actually observed in the S. cerevisiae nucleosome positioning data [81] suggesting that they are involved in the nucleosome organization in the so-called 30 nm chromatin fiber [82]. The fact that this second regime of LRC is also present in eubacterial sequences shows that it is likely to be a possible key to the understanding of the structure and dynamics of both eukaryotic and prokaryotic chromatin fibers. In regards to their potential role in regulating the hierarchical structure and dynamics of chromatin, the recent report [83] of sequence-induced LRC effects on the conformations of naked DNA molecules deposited onto mica surface under 2D thermodynamic equilibrium observed by Atomic Force Microscopy (AFM) is a definite experimental breakthrough. Our purpose here is to take advantage of the availability of fully sequenced genomes to generalize the application of the WTMM method to genome-wide multifractal sequence analysis when using codings that have a clear functional meaning. According to the second parity rule [84,85], under no strand-bias conditions, each genomic DNA strand should present equimolarities of adenines A and thymines T and of guanines G and cytosines C [86,87]. Deviations from intrastrand equimolarities have been extensively studied during the past decade and the observed skews have been attributed to asymmetries intrinsic to the replication and transcription processes that both require the opening of the double helix. Actually, during these processes mutational events can affect the two strands differently and an asymmetry can
result if one strand undergoes different mutations, or is repaired differently than the other strand. The existence of transcription and/or replication associated strand asymmetries has been mainly established for prokaryote, organelle and virus genomes [88,89,90,91,92,93,94]. For a long time the existence of compositional biases in eukaryotic genomes has been unclear and it is only recently that (i) the statistical analysis of eukaryotic gene introns have revealed the presence of transcription-coupled strand asymmetries [95,96,97] and (ii) the genome wide multiscale analysis of mammalian genomes has clearly shown some departure from intrastrand equimolarities in intergenic regions and further confirmed the existence of replication-associated strand asymmetries [98,99,100]. In this manuscript, we will review recent results obtained when using the WT microscope to explore the scale invariance properties of the TA and GC skew profiles in the 22 human autosomes [98,99,100]. These results will enlighten the richness of information that can be extracted from these functional codings of DNA sequences including the prediction of 1012 putative human replication origins. In particular, this study will reveal a remarkable human gene organization driven by the coordination of transcription and replication [101]. A Wavelet-Based Multifractal Formalism The Continuous Wavelet Transform The WT is a space-scale analysis which consists in expanding signals in terms of wavelets which are constructed from a single function, the analyzing wavelet , by means of translations and dilations. The WT of a real-valued function f is defined as [1,2]: T [ f ] (x0 ; a) D
1 a
C1 Z
f (x)
x x
1
0
a
dx ;
(1)
where x0 is the space parameter and a (> 0) the scale parameter. The analyzing wavelet is generally chosen to be well localized in both space and frequency. Usually is required to be of zero mean for the WT to be invertible. But for the particular purpose of singularity tracking that is of interest here, we will further require to be orthogonal to low-order polynomials in Fig. 1 [7,8,9,10,11,12,13, 14,15,16]: C1 Z
x m (x)dx ;
0m 0, so that for any point x in a neighborhood of x0 , one has [7,8,9,10,11,13,16]: j f (x) Pn (x x0 )j C jx x0 j h :
(3)
If f is n times continuously differentiable at the point x0 , then one can use for the polynomial Pn (x x0 ), the ordern Taylor series of f at x0 and thus prove that h(x0 ) > n. Thus h(x0 ) measures how irregular the function f is at the point x0 . The higher the exponent h(x0 ), the more regular the function f . The main interest in using the WT for analyzing the regularity of a function lies in its ability to be blind to polynomial behavior by an appropriate choice of the analyzing wavelet . Indeed, let us assume that according to Eq. (3), f has, at the point x0 , a local scaling (Hölder) exponent h(x0 ); then, assuming that the singularity is not oscillating [11,102,103], one can easily prove that the local behavior of f is mirrored by the WT which locally behaves
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 2 WT of monofractal and multifractal stochastic signals. Fractional Brownian motion: a a realization of B1/3 (L D 65 536); c WT of B1/3 as coded, independently at each scale a, using 256 colors from black (jT j D 0) to red (maxb jT j); e WT skeleton defined by the set of all the maxima lines. Log-normal random W -cascades: b a realization of the log-normal W -cascade model (L D 65 536) with the following parameter values m D 0:355 ln 2 and 2 D 0:02 ln 2 (see [116]); d WT of the realization in b represented with the same color coding as in c; f WT skeleton. The analyzing wavelet is g(1) (see Fig. 1a)
like [7,8,9,10,11,12,13,14,15,16,17,18]: T [ f ](x0 ; a) a h(x 0 ) ;
a ! 0C ;
(4)
provided n > h(x0 ), where n is the number of vanishing moments of (Eq. (2)). Therefore one can extract the exponent h(x0 ) as the slope of a log-log plot of the WT amplitude versus the scale a. On the contrary, if one chooses
n < h(x0 ), the WT still behaves as a power-law but with a scaling exponent which is n : T [ f ](x0 ; a) a n ;
a ! 0C :
(5)
Thus, around a given point x0 , the faster the WT decreases when the scale goes to zero, the more regular f is around
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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
that point. In particular, if f 2 C 1 at x0 (h(x0 ) D C1), then the WT scaling exponent is given by n , i. e. a value which is dependent on the shape of the analyzing wavelet. According to this observation, one can hope to detect the points where f is smooth by just checking the scaling behavior of the WT when increasing the order n of the analyzing wavelet [12,13,14,15,16]. Remark 1 A very important point (at least for practical purpose) raised by Mallat and Hwang [10] is that the local scaling exponent h(x0 ) can be equally estimated by looking at the value of the WT modulus along a maxima line converging towards the point x0 . Indeed one can prove that both Eqs. (4) and (5) still hold when following a maxima line from large down to small scales [10,11]. The Wavelet Transform Modulus Maxima Method As originally defined by Parisi and Frisch [104], the multifractal formalism of multi-affine functions amounts to compute the so-called singularity spectrum D(h) defined as the Hausdorff dimension of the set where the Hölder exponent is equal to h [12,13,16]: D(h) D dimH fx; h(x) D hg ;
(6)
where h can take, a priori, positive as well as negative real values (e. g., the Dirac distribution ı(x) corresponds to the Hölder exponent h(0) D 1) [17]. A natural way of performing a multifractal analysis of fractal functions consists in generalizing the “classical” multifractal formalism [105,106,107,108,109] using wavelets instead of boxes. By taking advantage of the freedom in the choice of the “generalized oscillating boxes” that are the wavelets, one can hope to get rid of possible smooth behavior that could mask singularities or perturb the estimation of their strength h. But the major difficulty with respect to box-counting techniques [48,106,110,111, 112] for singular measures, consists in defining a covering of the support of the singular part of the function with our set of wavelets of different sizes. As emphasized in [12,13, 14,15,16], the branching structure of the WT skeletons of fractal functions in the (x; a) half-plane enlightens the hierarchical organization of their singularities (Figs. 2e, 2f). The WT skeleton can thus be used as a guide to position, at a considered scale a, the oscillating boxes in order to obtain a partition of the singularities of f . The wavelet transform modulus maxima (WTMM) method amounts to compute the following partition function in terms of WTMM coefficients [12,13,14,15,16]: X ˇ q ˇ (7) Z(q; a) D sup ˇT [ f ](x; a0 )ˇ ; l 2L(a)
(x;a 0 )2l a 0 a
where q 2 R and the sup can be regarded as a way to define a scale adaptative “Hausdorff-like” partition. Now from the deep analogy that links the multifractal formalism to thermodynamics [12,113], one can define the exponent (q) from the power-law behavior of the partition function: Z(q; a) a(q) ;
a ! 0C ;
(8)
where q and (q) play respectively the role of the inverse temperature and the free energy. The main result of this wavelet-based multifractal formalism is that in place of the energy and the entropy (i. e. the variables conjugated to q and ), one has h, the Hölder exponent, and D(h), the singularity spectrum. This means that the singularity spectrum of f can be determined from the Legendre transform of the partition function scaling exponent (q) [13,17, 18]: D(h) D min(qh (q)) :
(9)
q
From the properties of the Legendre transform, it is easy to see that homogeneous monofractal functions that involve singularities of unique Hölder exponent h D @/@q, are characterized by a (q) spectrum which is a linear function of q. On the contrary, a nonlinear (q) curve is the signature of nonhomogeneous functions that exhibit multifractal properties, in the sense that the Hölder exponent h(x) is a fluctuating quantity that depends upon the spatial position x. Defining Our Battery of Analyzing Wavelets There are almost as many analyzing wavelets as applications of the continuous WT [3,4,5,12,13,14,15,16]. In the present work, we will mainly used the class of analyzing wavelets defined by the successive derivatives of the Gaussian function: g (N) (x) D
d N x 2 /2 e ; dx N
(10)
for which n D N and more specifically g (1) and g (2) that are illustrated in Figs. 1a, 1b. Remark 2 The WT of a signal f with g (N) (Eq. (10)) takes the following simple expression: 1 Tg (N) [ f ](x; a) D a D
C1 Z
f (y)g (N)
1 dN aN N dx
y x a
Tg (0) [ f ](x; a) :
dy ; (11)
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Equation (11) shows that the WT computed with g (N) at scale a is nothing but the Nth derivative of the signal f (x) smoothed by a dilated version g (0) (x/a) of the Gaussian function. This property is at the heart of various applications of the WT microscope as a very efficient multi-scale singularity tracking technique [20]. With the specific goal of disentangling the contributions to the nucleotide composition strand asymmetry coming respectively from transcription and replication processes, we will use in Sect. “A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome”, an adapted analyzing wavelet of the following form (Fig. 1c) [101,114]:
1 1 1 R (x) D x ; for x 2 ; 2 2 2 (12) D 0 elsewhere : By performing multi-scale pattern recognition in the (space, scale) half-plane with this analyzing wavelet, we will be able to define replication domains bordered by putative replication origins in the human genome and more generally in mammalian genomes [101,114]. Test Applications of the WTMM Method on Monofractal and Multifractal Synthetic Random Signals This section is devoted to test applications of the WTMM method to random functions generated either by additive models like fractional Brownian motions [115] or by multiplicative models like random W -cascades on wavelet dyadic trees [21,22,116,117]. For each model, we first wavelet transform 1000 realizations of length L D 65 536 with the first order (n D 1) analyzing wavelet g (1) . From the WT skeletons defined by the WTMM, we compute the mean partition function (Eq. (7)) from which we extract the annealed (q) (Eq. (8)) and, in turn, D(h) (Eq. (9)) multifractal spectra. We systematically test the robustness of our estimates with respect to some change of the shape of the analyzing wavelet, in particular when increasing the number n of zero moments, going from g (1) to g (2) (Eq. (10)). Fractional Brownian Signals Since its introduction by Mandelbrot and van Ness [115], the fractional Brownian motion (fBm) B H has become a very popular model in signal and image processing [16,20,39]. In 1D, fBm has proved useful for modeling various physical phenomena with long-range dependence, e. g., “1/ f ” noises. The fBm exhibits a power spectral density S(k) 1/k ˇ , where the
spectral exponent ˇ D 2H C 1 is related to the Hurst exponent H. fBm has been extensively used as test stochastic signals for Hurst exponent measurements. In Figs. 2, 3 and 4, we report the results of a statistical analysis of fBm’s using the WTMM method [12,13,14,15,16]. We mainly concentrate on B1/3 since it has a k 5/3 power-spectrum similar to the spectrum of the multifractal stochastic signal we will study next. Actually, our goal is to demonstrate that, where the power spectrum analysis fails, the WTMM method succeeds in discriminating unambiguously between these two fractal signals. The numerical signals were generated by filtering uniformly generated pseudo-random noise in Fourier space in order to have the required k 5/3 spectral density. A B1/3 fractional Brownian trail is shown in Fig. 2a. Figure 2c illustrates the WT coded, independently at each scale a, using 256 colors. The analyzing wavelet is g (1) (n D 1). Figure 3a displays some plots of log2 Z(q; a) versus log2 (a) for different values of q, where the partition function Z(q; a) has been computed on the WTMM skeleton shown in Fig. 2e, according to the definition (Eq. (7)). Using a linear regression fit, we then obtain the slopes (q) of these graphs. As shown in Fig. 3c, when plotted versus q, the data for the exponents (q) consistently fall on a straight line that is remarkably fitted by the theoretical prediction: (q) D qH 1 ;
(13)
with H D 1/3. From the Legendre transform of this linear (q) (Eq. (9)), one gets a D(h) singularity spectrum that reduces to a single point: D(h) D 1
if h D H ;
D 1 if h ¤ H :
(14)
Thus, as expected theoretically [16,115], one finds that the fBm B1/3 is a nowhere differentiable homogeneous fractal signal with a unique Hölder exponent h D H D 1/3. Note that similar good estimates are obtained when using analyzing wavelets of different order (e. g. g (2) ), and this whatever the value of the index H of the fBm [12,13,14,15, 16]. Within the perspective of confirming the monofractality of fBm’s, we have studied the probability density function (pdf) of wavelet coefficient values a (Tg (1) (:; a)), as computed at a fixed scale a in the fractal scaling range. According to the monofractal scaling properties, one expects these pdfs to satisfy the self-similarity relationship [20, 27,28]: a H a (a H T) D (T) ;
(15)
613
614
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 3 Determination of the (q) and D(h) multifractal spectra of fBm B1/3 (red circles) and log-normal random W -cascades (green dots) using the WTMM method. a log2 Z(q; a) vs. log2 a: B1/3 . b log2 Z(q; a) vs. log2 a: Log-normal W -cascades with the same parameters as in Fig. 2b. c (q) vs. q; the solid lines correspond respectively to the theoretical spectra (13) and (16). d D(h) vs. h; the solid lines correspond respectively to the theoretical predictions (14) and (17). The analyzing wavelet is g(1) . The reported results correspond to annealed averaging over 1000 realizations of L D 65 536
where (T) is a “universal” pdf (actually the pdf obtained at scale a D 1) that does not depend on the scale parameter a. As shown in Figs. 4a, 4a0 for B1/3 , when plotting a H a (a H T) vs. T, all the a curves corresponding to different scales (Fig. 4a) remarkably collapse on a unique curve when using a unique exponent H D 1/3 (Fig. 4a0 ). Furthermore the so-obtained universal curve cannot be distinguished from a parabola in semi-log representation
as the signature of the monofractal Gaussian statistics of fBm fluctuations [16,20,27]. Random W -Cascades Multiplicative cascade models have enjoyed increasing interest in recent years as the paradigm of multifractal objects [16,19,48,105,107, 108,118]. The notion of cascade actually refers to a selfsimilar process whose properties are defined multiplica-
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 4 Probability distribution functions of wavelet coefficient values of fBm B1/3 (open symbols) and log-normal random W -cascades (filled symbols) with the same parameters as in Fig. 2b. a a vs. Tg(1) for the set of scales a D 10 (4), 50 (), 100 (), 1000 (˙), 9000 (O); a0 aH a (aH Tg(1) ) vs. Tg(1) with H D 1/3; The symbols have the same meaning as in a. b a vs. Tg(1) for the set of scales a D 10 (▲), 50 (■), 100 (●), 1000 (◆), 9000 (▼); (b0 ) aH a (aH Tg(1) ) vs. Tg(1) with H D m/ ln 2 D 0:355. The analyzing wavelet is g(1) (Fig. 1a)
tively from coarse to fine scales. In that respect, it occupies a central place in the statistical theory of turbulence [48, 104]. Originally, the concept of self-similar cascades was introduced to model multifractal measures (e. g. dissipation or enstrophy) [48]. It has been recently generalized to the construction of scale-invariant signals (e. g. longitudinal velocity, pressure, temperature) using orthogonal wavelet basis [116,119]. Instead of redistributing the measure over sub-intervals with multiplicative weights, one allocates the wavelet coefficients in a multiplicative way on the dyadic grid. This method has been implemented to generate multifractal functions (with weights W) from a given deterministic or probabilistic multiplicative process. Along the line of the modeling of fully developed turbulent signals by log-infinitely divisible multiplicative pro-
cesses [120,121], we will mainly concentrate here on the log-normal W -cascades in order to calibrate the WTMM method. If m and 2 are respectively the mean and the variance of ln W (where W is a multiplicative random variable with log-normal probability distribution), then, as shown in [116], a straightforward computation leads to the following (q) spectrum: (q) D log2 hW q i 1 ; D
2 2 ln 2
q2
8q 2 R
m q1; ln 2
(16)
where h: : :i means ensemble average. The corresponding D(h) singularity spectrum is obtained by Legendre
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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
transforming (q) (Eq. (9)): D(h) D
(h C m/ ln 2)2 C1: 2 2 / ln 2
Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription (17)
According to the convergence criteria established in [116], m andp 2 have to satisfy the conditions: m < 0 and jmj/ > 2 ln 2. Moreover, by solving D(h) D 0, one gets the following bounds for the supportpof thepD(h) singularity spectrum: hmin m/ ln 2 ( 2)/ ln 2 p D p and hmax D m/ ln 2 C ( 2)/ ln 2. In Fig. 2b is illustrated a realization of a log-normal W -cascade for the parameter values m D 0:355 ln 2 and 2 D 0:02 ln 2. The corresponding WT and WT skeleton as computed with g (1) are shown in Figs. 2d and 2f respectively. The results of the application of the WTMM method are reported in Fig. 3. As shown in Fig. 3b, when plotted versus the scale parameter a in a logarithmic representation, the annealed average of the partition functions Z(q; a) displays a well defined scaling behavior over a range of scales of about 5 octaves. Note that scaling of quite good quality is found for a rather wide range of q values: 5 q 10. When processing to a linear regression fit of the data over the first four octaves, one gets the (q) spectrum shown in Fig. 3c. This spectrum is clearly a nonlinear function of q, the hallmark of multifractal scaling. Moreover, the numerical data are in remarkable agreement with the theoretical quadratic prediction (Eq. (16)). Similar quantitative agreement is observed on the D(h) singularity spectrum in Fig. 3d which displays a single humped parabola shape that characterizes intermittent fluctuations corresponding to Hölder exponents values ranging from hmin D 0:155 to hmax D 0:555. Unfortunately, to capture the strongest and the weakest singularities, one needs to compute the (q) spectrum for very large values of jqj. This requires the processing of many more realizations of the considered log-normal random W -cascade. The multifractal nature of log-normal W -cascade realizations is confirmed in Figs. 4b, 4b0 where the self-similarity relationship (Eq. (15)) is shown not to apply. Actually there does not exist a H value allowing to superimpose onto a single curve the WT pdfs computed at different scales. The test applications reported in this section demonstrate the ability of the WTMM method to resolve multifractal scaling of 1D signals, a hopeless task for classical power spectrum analysis. They were used on purpose to calibrate and to test the reliability of our methodology, and of the corresponding numerical tools, with respect to finite-size effects and statistical convergence.
During genome evolution, mutations do not occur at random as illustrated by the diversity of the nucleotide substitution rate values [122,123,124,125]. This non-randomness is considered as a by-product of the various DNA mutation and repair processes that can affect each of the two DNA strands differently. Asymmetries of substitution rates coupled to transcription have been mainly observed in prokaryotes [88,89,91], with only preliminary results in eukaryotes. In the human genome, excess of T was observed in a set of gene introns [126] and some large-scale asymmetry was observed in human sequences but they were attributed to replication [127]. Only recently, a comparative analysis of mammalian sequences demonstrated a transcription-coupled excess of G+T over A+C in the coding strand [95,96,97]. In contrast to the substitution biases observed in bacteria presenting an excess of C!T transitions, these asymmetries are characterized by an excess of purine (A!G) transitions relatively to pyrimidine (T!C) transitions. These might be a by-product of the transcription-coupled repair mechanism acting on uncorrected substitution errors during replication [128]. In this section, we report the results of a genome-wide multifractal analysis of strand-asymmetry DNA walk profiles in the human genome [129]. This study is based on the computation of the TA and GC skews in non-overlapping 1 kbp windows: nT nA nG nC STA D ; SGC D ; (18) nT C nA nG C nC where nA , nC , nG and nT are respectively the numbers of A, C, G and T in the windows. Because of the observed correlation between the TA and GC skews, we also considered the total skew S D STA C SGC :
(19)
From the skews STA (n), SGC (n) and S(n), obtained along the sequences, where n is the position (in kbp units) from the origin, we also computed the cumulative skew profiles (or skew walk profiles): ˙TA (n) D
n X
STA ( j) ;
jD1
˙GC (n) D
n X
SGC ( j) ; (20)
jD1
and ˙ (n) D
n X
S( j) :
(21)
jD1
Our goal is to show that the skew DNA walks of the 22 human autosomes display an unexpected (with respect
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
to previous monofractal diagnosis [27,28,29,30]) bifractal scaling behavior in the range 10 to 40 kbp as the signature of the presence of transcription-induced jumps in the LRC noisy S profiles. Sequences and gene annotation data (“refGene”) were retrieved from the UCSC Genome Browser (May 2004). We used RepeatMasker to exclude repetitive elements that might have been inserted recently and would not reflect long-term evolutionary patterns. Revealing the Bifractality of Human Skew DNA Walks with the WTMM Method As an illustration of our wavelet-based methodology, we show in Fig. 5 the S skew profile of a fragment of human chromosome 6 (Fig. 5a), the corresponding skew DNA walk (Fig. 5b) and its space-scale wavelet decomposition using the Mexican hat analyzing wavelet g (2) (Fig. 1b). When computing Z(q; a) (Eq. (7)) from the WT skeletons of the skew DNA walks ˙ of the 22 human autosomes, we get convincing power-law behavior for 1:5 q 3 (data not shown). In Fig. 6a are reported the (q) exponents obtained using a linear regression fit of ln Z(q; a) vs. ln a over the range of scales 10 kbp a 40 kbp. All the data points remarkably fall on two straight lines 1 (q) D 0:78q 1 and 2 (q) D q 1 which strongly suggests the presence of two types of singularities h1 D 0:78 and h2 D 1, respectively on two sets S1 and S2 with the same Haussdorf dimension D D 1 (0) D 2 (0) D 1, as confirmed when computing the D(h) singularity spectrum in Fig. 6b. This observation means that Z(q; a) can be split in two parts [12,16]: Z(q; a) D C1 (q)a q h1 1 C C2 (q)a q h2 1 ;
(22)
where C1 (q) and C2 (q) are prefactors that depend on q. Since h1 < h2 , in the limit a 7! 0C , the partition function is expected to behave like Z(q; a) C1 (q)a q h1 1 for q > 0 and like Z(q; a) C2 (q)a q h2 1 for q < 0, with a so-called phase transition [12,16] at the critical value q c D 0. Surprisingly, it is the contribution of the weakest singularities h2 D 1 that controls the scaling behavior of Z(q; a) for q > 0 while the strongest ones h1 D 0:78 actually dominate for q < 0 (Fig. 6a). This inverted behavior originates from finite (1 kbp) resolution which prevents the observation of the predicted scaling behavior in the limit a 7! 0C . The prefactors C1 (q) and C2 (q) in Eq. (22) are sensitive to (i) the number of maxima lines in the WT skeleton along which the WTMM behave as a h1 or a h2 and (ii) the relative amplitude of these WTMM. Over the range of scales used to estimate (q), the WTMM along the maxima lines pointing (at small scale) to h2 D 1 singularities are significantly larger than those along the maxima lines associated
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 5 a Skew profile S(n) (Eq. (19)) of a repeat-masked fragment of human chromosome 6; red (resp. blue) 1 kbp window points correspond to (+) genes (resp. () genes) lying on the Watson (resp. Crick) strand; black points to intergenic regions. b Cumulated skew profile ˙ (n) (Eq. (21)). c WT of ˙ ; Tg(2) (n; a) is coded from black (min) to red (max); the WT skeleton defined by the maxima lines is shown in solid (resp. dashed) lines corresponding to positive (resp. negative) WT values. For illustration yellow solid (resp. dashed) maxima lines are shown to point to the positions of 2 upward (resp. 2 downward) jumps in S (vertical dashed lines in a and b) that coincide with gene transcription starts (resp. ends). In green are shown maxima lines that persist above a 200 kbp and that point to sharp upward jumps in S (vertical solid lines in a and b) that are likely to be the locations of putative replication origins (see Sect. “From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes”) [98,100]; note that 3 out of those 4 jumps are co-located with transcription start sites [129]
to h1 D 0:78 (see Figs. 6c, 6d). This implies that the larger q > 0, the stronger the inequality C2 (q) C1 (q) and the more pronounced the relative contribution of the second term in the r.h.s. of Eq. (22). On the opposite for q < 0, C1 (q) C2 (q) which explains that the strongest singularities h1 D 0:78 now control the scaling behavior of Z(q; a) over the explored range of scales. In Figs. 6c, 6d are shown the WTMM pdfs computed at scales a D 10, 20 and 40 kbp after rescaling by a h1 and a h2 respectively. We note that there does not exist a value of H such that all the pdfs collapse on a single curve as expected from Eq. (15) for monofractal DNA walks. Consistently with the (q) data in Fig. 6a and with the inverted
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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 6 Multifractal analysis of ˙ (n) of the 22 human (filled symbols) and 19 mouse (open circle) autosomes using the WTMM method with g(2) over the range 10 kbp a 40 kbp [129]. a (q) vs. q. b D(h) vs. h. c WTMM pdf: is plotted versus jTj/aH where H D h1 D 0:78, in semi-log representation; the inset is an enlargement of the pdf central part in linear representation. d Same as in c but with H D h2 D 1. In c and d, the symbols correspond to scales a D 10 (●), 20 (■) and 40 kbp (▲)
scaling behavior discussed above, when using the two exponents h1 D 0:78 and h2 D 1, one succeeds in superimposing respectively the central (bump) part (Fig. 6c) and the tail (Fig. 6d) of the rescaled WTMM pdfs. This corroborates the bifractal nature of the skew DNA walks that display two competing scale-invariant components of Hölder exponents: (i) h1 D 0:78 corresponds to LRC homogeneous fluctuations previously observed over the range 200 bp . a . 20 kbp in DNA walks generated with structural codings [29,30] and (ii) h2 D 1 is associated to convex _ and concave ^ shapes in the DNA walks ˙ indicating the presence of discontinuities in the derivative of ˙ , i. e., of jumps in S (Figs. 5a, 5b). At a given scale a, according to Eq. (11), a large value of the WTMM in Fig. 5c corresponds to a strong derivative of the smoothed S profile and the maxima line to which it belongs is likely to point to a jump location in S. This is particularly the case for the
colored maxima lines in Fig. 5c: Upward (resp. downward) jumps (Fig. 5a) are so-identified by the maxima lines corresponding to positive (resp. negative) values of the WT. Transcription-Induced Step-like Skew Profiles in the Human Genome In order to identify the origin of the jumps observed in the skew profiles, we have performed a systematic investigation of the skews observed along 14 854 intron containing genes [96,97]. In Fig. 7 are reported the mean values of STA and SGC skews for all genes as a function of the distance to the 50 - or 30 - end. At the 50 gene extremities (Fig. 7a), a sharp transition of both skews is observed from about zero values in the intergenic regions to finite positive values in transcribed regions ranging between 4 and 6% for S¯TA and between 3 and 5% for S¯GC . At the
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 7 TA (●) and GC (green ●) skew profiles in the regions surrounding 50 and 30 gene extremities [96]. STA and SGC were calculated in 1 kbp windows starting from each gene extremities in both directions. In abscissa is reported the distance (n) of each 1 kbp window to the indicated gene extremity; zero values of abscissa correspond to 50 - (a) or 30 - (b) gene extremities. In ordinate is reported the mean value of the skews over our set of 14 854 intron-containing genes for all 1 kbp windows at the corresponding abscissa. Error bars represent the standard error of the means
gene 30 - extremities (Fig. 7b), the TA and GC skews also exhibit transitions from significantly large values in transcribed regions to very small values in untranscribed regions. However, in comparison to the steep transitions observed at 50 - ends, the 30 - end profiles present a slightly smoother transition pattern extending over 5 kbp and including regions downstream of the 30 - end likely reflecting the fact that transcription continues to some extent downstream of the polyadenylation site. In pluricellular organisms, mutations responsible for the observed biases are expected to have mostly occurred in germ-line cells. It could happen that gene 30 - ends annotated in the databank differ from the poly-A sites effectively used in the germline cells. Such differences would then lead to some broadening of the skew profiles. From Skew Multifractal Analysis to Gene Detection In Fig. 8 are reported the results of a statistical analysis of the jump amplitudes in human S profiles [129]. For maxima lines that extend above a D 10 kbp in the WT skeleton (see Fig. 5c), the histograms obtained for upward and downward variations are quite similar, especially their tails that are likely to correspond to jumps in the S profiles (Fig. 8a). When computing the distance between upward or downward jumps (jSj 0:1) to the closest transcription start (TSS) or end (TES) sites (Fig. 8b), we reveal that the number of upward jumps in close proximity (jnj . 3 kpb) to TSS over-exceeds the number of such jumps close to TES. Similarly, downward jumps are preferentially located at TES. These observations are consistent with the step-like shape of skew profiles induced
by transcription: S > 0 (resp. S < 0) is constant along a (+) (resp. ()) genes and S D 0 in the intergenic regions (Fig. 7) [96]. Since a step-like pattern is edged by one upward and one downward jump, the set of human genes that are significantly biased is expected to contribute to an even number of S > 0 and S < 0 jumps when exploring the range of scales 10 . a . 40 kbp, typical of human gene size. Note that in Fig. 8a, the number of sharp upward jumps actually slightly exceeds the number of sharp downward jumps, consistently with the experimental observation that whereas TSS are well defined, TES may extend over 5 kbp resulting in smoother downward skew transitions (Fig. 7b). This TES particularity also explains the excess of upward jumps found close to TSS as compared to the number of downward jumps close to TES (Fig. 8b). In Fig. 9a, we report the analysis of the distance of TSS to the closest upward jump [129]. For a given upward jump amplitude, the number of TSS with a jump within jnj increases faster than expected (as compared to the number found for randomized jump positions) up to jnj ' 2 kbp. This indicates that the probability to find an upward jump within a gene promoter region is significantly larger than elsewhere. For example, out of 20 023 TSS, 36% (7228) are delineated within 2 kbp by a jump with S > 0:1. This provides a very reasonable estimate for the number of genes expressed in germline cells as compared to the 31.9% recently experimentally found to be bound to Pol II in human embryonic stem cells [130]. Combining the previous results presented in Figs. 8b and 9a, we report in Fig. 9b an estimate of the efficiency/coverage relationship by plotting the proportion of upward jumps (S > S ) lying in TSS proximity as
619
620
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 8 Statistical analysis of skew variations at the singularity positions determined at scale 1 kbp from the maxima lines that exist at scales a 10 kbp in the WT skeletons of the 22 human autosomes [129]. For each singularity, we computed the variation amplitudes ¯ 0 ) over two adjacent 5 kbp windows, respectively in the 30 and 50 directions and the distances n to the closest ¯ 0 ) S(5 S D S(3 TSS (resp. TES). a Histograms N(jSj) for upward (S > 0, red) and downward (S < 0, black) skew variations. b Histograms of the distances n of upward (red) or downward (black) jumps with jSj 0:1 to the closest TSS (●, red ●) and TES (, red )
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 9 a Number of TSS with an upward jump within jnj (abscissa) for jump amplitudes S > 0.1 (black), 0.15 (dark gray) and 0.2 (light gray). Solid lines correspond to true jump positions while dashed lines to the same analysis when jump positions were randomly drawn along each chromosome [129]. b Among the Ntot (S ) upward jumps of amplitude larger than some threshold S , we plot the proportion of those that are found within 1 kbp (●), 2 kbp (■) or 4 kbp (▲) of the closest TSS vs. the number NTSS of the sodelineated TSS. Curves were obtained by varying S from 0.1 to 0.3 (from right to left). Open symbols correspond to similar analyzes performed on random upward jump and TSS positions
a function of the number of so-delineated TSS [129]. For a given proximity threshold jnj, increasing S results in a decrease of the number of delineated TSS, characteristic of the right tail of the gene bias pdf. Concomitant to this decrease, we observe an increase of the efficiency up to a maximal value corresponding to some optimal value for S . For jnj < 2 kbp, we reach a maximal efficiency of 60% for S D 0:225; 1403 out of 2342 up-
ward jumps delineate a TSS. Given the fact that the actual number of human genes is estimated to be significantly larger ( 30 000) than the number provided by refGene, a large part of the the 40% (939) of upward jumps that have not been associated to a refGene could be explained by this limited coverage. In other words, jumps with sufficiently high amplitude are very good candidates for the location of highly-biased gene promoters. Let us point that out of
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
the above 1403 (resp. 2342) upward jumps, 496 (resp. 624) jumps are still observed at scale a D 200 kbp. We will see in the next section that these jumps are likely to also correspond to replication origins underlying the fact that large upward jumps actually result from the cooperative contributions of both transcription- and replication- associated biases [98,99,100,101]. The observation that 80% (496/624) of the predicted replication origins are co-located with TSS enlightens the existence of a remarkable gene organization at replication origins [101]. To summarize, we have demonstrated the bifractal character of skew DNA walks in the human genome. When using the WT microscope to explore (repeatmasked) scales ranging from 10 to 40 kbp, we have identified two competing homogeneous scale-invariant components characterized by Hölder exponents h1 D 0:78 and h2 D 1 that respectively correspond to LRC colored noise and sharp jumps in the original DNA compositional asymmetry profiles. Remarkably, the so-identified upward (resp. downward) jumps are mainly found at the TSS (resp. TES) of human genes with high transcription bias and thus very likely highly expressed. As illustrated in Fig. 6a, similar bifractal properties are also observed when investigating the 19 mouse autosomes. This suggests that the results reported in this section are general features of mammalian genomes [129]. From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes DNA replication is an essential genomic function responsible for the accurate transmission of genetic information through successive cell generations. According to the so-called “replicon” paradigm derived from prokaryotes [131], this process starts with the binding of some “initiator” protein to a specific “replicator” DNA sequence called origin of replication. The recruitment of additional factors initiate the bi-directional progression of two divergent replication forks along the chromosome. One strand is replicated continuously (leading strand), while the other strand is replicated in discrete steps towards the origin (lagging strand). In eukaryotic cells, this event is initiated at a number of replication origins and propagates until two converging forks collide at a terminus of replication [132]. The initiation of different replication origins is coupled to the cell cycle but there is a definite flexibility in the usage of the replication origins at different developmental stages [133,134,135,136, 137]. Also, it can be strongly influenced by the distance and timing of activation of neighboring replication ori-
gins, by the transcriptional activity and by the local chromatin structure [133,134,135,137]. Actually, sequence requirements for a replication origin vary significantly between different eukaryotic organisms. In the unicellular eukaryote Saccharomyces cerevisiae, the replication origins spread over 100–150 bp and present some highly conserved motifs [132]. However, among eukaryotes, S. cerevisiae seems to be the exception that remains faithful to the replicon model. In the fission yeast Schizosaccharomyces pombe, there is no clear consensus sequence and the replication origins spread over at least 800 to 1000 bp [132]. In multicellular organisms, the nature of initiation sites of DNA replication is even more complex. Metazoan replication origins are rather poorly defined and initiation may occur at multiple sites distributed over a thousand of base pairs [138]. The initiation of replication at random and closely spaced sites was repeatedly observed in Drosophila and Xenopus early embryo cells, presumably to allow for extremely rapid S phase, suggesting that any DNA sequence can function as a replicator [136,139,140]. A developmental change occurs around midblastula transition that coincides with some remodeling of the chromatin structure, transcription ability and selection of preferential initiation sites [136,140]. Thus, although it is clear that some sites consistently act as replication origins in most eukaryotic cells, the mechanisms that select these sites and the sequences that determine their location remain elusive in many cell types [141,142]. As recently proposed by many authors [143,144,145], the need to fulfill specific requirements that result from cell diversification may have led multicellular eukaryotes to develop various epigenetic controls over the replication origin selection rather than to conserve specific replication sequence. This might explain that only very few replication origins have been identified so far in multicellular eukaryotes, namely around 20 in metazoa and only about 10 in human [146]. Along the line of this epigenetic interpretation, one might wonder what can be learned about eukaryotic DNA replication from DNA sequence analysis. Replication Induced Factory-Roof Skew Profiles in Mammalian Genomes The existence of replication associated strand asymmetries has been mainly established in bacterial genomes [87,90, 92,93,94]. SGC and STA skews abruptly switch sign (over few kbp) from negative to positive values at the replication origin and in the opposite direction from positive to negative values at the replication terminus. This step-like profile is characteristic of the replicon model [131] (see Fig. 13, left panel). In eukaryotes, the existence of compo-
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sitional biases is unclear and most attempts to detect the replication origins from strand compositional asymmetry have been inconclusive. Several studies have failed to show compositional biases related to replication, and analysis of nucleotide substitutions in the region of the ˇ-globin replication origin in primates does not support the existence of mutational bias between the leading and the lagging strands [92,147,148]. Other studies have led to rather opposite results. For instance, strand asymmetries associated with replication have been observed in the subtelomeric regions of Saccharomyces cerevisiae chromosomes, supporting the existence of replication-coupled asymmetric mutational pressure in this organism [149]. As shown in Fig. 10a for the TOP1 replication origin [146], most of the known replication origins in the human genome correspond to rather sharp (over several kbp) transitions from negative to positive S (STA as well as SGC ) skew values that clearly emerge from the noisy background. But when examining the behavior of the skews at larger distances from the origin, one does not observe a step-like pattern with upward and downward jumps at the origin and termination positions, respectively, as expected for the bacterial replicon model (Fig. 13, left panel). Surprisingly, on both sides of the upward jump, the noisy S profile decreases steadily in the 50 to 30 direction without clear evidence of pronounced downward jumps. As shown in Figs. 10b–10d, sharp upward jumps of amplitude S & 15%, similar to the ones observed for the known replication origins (Fig. 10a), seem to exist also at many other locations along the human chromosomes. But the most striking feature is the fact that in between two neighboring major upward jumps, not only the noisy S profile does not present any comparable downward sharp transition, but it displays a remarkable decreasing linear behavior. At chromosome scale, we thus get jagged S profiles that have the aspect of “factory roofs” [98,100,146]. Note that the jagged S profiles shown in Figs. 10a–10d look somehow disordered because of the extreme variability in the distance between two successive upward jumps, from spacing 50–100 kbp ( 100–200 kbp for the native sequences) mainly in GC rich regions (Fig. 10d), up to 1–2 Mbp ( 2–3 Mbp for native sequences) (Fig. 10c) in agreement with recent experimental studies [150] that have shown that mammalian replicons are heterogeneous in size with an average size 500 kbp, the largest ones being as large as a few Mbp. But what is important to notice is that some of these segments between two successive skew upward jumps are entirely intergenic (Figs. 10a, 10c), clearly illustrating the particular profile of a strand bias resulting solely from replication [98,100,146]. In most other cases, we observe the superimposition of this replication
profile and of the step-like profiles of (+) and () genes (Fig. 7), appearing as upward and downward blocks standing out from the replication pattern (Fig. 10c). Importantly, as illustrated in Figs. 10e, 10f, the factory-roof pattern is not specific to human sequences but is also observed in numerous regions of the mouse and dog genomes [100]. Hence, the presence of strand asymmetry in regions that have strongly diverged during evolution further supports the existence of compostional bias associated with replication in mammalian germ-line cells [98,100,146]. Detecting Replication Origins from the Skew WT Skeleton We have shown in Fig. 10a that experimentally determined human replication origins coincide with large-amplitude upward transitions in noisy skew profiles. The corresponding S ranges between 14% and 38%, owing to possible different replication initiation efficiencies and/or different contributions of transcriptional biases (Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”). Along the line of the jump detection methodology described in Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”, we have checked that upward jumps observed in the skew S at these known replication origins correspond to maxima lines in the WT skeleton that extend to rather large scales a > a D 200 kbp. This observation has led us to select the maxima lines that exist above a D 200 kbp, i. e. a scale which is smaller than the typical replicon size and larger than the typical gene size [98,100]. In this way, we not only reduce the effect of the noise but we also reduce the contribution of the upward (50 extremity) and backward (30 extremity) jumps associated to the step-like skew pattern induced by transcription only (Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”), to the benefit of maintaining a good sensitivity to replication induced jumps. The detected jump locations are estimated as the positions at scale 20 kbp of the so-selected maxima lines. According to Eq. (11), upward (resp. downward) jumps are identified by the maxima lines corresponding to positive (resp. negative) values of the WT as illustrated in Fig. 5c by the green solid (resp. dashed) maxima lines. When applying this methodology to the total skew S along the repeat-masked DNA sequences of the 22 human autosomal chromosomes, 2415 upward jumps are detected and, as expected, a similar number (namely 2686) of downward jumps. In Fig. 11a are reported the histograms of the amplitude jSj of the so-identified upward (S > 0) and downward (S < 0) jumps respec-
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 10 S profiles along mammalian genome fragments [100,146]. a Fragment of human chromosome 20 including the TOP1 origin (red vertical line). b and c Human chromosome 4 and chromosome 9 fragments, respectively, with low GC content (36%). d Human chromosome 22 fragment with larger GC content (48%). In a and b, vertical lines correspond to selected putative origins (see Subsect. “Detecting Replication Origins from the Skew WT Skeleton”); yellow lines are linear fits of the S values between successive putative origins. Black intergenic regions; red, (+) genes; blue, () genes. Note the fully intergenic regions upstream of TOP1 in a and from positions 5290–6850 kbp in c. e Fragment of mouse chromosome 4 homologous to the human fragment shown in c. f Fragment of dog chromosome 5 syntenic to the human fragment shown in c. In e and f, genes are not represented
tively. These histograms no longer superimpose as previously observed at smaller scales in Fig. 8a, the former being significantly shifted to larger jSj values. When plotting N(jSj > S ) versus S in Fig. 11b, we can see that the number of large amplitude upward jumps overexceeds the number of large amplitude downward jumps.
These results confirm that most of the sharp upward transitions in the S profiles in Fig. 10 have no sharp downward transition counterpart [98,100]. This excess likely results from the fact that, contrasting with the prokaryote replicon model (Fig. 13, left panel) where downward jumps result from precisely positioned replication terminations,
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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 11 Statistical analysis of the sharp jumps detected in the S profiles of the 22 human autosomal chromosomes by the WT microscope at ¯ 0 ) S(5 ¯ 0 )j, where the averages were computed over the two scale a D 200 kbp for repeat-masked sequences [98,100]. jSj D jS(3 adjacent 20 kbp windows, respectively, in the 30 and 50 direction from the detected jump location. a Histograms N(jSj) of jSj values. b N(jSj > S ) vs. S . In a and b, the black (resp. red) line corresponds to downward S < 0 (resp. upward S > 0) jumps. R D 3 corresponds to the ratio of upward over downward jumps presenting an amplitude jSj 12:5% (see text)
in mammals termination appears not to occur at specific positions but to be randomly distributed. Accordingly the small number of downward jumps with large jSj is likely to result from transcription (Fig. 5) and not from replication. These jumps are probably due to highly biased genes that also generate a small number of large-amplitude upward jumps, giving rise to false-positive candidate replication origins. In that respect, the number of large downward jumps can be taken as an estimation of the number of false positives. In a first step, we have retained as acceptable a proportion of 33% of false positives. As shown in Fig. 11b, this value results from the selection of upward and downward jumps of amplitude jSj 12:5%, corresponding to a ratio of upward over downward jumps R D 3. Let us notice that the value of this ratio is highly variable along the chromosome [146] and significantly larger than 1 for GCC . 42%. In a final step, we have decided [98,100,146] to retain as putative replication origins upward jumps with jSj 12:5% detected in regions with G+C 42%. This selection leads to a set of 1012 candidates among which our estimate of the proportion of true replication origins is 79% (R D 4:76). In Fig. 12 is shown the mean skew profile calculated in intergenic windows on both sides of the 1012 putative replication origins [100]. This mean skew profile presents a rather sharp transition from negative to positive values when crossing the origin position. To avoid any bias in the skew values that could result from incompletely annotated gene extremities (e. g. 50 and 30 UTRs), we have removed 10-kbp sequences at both ends of all annotated transcripts. As shown in Fig. 12, the removal of
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 12 Mean skew profile of intergenic regions around putative replication origins [100]. The skew S was calculated in 1 kbp windows (Watson strand) around the position (˙300 kbp without repeats) of the 1012 detected upward jumps; 50 and 30 transcript extremities were extended by 0.5 and 2 kbp, respectively (●), or by 10 kbp at both ends ( ). The abscissa represents the distance (in kbp) to the corresponding origin; the ordinate represents the skews calculated for the windows situated in intergenic regions (mean values for all discontinuities and for 10 consecutive 1 kbp window positions). The skews are given in percent (vertical bars, SEM). The lines correspond to linear fits of the values of the skew ( ) for n < 100 kbp and n > 100 kbp
these intergenic sequences does not significantly modifies the mean skew profile, indicating that the observed values do not result from transcription. On both sides of the jump, we observe a linear decrease of the bias with some
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 13 Model of replication termination [98,100]. Schematic representation of the skew profiles associated with three replication origins O1 , O2 , and O3 ; we suppose that these replication origins are adjacent, bidirectional origins with similar replication efficiency. The abscissa represents the sequence position; the ordinate represents the S value (arbitrary units). Upward (or downward) steps correspond to origin (or termination) positions. For convenience, the termination sites are symmetric relative to O2 . (Left) Three different termination positions T i , T j , and T k , leading to elementary skew profiles Si , Sj , and Sk as predicted by the replicon model [146]. (Center) Superposition of these three profiles. (Right) Superposition of a large number of elementary profiles leading to the final factoryroof pattern. In the simple model, termination occurs with equal probability on both sides of the origins, leading to the linear profile (thick line). In the alternative model, replication termination is more likely to occur at lower rates close to the origins, leading to a flattening of the profile (gray line)
flattening of the profile close to the transition point. Note that, due to (i) the potential presence of signals implicated in replication initiation and (ii) the possible existence of dispersed origins [151], one might question the meaningfulness of this flattening that leads to a significant underestimate of the jump amplitude. Furthermore, according to our detection methodology, the numerical uncertainty on the putative origin position estimate may also contribute to this flattening. As illustrated in Fig. 12, when extrapolating the linear behavior observed at distances > 100 kbp from the jump, one gets a skew of 5.3%, i. e. a value consistent with the skew measured in intergenic regions around the six experimentally known replication origins namely 7:0 ˙ 0:5%. Overall, the detection of sharp upward jumps in the skew profiles with characteristics similar to those of experimentally determined replication origins and with no downward counterpart further supports the existence, in human chromosomes, of replication-associated strand asymmetries, leading to the identification of numerous putative replication origins active in germ-line cells. A Model of Replication in Mammalian Genomes Following the observation of jagged skew profiles similar to factory roofs in Subsect. “Replication Induced Factory-Roof Skew Profiles in Mammalian Genomes”, and the quantitative confirmation of the existence of such (piecewise linear) profiles in the neighborhood of 1012 putative origins in Fig. 12, we have proposed, in Tou-
chon et al. [100] and Brodie of Brodie et al. [98], a rather crude model for replication in the human genome that relies on the hypothesis that the replication origins are quite well positioned while the terminations are randomly distributed. Although some replication terminations have been found at specific sites in S. cerevisiae and to some extent in Schizosaccharomyces pombe [152], they occur randomly between active origins in Xenopus egg extracts [153, 154]. Our results indicate that this property can be extended to replication in human germ-line cells. As illustrated in Fig. 13, replication termination is likely to rely on the existence of numerous potential termination sites distributed along the sequence. For each termination site (used in a small proportion of cell cycles), strand asymmetries associated with replication will generate a steplike skew profile with a downward jump at the position of termination and upward jumps at the positions of the adjacent origins (as in bacteria). Various termination positions will thus correspond to classical replicon-like skew profiles (Fig. 13, left panel). Addition of these profiles will generate the intermediate profile (Fig. 13, central panel). In a simple picture, we can reasonably suppose that termination occurs with constant probability at any position on the sequence. This behavior can, for example, result from the binding of some termination factor at any position between successive origins, leading to a homogeneous distribution of termination sites during successive cell cycles. The final skew profile is then a linear segment decreasing between successive origins (Fig. 13, right panel).
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Let us point out that firing of replication origins during time interval of the S phase [155] might result in some flattening of the skew profile at the origins as sketched in Fig. 13 (right panel, gray curve). In the present state, our results [98,100,146] support the hypothesis of random replication termination in human, and more generally in mammalian cells (Fig. 10), but further analyzes will be necessary to determine what scenario is precisely at work. A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome During the duplication of eukaryotic genomes that occurs during the S phase of the cell cycle, the different replication origins are not all activated simultaneously [132,135, 138,150,155,156]. Recent technical developments in genomic clone microarrays have led to a novel way of detecting the temporal order of DNA replication [155,156]. The arrays are used to estimate replication timing ratios i. e. ratios between the average amount of DNA in the S phase at a locus along the genome and the usual amount of DNA present in the G1 phase for that locus. These ratios should vary between 2 (throughout the S phase, the amount of DNA for the earliest replicating regions is twice the amount during G1 phase) and 1 (the latest replicating regions are not duplicated until the end of S phase). This approach has been successfully used to generate genomewide maps of replication timing for S. cerevisiae [157], Drosophila melanogaster [137] and human [158]. Very recently, two new analyzes of human chromosome 6 [156] and 22 [155] have improved replication timing resolution from 1 Mbp down to 100 kbp using arrays of overlapping tile path clones. In this section, we report on a very promising first step towards the experimental confirmation of the thousand putative replication origins described in Sect. “From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes”. The strategy will consist in mapping them on the recent high-resolution timing data [156] and in checking that these regions replicate earlier than their surrounding [114]. But to provide a convincing experimental test, we need as a prerequisite to extract the contribution of the compositional skew specific to replication. Disentangling Transcription- and Replication-Associated Strand Asymmetries The first step to detect putative replication domains consists in developing a multi-scale pattern recognition
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 14 Wavelet-based analysis of genomic sequences. a Skew profile S of a 9 Mbp repeat-masked fragment of human chromosome 21. b WT of S using 'R (Fig. 1c); TR [S](n; a) is color-coded from dark-blue (min; negative values) to red (max; positive values) through green (null values). Light-blue and purple lines illustrate the detection of two replication domains of significantly different sizes. Note that in b, blue cone-shape areas signing upward jumps point at small scale (top) towards the putative replication origins and that the vertical positions of the WT maxima (red areas) corresponding to the two indicated replication domains match the distance between the putative replication origins (1.6 Mbp and 470 kbp respectively)
methodology based on the WT of the strand compositional asymmetry S using as analyzing wavelet R (x) (Eq. (12)) that is adapted to perform an objective segmentation of factory-roof skew profiles (Fig. 1c). As illustrated in Fig. 14, the space-scale location of significant maxima values in the 2D WT decomposition (red areas in Fig. 14b) indicates the middle position (spatial location) of candidate replication domains whose size is given by the scale location. In order to avoid false positives, we then check that there does exist a well-defined upward jump at each domain extremity. These jumps appear in Fig. 14b as blue cone-shape areas pointing at small scale to the jumps positions where are located the putative replication origins. Note that because the analyzing wavelet is of zero mean (Eq. (2)), the WT decomposition is insensitive to (global) asymmetry offset. But as discussed in Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”, the overall observed skew S also contains some contribution induced by transcription that generates step-like
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
blocks corresponding to (+) and () genes [96,97,129]. Hence, when superimposing the replication serrated and transcription step-like skew profiles, we get the following theoretical skew profile in a replication domain [114]: S(x 0 ) D S R (x 0 ) C S T (x 0 ) X 1 D 2ı x 0 cg g x0 ; C 2 gene
(23)
where position x 0 within the domain has been rescaled between 0 and 1, ı > 0 is the replication bias, g is the characteristic function for the g th gene (1 when x 0 points within the gene and 0 elsewhere) and cg is its transcriptional bias calculated on the Watson strand (likely to be positive for (+) genes and negative for () genes). The objective is thus to detect human replication domains by delineating, in the noisy S profile obtained at 1 kbp resolution (Fig. 15a), all chromosomal loci where S is well fitted by the theoretical skew profile Eq. (23). In order to enforce strong compatibility with the mammalian replicon model (Subsect. “A Model of Replication in Mammalian Genomes”), we will only retain the domains the most likely to be bordered by putative replication origins, namely those that are delimited by upward jumps corresponding to a transition from a negative S value < 3% to a positive S value > C3%. Also, for each domain so-identified, we will use a least-square fitting procedure to estimate the replication bias ı, and each of the gene transcription bias cg . The resulting 2 value will then be used to select the candidate domains where the noisy S profile is well described by Eq. (23). As illustrated in Fig. 15 for a fragment of human chromosome 6 that contains 4 adjacent replication domains (Fig. 15a), this method provides a very efficient way of disentangling the step-like transcription skew component (Fig. 15b) from the serrated component induced by replication (Fig. 15c). Applying this procedure to the 22 human autosomes, we delineated 678 replication domains of mean length hLi D 1:2 ˙ 0:6 Mbp, spanning 28.3% of the genome and predicted 1060 replication origins. DNA Replication Timing Data Corroborate in silico Human Replication Origin Predictions Chromosome 22 being rather atypical in gene and GC contents, we mainly report here on the correlation analysis [114] between nucleotide compositional skew and timing data for chromosome 6 which is more representative of the whole human genome. Note that timing data for clones completely included in another clone have been removed after checking for timing ratio value consistency leaving
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 15 a Skew profile S of a 4.3 Mbp repeat-masked fragment of human chromosome 6 [114]; each point corresponds to a 1 kbp window: Red, (+) genes; blue, () genes; black, intergenic regions (the color was defined by majority rule); the estimated skew profile (Eq. (23)) is shown in green; vertical lines correspond to the locations of 5 putative replication origins that delimit 4 adjacent domains identified by the wavelet-based methodology. b Transcription-associated skew ST obtained by subtracting the estimated replication-associated profile (green lines in c) from the original S profile in a; the estimated transcription step-like profile (second term on the rhs of Eq. (23)) is shown in green. c Replication-associated skew SR obtained by subtracting the estimated transcription step-like profile (green lines in b) from the original S profile in a; the estimated replication serrated profile (first term in the rhs of Eq. (23)) is shown in green; the light-blue dots correspond to high-resolution tr data
1648 data points. The timing ratio value at each point has been chosen as the median over the 4 closest data points to remove noisy fluctuations resulting from clone heterogeneity (clone length 100 ˙ 51 kbp and distance between successive clone mid-points 104 ˙ 89 kbp), so that the spatial resolution is rather inhomogeneous 300 kbp. Note that using asynchronous cells also results in some smoothing of the data, possibly masking local maxima. Our wavelet-based methodology has identified 54 replication domains in human chromosome 6 [114]; these domains are bordered by 83 putative replication origins among which 25 are common to two adjacent domains. Four of these contiguous domains are shown in Fig. 15. In Fig. 15c, on top of the replication skew profile SR , are
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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 16 a Average replication timing ratio (˙SEM) determined around the 83 putative replication origins (●), 20 origins with well-defined local maxima () and 10 late replicating origins (4). x is the native distance to the origins in Mbp units [114]. b Histogram of Pearson’s correlation coefficient values between tr and the absolute value of SR over the 38 predicted domains of length L 1 Mbp. The dotted line corresponds to the expected histogram computed with the correlation coefficients between tr and jSj profiles over independent windows randomly positioned along chromosome 6 and with the same length distribution as the 38 detected domains
reported for comparison the high-resolution timing ratio tr data from [156]. The histogram of tr values obtained at the 83 putative origin locations displays a maximum at tr ' htr i ' 1:5 (data not shown) and confirms what is observed in Fig. 15c, namely that a majority of the predicted origins are rather early replicating with tr & 1:4. This contrasts with the rather low tr ('1.2) values observed in domain central regions (Fig. 15c). But there is an even more striking feature in the replication timing profile in Fig. 15c: 4 among the 5 predicted origins correspond, relatively to the experimental resolution, to local maxima of the tr profile. As shown in Fig. 16a, the average tr profile around the 83 putative replication origins decreases regularly on both sides of the origins over a few (4– 6) hundreds kbp confirming statistically that domain borders replicate earlier than their left and right surroundings which is consistent with these regions being true replication origins mostly active early in S phase. In fact, when averaging over the top 20 origins with a well-defined local maximum in the tr profile, htr i displays a faster decrease on both sides of the origin and a higher maximum value 1.55 corresponding to the earliest replicating origins. On the opposite, when averaging tr profiles over the top 10 late replicating origins, we get, as expected, a rather flat mean profile (tr 1:2) (Fig. 16a). Interestingly, these origins are located in rather wide regions of very low GC
content (. 34%, not shown) correlating with chromosomal G banding patterns predominantly composed of GCpoor isochores [159,160]. This illustrates how the statistical contribution of rather flat profiles observed around late replicating origins may significantly affect the overall mean tr profile. Individual inspection of the 38 replication domains with L 1 Mbp shows that, in those domains that are bordered by early replicating origins (tr & 1:4 1:5), the replication timing ratio tr and the absolute value of the replication skew jS R j turn out to be strongly correlated. This is quantified in Fig. 16b by the histogram of the Pearson’s correlation coefficient values that is clearly shifted towards positive values with a maximum at 0.4. Altogether the results of this comparative analysis provide the first experimental verification of in silico replication origins predictions: The detected putative replication domains are bordered by replication origins mostly active in the early S phase, whereas the central regions replicate more likely in late S phase. Gene Organization in the Detected Replication Domains Most of the 1060 putative replication origins that border the detected replication domains are intergenic (77%) and are located near to a gene promoter more often than
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 17 Analysis of the genes located in the identified replication domains [101]. a Arrows indicate the R+ orientation, i. e. the same orientation as the most frequent direction of putative replication fork progression; R orientation (opposed direction); red, (+) genes; blue, () genes. b Gene density. The density is defined as the number of 50 ends (for (+) genes) or of 30 ends (for () genes) in 50-kbp adjacent windows, divided by the number of corresponding domains. In abscissa, the distance, d, in Mbp, to the closest domain extremity. c Mean gene length. Genes are ranked by their distance, d, from the closest domain extremity, grouped by sets of 150 genes, and the mean length (kbp) is computed for each set. d Relative number of base pairs transcribed in the + direction (red), direction (blue) and non-transcribed (black) determined in 10-kbp adjacent sequence windows. e Mean expression breadth using EST data [101]
would be expected by chance (data not shown) [101]. The replication domains contain approximately equal numbers of genes oriented in each direction (1511 (+) genes and 1507 () genes). Gene distributions in the 50 halves of domains contain more (+) genes than () genes, regardless of the total number of genes located in the halfdomains (Fig. 17b). Symmetrically, the 30 halves contain more () genes than (+) genes (Fig. 17b). 32.7% of halfdomains contain one gene, and 50.9% contain more than one gene. For convenience, (+) genes in the 50 halves and () genes in the 30 halves are defined as R+ genes (Fig. 17a): their transcription is, in most cases, oriented in the same direction as the putative replication fork progression (genes transcribed in the opposite direction are defined as R genes). The 678 replication domains contain significantly more R+ genes (2041) than R genes (977). Within 50 kbp of putative replication origins, the mean density of R+ genes is 8.2 times greater than that of R genes. This asymmetry weakens progressively with the distance from the putative origins, up to 250 kbp (Fig. 17b). A similar asymmetric pattern is observed when the do-
mains containing duplicated genes are eliminated from the analysis, whereas control domains obtained after randomization of domain positions present similar R+ and R gene density distributions (Supplementary in [101]). The mean length of the R+ genes near the putative origins is significantly greater ( 160 kbp) than that of the R genes ( 50 kbp), however both tend towards similar values ( 70 kbp) at the center of the domain (Fig. 17c). Within 50 kbp of the putative origins, the ratio between the numbers of base pairs transcribed in the R+ and R directions is 23.7; this ratio falls to 1 at the domain centers (Fig. 17d). In Fig. 17e are reported the results of the analysis of the breadth of expression, N t (number of tissues in which a gene is expressed) of genes located within the detected domains [101]. As measured by EST data (similar results are obtained by SAGE or microarray data [101]), N t is found to decrease significantly from the extremities to the center in a symmetrical manner in the 50 and 30 halfdomains (Fig. 17e). Thus, genes located near the putative replications origins tend to be widely expressed whereas those located far from them are mostly tissue-specific.
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To summarize, the results reported in this section provide the first demonstration of quantitative relationships in the human genome between gene expression, orientation and distance from putative replication origins [101]. A possible key to the understanding of this complex architecture is the coordination between replication and transcription [101]. The putative replication origins would mostly be active early in the S phase in most tissues. Their activity could result from particular genomic context involving transcription factor binding sites and/or from the transcription of their neighboring housekeeping genes. This activity could also be associated with an open chromatin structure, permissive to early replication and gene expression in most tissues [161,162,163,164]. This open conformation could extend along the first gene, possibly promoting the expression of further genes. This effect would progressively weaken with the distance from the putative replication origin, leading to the observed decrease in expression breadth. This model is consistent with a number of data showing that in metazoans, ORC and RNA polymerase II colocalize at transcriptional promoter regions [165], and that replication origins are determined by epigenetic information such as transcription factor binding sites and/or transcription [166,167,168,169]. It is also consistent with studies in Drosophila and humans that report correlation between early replication timing and increased probability of expression [137,155,156,165,170]. Furthermore, near the putative origins bordering the replication domains, transcription is preferentially oriented in the same direction as replication fork progression. This coorientation is likely to reduce head-on collisions between the replication and transcription machineries, which may induce deleterious recombination events either directly or via stalling of the replication fork [171,172]. In bacteria, co-orientation of transcription and replication has been observed for essential genes, and has been associated with a reduction in head-on collisions between DNA and RNA polymerases [173]. It is noteworthy that in human replication domains such co-orientation usually occurs in widely-expressed genes located near putative replication origins. Near domain centers, head-on collisions may occur in 50% of replication cycles, regardless of the transcription orientation, since there is no preferential orientation of the replication fork progression in these regions. However, in most cell types, there should be few head-on collisions due to the low density and expression breadth of the corresponding genes. Selective pressure to reduce head-on collisions may thus have contributed to the simultaneous and coordinated organization of gene orientation and expression breadth along the detected replication domains [101].
Future Directions From a statistical multifractal analysis of nucleotide strand asymmetries in mammalian genomes, we have revealed the existence of jumps in the noisy skew profiles resulting from asymmetries intrinsic to the transcription and replication processes [98,100]. This discovery has led us to extend our 1D WTMM methodology to an adapted multi-scale pattern recognition strategy in order to detect putative replication domains bordered by replication origins [101,114]. The results reported in this manuscript show that directly from the DNA sequence, we have been able to reveal the existence in the human genome (and very likely in all mammalian genomes), of regions bordered by early replicating origins in which gene position, orientation and expression breadth present a high level of organization, possibly mediated by the chromatin structure. These results open new perspectives in DNA sequence analysis, chromatin modeling as well as in experiment. From a bioinformatic and modeling point of view, we plan to study the lexical and structural characteristics of our set of putative origins. In particular we will search for conserved sequence motifs in these replication initiation zones. Using a sequence-dependent model of DNA-histones interactions, we will develop physical studies of nucleosome formation and diffusion along the DNA fiber around the putative replication origins. These bioinformatic and physical studies, performed for the first time on a large number of replication origins, should shed light on the processes at work during the recognition of the replication initiation zone by the replication machinery. From an experimental point of view, our study raises new opportunities for future experiments. The first one concerns the experimental validation of the predicted replication origins (e. g. by molecular combing of DNA molecules [174]), which will allow us to determine precisely the existence of replication origins in given genome regions. Large scale study of all candidate origins is in current progress in the laboratory of O. Hyrien (École Normale Supérieure, Paris). The second experimental project consists in using Atomic Force Microscopy (AFM) [175] and Surface Plasmon Resonance Microscopy (SPRM) [176] to visualize and study the structural and mechanical properties of the DNA double helix, the nucleosomal string and the 30 nm chromatin fiber around the predicted replication origins. This work is in current progress in the experimental group of F. Argoul at the Laboratoire Joliot– Curie (ENS, Lyon) [83]. Finally the third experimental perspective concerns in situ studies of replication origins. Using fluorescence techniques (FISH chromosome painting [177]), we plan to study the distributions and dynam-
Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .
ics of origins in the cell nucleus, as well as chromosome domains potentially associated with territories and their possible relation to nuclear matrix attachment sites. This study is likely to provide evidence of chromatin rosette patterns as suggested in [146]. This study is under progress in the molecular biology experimental group of F. Mongelard at the Laboratoire Joliot–Curie. Acknowledgments We thank O. Hyrien, F. Mongelard and C. Moskalenko for interesting discussions. This work was supported by the Action Concertée Incitative Informatique, Mathématiques, Physique en Biologie Moléculaire 2004 under the project “ReplicOr”, the Agence Nationale de la Recherche under the project “HUGOREP” and the program “Emergence” of the Conseil Régional Rhônes-Alpes and by the Programme d’Actions Intégrées Tournesol. Bibliography Primary Literature 1. Goupillaud P, Grossmann A, Morlet J (1984) Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23:85–102 2. Grossmann A, Morlet J (1984) Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J Math Anal 15:723–736 3. Arneodo A, Argoul F, Bacry E, Elezgaray J, Freysz E, Grasseau G, Muzy J-F, Pouligny B (1992) Wavelet transform of fractals. In: Meyer Y (ed) Wavelets and applications. Springer, Berlin, pp 286–352 4. Arneodo A, Argoul F, Elezgaray J, Grasseau G (1989) Wavelet transform analysis of fractals: Application to nonequilibrium phase transitions. In: Turchetti G (ed) Nonlinear dynamics. World Scientific, Singapore, pp 130–180 5. Arneodo A, Grasseau G, Holschneider M (1988) Wavelet transform of multifractals. Phys Rev Lett 61:2281–2284 6. Holschneider M (1988) On the wavelet transform of fractal objects. J Stat Phys 50:963–993 7. Holschneider M, Tchamitchian P (1990) Régularité locale de la fonction non-différentiable de Riemann. In: Lemarié PG (ed) Les ondelettes en 1989. Springer, Berlin, pp 102–124 8. Jaffard S (1989) Hölder exponents at given points and wavelet coefficients. C R Acad Sci Paris Sér. I 308:79–81 9. Jaffard S (1991) Pointwise smoothness, two-microlocalization and wavelet coefficients. Publ Mat 35:155–168 10. Mallat S, Hwang W (1992) Singularity detection and processing with wavelets. IEEE Trans Info Theory 38:617–643 11. Mallat S, Zhong S (1992) Characterization of signals from multiscale edges. IEEE Trans Patt Recog Mach Intell 14:710–732 12. Arneodo A, Bacry E, Muzy J-F (1995) The thermodynamics of fractals revisited with wavelets. Physica A 213:232–275 13. Bacry E, Muzy J-F, Arneodo A (1993) Singularity spectrum of fractal signals from wavelet analysis: Exact results. J Stat Phys 70:635–674
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154. Santamaria D, Viguera E, Martinez-Robles ML, Hyrien O, Hernandez P, Krimer DB, Schvartzman JB (2000) Bi-directional replication and random termination. Nucleic Acids Res 28:2099–2107 155. White EJ, Emanuelsson O, Scalzo D, Royce T, Kosak S, Oakeley EJ, Weissman S, Gerstein M, Groudine M, Snyder M, Schübeler D (2004) DNA replication-timing analysis of human chromosome 22 at high resolution and different developmental states. Proc Natl Acad Sci USA 101:17771–17776 156. Woodfine K, Beare DM, Ichimura K, Debernardi S, Mungall AJ, Fiegler H, Collins VP, Carter NP, Dunham I (2005) Replication timing of human chromosome 6. Cell Cycle 4:172–176 157. Raghuraman MK, Winzeler EA, Collingwood D, Hunt S, Wodicka L, Conway A, Lockhart DJ, Davis RW, Brewer BJ, Fangman WL (2001) Replication dynamics of the yeast genome. Science 294:115–121 158. Watanabe Y, Fujiyama A, Ichiba Y, Hattori M, Yada T, Sakaki Y, Ikemura T (2002) Chromosome-wide assessment of replication timing for human chromosomes 11q and 21q: Disease-related genes in timing-switch regions. Hum Mol Genet 11:13–21 159. Costantini M, Clay O, Federico C, Saccone S, Auletta F, Bernardi G (2007) Human chromosomal bands: Nested structure, high-definition map and molecular basis. Chromosoma 116:29–40 160. Schmegner C, Hameister H, Vogel W, Assum G (2007) Isochores and replication time zones: A perfect match. Cytogenet Genome Res 116:167–172 161. Chakalova L, Debrand E, Mitchell JA, Osborne CS, Fraser P (2005) Replication and transcription: shaping the landscape of the genome. Nat Rev Genet 6:669–677 162. Gilbert N, Boyle S, Fiegler H, Woodfine K, Carter NP, Bickmore WA (2004) Chromatin architecture of the human genome: Gene-rich domains are enriched in open chromatin fibers. Cell 118:555–566 163. Hurst LD, Pál C, Lercher MJ (2004) The evolutionary dynamics of eukaryotic gene order. Nat Rev Genet 5:299–310 164. Sproul D, Gilbert N, Bickmore WA (2005) The role of chromatin structure in regulating the expression of clustered genes. Nat Rev Genet 6:775–781 165. MacAlpine DM, Rodriguez HK, Bell SP (2004) Coordination of replication and transcription along a Drosophila chromosome. Genes Dev 18:3094–3105 166. Danis E, Brodolin K, Menut S, Maiorano D, Girard-Reydet C, Méchali M (2004) Specification of a DNA replication origin by a transcription complex. Nat Cell Biol 6:721–730 167. DePamphilis ML (2005) Cell cycle dependent regulation of the origin recognition complex. Cell Cycle 4:70–79 168. Ghosh M, Liu G, Randall G, Bevington J, Leffak M (2004) Transcription factor binding and induced transcription alter chromosomal c-myc replicator activity. Mol Cell Biol 24: 10193–10207 169. Lin CM, Fu H, Martinovsky M, Bouhassira E, Aladjem MI (2003) Dynamic alterations of replication timing in mammalian cells. Curr Biol 13:1019–1028 170. Jeon Y, Bekiranov S, Karnani N, Kapranov P, Ghosh S, MacAlpine D, Lee C, Hwang DS, Gingeras TR, Dutta A (2005) Temporal profile of replication of human chromosomes. Proc Natl Acad Sci USA 102:6419–6424 171. Deshpande AM, Newlon CS (1996) DNA replication fork pause sites dependent on transcription. Science 272:1030–1033
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Books and Reviews Fractals Aharony A, Feder J (eds) (1989) Fractals in Physics, Essays in Honour of BB Mandelbrot. Physica D 38. North-Holland, Amsterdam Avnir D (ed) (1988) The fractal approach to heterogeneous chemistry: surfaces, colloids, polymers. Wiley, New-York Barabàsi AL, Stanley HE (1995) Fractals concepts in surface growth. Cambridge University Press, Cambridge Ben Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge Bouchaud J-P, Potters M (1997) Théorie des risques financiers. Cambridge University Press, Cambridge Bunde A, Havlin S (eds) (1991) Fractals and disordered systems. Springer, Berlin Bunde A, Havlin S (eds) (1994) Fractals in science. Springer, Berlin Bunde A, Kropp J, Schellnhuber HJ (eds) (2002) The science of disasters: Climate disruptions, heart attacks and market crashes. Springer, Berlin Family F, Meakin P, Sapoval B, Wood R (eds) (1995) Fractal aspects of materials. Material Research Society Symposium Proceedings, vol 367. MRS, Pittsburg Family F, Vicsek T (1991) Dynamics of fractal surfaces. World Scientific, Singapore Feder J (1988) Fractals. Pergamon, New-York Frisch U (1995) Turbulence. Cambridge University Press, Cambridge Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco Mantegna RN, Stanley HE (2000) An introduction to econophysics. Cambridge University Press, Cambridge Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge Peitgen HO, Jürgens H, Saupe D (1992) Chaos and fractals: New frontiers of science. Springer, New York Peitgen HO, Saupe D (eds) (1987) The science of fractal images. Springer, New-York Pietronero L, Tosatti E (eds) (1986) Fractals in physics. North-Holland, Amsterdam Stanley HE, Osbrowski N (eds) (1986) On growth and form: Fractal and non-fractal patterns in physics. Martinus Nijhof, Dordrecht
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Stanley HE, Ostrowski N (eds) (1988) Random fluctuations and pattern growth. Kluwer, Dordrecht Vicsek T (1989) Fractal growth phenomena. World Scientific, Singapore Vicsek T, Schlesinger M, Matsuchita M (eds) (1994) Fractals in natural science. World Scientific, Singapore West BJ (1990) Fractal physiology and chaos in medicine. World Scientific, Singapore West BJ, Deering W (1994) Fractal physiology for physicists: Levy statistics. Phys Rep 246:1–100 Wilkinson GG, Kanellopoulos J, Megier J (eds) (1995) Fractals in geoscience and remote sensing, image understanding research senes, vol 1. ECSC-EC-EAEC, Brussels
Mallat S (1998) A wavelet tour in signal processing. Academic Press, New-York Meyer Y (1990) Ondelettes. Herman, Paris Meyer Y (ed) (1992) Wavelets and applications. Springer, Berlin Meyer Y, Roques S (eds) (1993) Progress in wavelets analysis and applications. Éditions Frontières, Gif-sur-Yvette Ruskai MB, Beylkin G, Coifman R, Daubechies I, Mallat S, Meyer Y, Raphael L (eds) (1992) Wavelets and their applications. Jones and Barlett, Boston Silverman BW, Vassilicos JC (eds) (2000) Wavelets: The key to intermittent information? Oxford University Press, Oxford Torresani B (1998) Analyse continue par ondelettes. Éditions de Physique, Les Ulis
Wavelets Abry P (1997) Ondelettes et turbulences. Diderot Éditeur, Art et Sciences, Paris Arneodo A, Argoul F, Bacry E, Elezgaray J, Muzy J-F (1995) Ondelettes, multifractales et turbulences: de l’ADN aux croissances cristallines. Diderot Éditeur, Art et Sciences, Paris Chui CK (1992) An introduction to wavelets. Academic Press, Boston Combes J-M, Grossmann A, Tchamitchian P (eds) (1989) Wavelets. Springer, Berlin Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia Erlebacher G, Hussaini MY, Jameson LM (eds) (1996) Wavelets: Theory and applications. Oxford University Press, Oxford Farge M, Hunt JCR, Vassilicos JC (eds) (1993) Wavelets, fractals and Fourier. Clarendon Press, Oxford Flandrin P (1993) Temps-Fréquence. Hermès, Paris Holschneider M (1996) Wavelets: An analysis tool. Oxford University Press, Oxford Jaffard S, Meyer Y, Ryan RD (eds) (2001) Wavelets: Tools for science and technology. SIAM, Philadelphia Lemarie PG (ed) (1990) Les ondelettes en 1989. Springer, Berlin
DNA and Chromatin Alberts B, Watson J (1994) Molecular biology of the cell, 3rd edn. Garland Publishing, New-York Calladine CR, Drew HR (1999) Understanding DNA. Academic Press, San Diego Graur D, Li WH (1999) Fundamentals of molecular evolution. Sinauer Associates, Sunderland Hartl DL, Jones EW (2001) Genetics: Analysis of genes and genomes. Jones and Bartlett, Sudbury Kolchanov NA, Lim HA (1994) Computer analysis of genetic macromolecules: Structure, function and evolution. World Scientific, Singapore Kornberg A, Baker TA (1992) DNA Replication. WH Freeman, NewYork Lewin B (1994) Genes V. Oxford University Press, Oxford Sudbery P (1998) Human molecular genetics. Addison Wesley, Singapore Van Holde, KE (1988) Chromatin. Springer, New-York Watson JD, Gilman M, Witkowski J, Zoller M (1992) Recombinant DNA. Freeman, New-York Wolfe AP (1998) Chromatin structure and function, 3rd edn. Academic Press, London
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Fractals in Biology SERGEY V. BULDYREV Department of Physics, Yeshiva University, New York, USA
Article Outline Glossary Definition of the Subject Introduction Self-similar Branching Structures Fractal Metabolic Rates Physical Models of Biological Fractals Diffusion Limited Aggregation and Bacterial Colonies Measuring Fractal Dimension of Real Biological Fractals Percolation and Forest Fires Critical Point and Long-Range Correlations Lévy Flight Foraging Dynamic Fractals Fractals and Time Series SOC and Biological Evolution Fractal Features of DNA Sequences Future Directions Bibliography Glossary Allometric laws An allometric law describes the relationship between two attributes of living organisms y and x, and is usually expressed as a power-law: y x ˛ , where ˛ is the scaling exponent of the law. For example, x can represent total body mass M and y can represent the mass of a brain mb . In this case mb M 3/4 . Another example of an allometric law: B M 3/4 where B is metabolic rate and M is body mass. Allometric laws can be also found in ecology: the number of different species N found in a habitat of area A scales as N A1/4 . Radial distribution function Radial distribution function g(r) describes how the average density of points of a set behaves as function of distance r from a point of this set. For an empirical set of N data points, the distances between all pair of points are computed and the number of pairs Np (r) such that their distance is less then r is found. Then M(r) D 2Np (r)/N gives the average number of the neighbors (mass) of the set within a distance r. For a certain distance bin r1 < r < r2 , we define g[(r2 C r1 )/2] D [M(r2 ) M(r1 )]/[Vd (r2 ) Vd (r1 )], where
Vd (r) D 2 d/2 r d /[d (d/2)] is the volume/area/length of a d-dimensional sphere/circle/interval of radius r. Fractal set We define a fractal set with the fractal dimension 0 < df < d as a set for which M(r) r d f for r ! 1. Accordingly, for such a set g(r) decreases as a power law of the distance g(r) r , where D d df . Correlation function For a superposition of a fractal set and a set with a finite density defined as D limr!1 M(r)/Vd (r), the correlation function is defined as h(r) g(r) . Long-range power law correlations The set of points has long-range power law correlations (LRPLC) if h(r) r for r ! 1 with 0 < < d. LRPLC indicate the presence of a fractal set with fractal dimension df D d superposed with a uniform set. Critical point Critical point is defined as a point in the system parameter space (e. g. temperature, T D Tc , and pressure P D Pc ), near which the system acquires LRPLC h(r)
1 r d2C
exp(r/) ;
(1)
where is the correlation length which diverges near the critical point as jT Tc j . Here > 0 and > 0 are critical exponents which depend on the few system characteristics such as dimensionality of space. Accordingly, the system is characterized by fractal density fluctuations with df D 2 . Self-organized criticality Self-organized criticality (SOC) is a term which describes a system for which the critical behavior characterized by a large correlation length is achieved for a wide range of parameters and thus does not require special tuning. This usually occurs when a critical point corresponds to an infinite value of a system parameter, such as a ratio of the characteristic time of the stress build up and a characteristic time of the stress release. Morphogenesis Morphogenesis is a branch of developmental biology concerned with the shapes of organs and the entire organisms. Several types of molecules are particularly important during morphogenesis. Morphogens are soluble molecules that can diffuse and carry signals that control cell differentiation decisions in a concentration-dependent fashion. Morphogens typically act through binding to specific protein receptors. An important class of molecules involved in morphogenesis are transcription factor proteins that determine the fate of cells by interacting with DNA. The morphogenesis of the branching fractal-like structures such as lungs involves a dozen of morpho-
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genes. The mechanism for keeping self-similarity of the branches at different levels of branching hierarchy is not yet fully understood. The experiments with transgenic mice with certain genes knocked-out produce mice without limbs and lungs or without terminal buds.
for each model and therefore may shed light on the origin of a particular biological phenomenon. In Sect. “Diffusion Limited Aggregation and Bacterial Colonies”, we discuss the techniques for measuring fractal dimension and their limitations. Introduction
Definition of the Subject Fractals occur in a wide range of biological applications: 1) In morphology when the shape of an organism (tree) or an organ (vertebrate lung) has a self-similar brunching structure which can be approximated by a fractal set (Sect. “Self-Similar Branching Structures”). 2) In allometry when the allometric power laws can be deduced from the fractal nature of the circulatory system (Sect. “Fractal Metabolic Rates”). 3) In ecology when a colony or a habitat acquire fractal shapes due to some SOC processes such as diffusion limited aggregation (DLA) or percolation which describes forest fires (Sects. “Physical Models of Biological Fractals”–“Percolation and Forest Fires”). 4) In epidemiology when some of the features of the epidemics is described by percolation which in turn leads to fractal behavior (Sect. “Percolation and Forest Fires”). 5) In behavioral sciences, when a trajectory of foraging animal acquires fractal features (Sect. “Lévy Flight Foraging”). 6) In population dynamics, when the population size fluctuates chaotically (Sect. “Dynamic Fractals”). 7) In physiology, when time series have LRPLC (Sect. “Fractals and Time Series”). 8) In evolution theory, which may have some features described by SOC (Sect. “SOC and Biological Evolution”). 9) In bioinformatics when a DNA sequence has a LRPLC or a network describing protein interactions has a selfsimilar fractal behavior (Sect. “Fractal Features of DNA Sequences”). Fractal geometry along with Euclidian geometry became a part of general culture which any scientist must be familiar with. Fractals often originate in the theory of complex systems describing the behavior of many interacting elements and therefore have a great number of biological applications. Complex systems have a general tendency for self-organization and complex pattern formation. Some of these patterns have certain nontrivial symmetries, for example fractals are characterized by scale invariance i. e. they look similarly on different magnification. Fractals are characterized by their fractal dimension, which is specific
The fact that simple objects of Euclidian geometry such as straight lines, circles, cubes, and spheres are not sufficient to describe complex biological shapes has been known for centuries. Physicists were always accused by biologists for introducing a “spherical cow”. Nevertheless people from antiquity to our days were fascinated by finding simple mathematical regularities which can describe the anatomy and physiology of leaving creatures. Five centuries ago, Leonardo da Vinci observed that “the branches of a tree at every stage of its height when put together are equal in thickness to the trunk” [107]. Another famous but still poorly understood phenomenon is the emergence of the Fibonacci numbers in certain types of pine cones and composite flowers [31,134]. In the middle of the seventies a new concept of fractal geometry was introduced by Mandelbrot [79]. This concept was readily accepted for analysis of the complex shapes in the biological world. However, after initial splash of enthusiasm, [14,40,41,74,75,76,88,122] the application of fractals in biology significantly dwindled and today the general consensus is that a “fractal cow” is often not much better than a “spherical cow”. Nature is always more complex than mathematical abstractions. Strictly speaking the fractal is an object whose mass, M, grows as a fractional power law of its linear dimension, L, M Ldf ;
(2)
where df is a non-integer quantity called fractal dimension. Simple examples of fractal objects of various fractal dimensions are given by iterative self-similar constructs such as a Cantor set, a Koch curve, a Sierpinski gasket, and a Menger sponge [94]. In all this constructs a next iteration of the object is created by arranging p exact copies of the previous iteration of the object (Fig. 1) in such a way that the linear size of the next iteration is q times larger than the linear size of the previous iteration. Thus the mass, M n , and the length, Ln , of the nth iteration scale as M n D p n M0 L n D q n L0 ;
(3)
where M 0 and L0 are mass and length of the zero order
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Fractals in Biology, Figure 1 a Cantor set (p D 2, q D 3, df D ln 2/ ln 3 0:63) is an example of fractal “dust” with fractal dimension less then 1; b Fractal tree (p D 3, q D 2, df D ln 3/ ln 2 1:58) with branches removed so that only the terminal points (leaves) can be seen. c The same tree with the trunks added. Both trees are produced by recursive combining of the three smaller trees: one serving as the top of the tree and the other two rotated by 90ı clockwise and counter-clockwise serving as two brunches joined with the top at the middle. In c a vertical segment representing a trunk of the length equal to the diagonal of the branches is added at each recursive step. Mathematically, the fractal dimensions of sets b and c are the same, because the mass of the tree with trunks (number of black pixels) for the system of linear size 2n grows as 3nC1 2nC1 , while for the tree without trunks mass scales simply as 3n . In the limit n ! 1 this leads to the same fractal dimension ln 3/ ln 2. However, visual inspection suggests that the tree with branches has a larger fractal dimension. This is in accord with the box counting method, which produces a higher value of the slope for the finite tree with the trunks. The slope slowly converges to the theoretical value as the number of recursive steps increases. d Sierpinski gasket has the same fractal dimension as the fractal tree (p D 3, q D 2) but has totally different topology and visual appearance than the tree
iteration. Excluding n from Eq. (3), we get d f Ln ; M n D M0 L0
Self-similar Branching Structures (4)
where df D ln p/ ln q
(5)
can be identified as fractal dimension. The above-described objects have the property of self-similarity, the previous iteration magnified q times looks exactly like the next iteration once we neglect coarse-graining on the lowest iteration level, which can be assumed to be infinitely small. An interesting feature of such fractals is the power law distribution of their parts. For example, the cumulative distribution of distances L between the points of Cantor set and the branch lengths of the fractal tree (Fig. 1) follows a power law: P(L > x) x d f :
(6)
Thus the emergence of power law distributions and other power law dependencies is often associated with fractals and often these power law regularities not necessarily related to geometrical fractals are loosely referred as fractal properties.
Perfect deterministic fractals described above are never observed in nature, where all shapes are subjects to random variations. The natural examples closest to the deterministic fractals are structures of trees, certain plants such as a cauliflower, lungs and a cardiovascular system. The control of branching morphogenesis [85,114,138] involves determining when and where a branch will occur, how long the tube grows before branching again, and at what angle the branch will form. The development of different organs (such as salivary gland, mammary gland, kidney, and lung) creates branching patterns easily distinguished from each other. Moreover, during the development of a particular organ, the form of branching often changes, depending on the place or time when the branching occurs. Morphogenesis is controlled by complex molecular interactions of morphogenes. Let us illustrate the idea of calculating fractal dimension of a branching object such as a cauliflower [51,64]. If we assume (which is not quite correct, see Fig. 2) that each branch of a cauliflower give rise to exactly p branches of the next generation exactly q times smaller than the original branch applying Eq. (5), we get df D ln q/ ln p. Note
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Fractals in Biology, Figure 2 a Cauliflower anatomy. A complete head of a cauliflower (left) and after it is taken apart (right). We remove 17 branches until the diameter of the remaining part becomes half of the diameter of the original head. b Diameters of the branches presented in a. c Estimation of the fractal dimension for asymmetric lungs ( D 3) and trees ( D 2) with branching parameter r using Eq. (7). For r > 0:855 or r < 0:145 the fractal dimension is not defined due to the “infrared catastrophe”: the number of large branches diverge in the limit of an infinite tree
that this is not the fractal dimension of the cauliflower itself, but of its skeleton in which each brunch is represented by the elements of its daughter branches, with an addition of a straight line connecting the daughter branches as in the example of a fractal tree (Fig. 1b, c). Because this addition does not change the fractal dimension formula, the fractal dimension of the skeleton is equal to the
fractal dimension of the surface of a cauliflower, which can be represented as the set of the terminal branches. As a physical object, the cauliflower is not a fractal but a three-dimensional body so that the mass of a branch of length L scales as L3 . This is because the diameter of each branch is proportional to the length of the branch.
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The distribution of lengths of the collection of all the branches of a cauliflower is also a power law given by P k n a simple idea that there are N n D n1 kD0 p p /(p 1) branches of length larger than L n D L0 qn . Excluding n d gives N n (L > L n ) L n f . In reality, however (Fig. 2a, b), the branching structure of a cauliflower is more complex. Simple measurements show that there are about p D 18 branches of sizes L k D L0 L0 [0:5 0:2(k 1)] for k D 1; 2; : : :; p, where L0 is the diameter of the complete head of the cauliflower. (We keep removing branches until the diameter of the central remaining part is equal to the half diameter of the complete head and we assume that it is similar to the rest of the branches). Subsequent generations (we count at least eight) obey similar rules, however p has a tendency to decrease with the generation number. To find fractal dimension of a cauliflower we will count the number of branches N(L) larger than certain size L. As in case of equal branches, this number scales as Ld f . Calculations, similar to those presented in [77] show that in this case the fractal dimension is equal to 0s 1 @ 2 2 1 ln p 2 1A ; (7) df D 2 where and are the mean and the standard deviation of ln q k ln(L0 /L k ). For the particular cauliflower shown in Fig. 2a the measurements presented in Fig. 2b give df D 2:75, which is very close to the estimate df D 2:8, of [64]. The physiological reason for such a peculiar branching pattern of cauliflower is not known. It is probably designed to store energy for a quick production of a dense inflorescence. For a tree, [142] the sum of the cross-section areas of the two daughter branches according to Leonardo is equal to the cross-section area of the mother branch. This can be understood because the number of capillary bundles going from the mother to the daughters is conserved. Accordingly, we have a relation between the daughter branch diameters d1 and d2 and the mother branch diameter d0 ,
d1 C d2 D d0 ;
(8)
where D 2. We can assume that the branch diameter of the largest daughter branch scales as d1 D rd0 , where r is asymmetry ratio maintaining for each generation of branches. In case of a symmetric tree d1 D d2 , r D 1/2. If we assume that the branch length L D sd, where s is a constant which is the same for any branch of the tree, then we can relate our tree to a fractal model with p D 2 and q D 21/ . Using Eq. (5) we get df D D 2, i. e. the tree skeleton or the surface of the terminal branches is
an object of fractal dimension two embedded in the threedimensional space. This is quite natural because all the leaves whose number is proportional to the number of terminal branches must be exposed to the sunlight and the easiest way to achieve this is to place all the leaves on the surface of a sphere which is a two-dimensional object. For an asymmetric tree the fractal dimension can be computed using Eq. (7) with D j ln[r(1 r)]/2j
(9)
and D j ln[r/(1 r)]/2j :
(10)
This value is slightly larger than 2 for a wide range of r (Fig. 2c). This property may be related to a tendency of a tree to maximize the surface of it leaves, which must not be necessarily exposed to the direct sunlight but can suffice on the light reflected by the outer leaves. For a lung [56,66,77,102,117], the flow is not a capillary but can be assumed viscous. According to flow conservation, the sum of the air flows of the daughter branches must be equal to the flow of the mother branch: Q1 C Q2 D Q0 :
(11)
For the Poiseuille flow, Q Pd 4 /L, where P is the pressure drop, which is supposed to be the same for the airways of all sizes. Assuming that the lung maintains the ratio s D L/d in all generations of branches, we conclude that the diameters of the daughter and mother branches must satisfy Eq. (8) with D 4 1 D 3. Accordingly, for a symmetrically branching lung df D D 3, which means that the surface of the alveoli which is proportional to the number of terminal airways scales as L3 , i. e. it is a space filling object. Again this prediction is quite reasonable because nature tends to maximize the gas exchange area so that it completely fills the volume of the lung. In reality, the flow in the large airways is turbulent, and the parameter of lungs of different species varies between 2 and 3 [77]. Also the lungs are known to be asymmetric and the ratio r D Q1 /Q0 ¤ 1/2 changes from one generation of the airways to the next [77]. However, Eq. (7) with and given by Eqs. (9) and (10) shows that the fractal dimension of an asymmetric tree remains very close to 3 for a wide range of 0:146 < r < 0:854 (Fig. 2c). The fact that the estimated fractal dimension is slightly larger than 3 does not contradict common sense because the branching stops when the airway diameter becomes smaller than some critical cutoff. Other implications of the lung asymmetry are discussed in [77]. An interesting idea
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was proposed in [117], according to which the dependence of the branching pattern on the generation of the airways can be derived from optimization principles and give rise to the complex value of the fractal dimension. An interesting property of the crackle sound produced by diseased lungs is that during inflation the resistance to airflow of the small airways decreases in discrete jumps. Airways do not open individually, but in a sequence of bursts or avalanches involving many airways; both the size of these jumps and the time intervals between jumps follow power-law distributions [128]. These avalanches are not related to the SOC as one might expect, but their power law distributions directly follow from the branching structure of lungs (see [3] and references therein).
Fractal Metabolic Rates An important question in biology is the scaling of the metabolic rate with respect to the body mass (Kleiber’s Law). It turns out that for almost all species of animals, which differ in terms of mass by 21 orders of magnitudes, the metabolic rate B scales as B M 3/4 [28,112,140]. The scatter plot ln B vs ln M is a narrow cloud concentrated around the straight line with a slope 3/4 (Fig. 3). This is one of the many examples of the allometric laws which describe the dependence of various biological parameters on the body mass or population size [1,9,87,105,106]. A simple argument based on the idea that the thermal energy loss and hence the metabolic rate should be propor-
Fractals in Biology, Figure 3 Dependence of metabolic rate on body mass for different types of organisms (Kleiber’s Law, after [112]). The least square fit lines have slopes 0:76 ˙ 0:01
tional to the surface area predicts however that B L2 M 2/3 . Postulating that the metabolic rate is proportional to some effective metabolic area leads to a strange prediction that this area should have a fractal dimension of 9/4. Many attempts has been made to explain this interesting fact [10,32,142,143]. One particular attempt [142] was based on the ideas of energy optimization and the fractal organization of the cardiovascular system. However, the arguments were crucially dependent on such details as turbulent vs laminar flow, the elasticity of the blood vessels and the pulsatory nature of the cardiovascular system. The fact that this derivation does not work for species with different types of circulatory systems suggests that there might be a completely different and quite general explanation of this phenomenon. Recently [10], as the theory of networks has became a hot subject, a general explanation of the metabolic rates was provided using a mathematical theorem that the total flow Q (e. g. of the blood) in the most efficient supply network must scale as Q BL/u, where L is the linear size of the network, B is the total consumption rate (e. g. metabolic rate), and u is the linear size of each consumption unit (e. g. cell). The authors argue that the total flow is proportional to the total amount of the liquid in the network, which must scale as the body mass M. On the other hand, L M 1/3 . If one assumes that u is independent of the body mass, then B M 2/3 , which is identical to the simple but incorrect prediction based on the surface area. In order to make ends meet, the authors postulate that some combination of parameters must be independent of the body size, from where it follows that u M 1/12 . If one identifies a consumption unit with the cell, it seems that the cell size must scale as L1/4 . Thus the cells of a whale (L D 30 m) must be about 12 times larger than those of a C. elegance (L D 1 mm), which more or less consistent with the empirical data [110] for slowly dividing cells such as neurons in mammals. However, the issue of the metabolic scaling is still not fully resolved [11,33,144]. It is likely that the universal explanation of Kleiber’s law is impossible. Moreover, recently it was observed that Kleiber’s law does not hold for plants [105]. An interesting example of an allometric law which may have some fractal implications is the scaling of the brain size with the body mass Mb M 3/4 [1]. Assuming that the mass of the brain is proportional to the mass of the neurons in the body, we can conclude that if the average mass of a neuron does not depend on the body mass, neurons must form a fractal set with the fractal dimension 9/4. However, if we assume that the mass of a neuron must scale with the body mass as M 1/12 , as it follows from [10,110], we can conclude that the number of neu-
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rons in the body scales simply as M 2/3 L2 , which means that the number of neurons is proportional to the surface area of the body. The latter conclusion is physiologically more plausible than the former, since it is obvious that the neurons are more likely to be located near the surface of an organism. From the point of view of comparative zoology, the universal scaling of the brain mass is not as important as the deviations from it. It is useful [1] to characterize an organism by the ratio of its actual brain mass to its expected brain mass which is defined as Eb D AM 3/4 , where A is a constant measured by the intercept of the log–log graph of brain mass versus body mass. Homo sapiens have the largest Mb /Eb D 8, several times larger than those of gorillas and chimpanzees. Physical Models of Biological Fractals The fractals discussed in Sect. “Self-Similar Branching Structures”, which are based on the explicit rules of construction and self-similarity, are useful concepts for analyzing branching structures in the living organisms whose development and growth are programmed according to these rules. However, in nature, there are many instances when fractal shapes emerge just from general physical principles without a special fractal “blueprint” [26,34, 58,124,125,131]. One of such examples particularly relevant to biology is the diffusion limited aggregation (DLA) [145]. Another phenomenon, described by the same equations as DLA and thus producing similar shapes is viscous fingering [17,26,131]. Other examples are a random walk (RW) [42,58,104,141], a self-avoiding walk (SAW) [34,58], and a percolation cluster [26,58,127]. DLA explains the growth of ramified inorganic aggregates which sometimes can be found in rock cracks. The aggregates grow because certain ions or molecules deposit on the surface of the aggregate. These ions come to the surface due to diffusion which is equivalent on the microscopic level to Brownian motion of individual ions. The Brownian trajectories can be modeled as random walks in which the direction of each next step is independent of the previous steps. The Brownian trajectory itself has fractal properties expressed by the Einstein formula: r2 D 2dtD ;
(12)
where r2 is the average square displacement of the Brownian particle during time t, D is the diffusion coefficient and d is the dimensionality of the embedding space. The number of steps, n, of the random walk is proportional to time of the Brownian motion t D n, where is the
Fractals in Biology, Figure 4 A random walk of n D 216 steps is surrounded by its p D p 4 parts of m D n/p D 214 steps magnified by factor of q D 4 D 2. One can see that the shapes and sizes of each of the magnified parts are similar to the shape and size of the whole
average duration of the random walk step. The diffusion coefficient can be expressed as D D r2 (m)/(2dm), where r2 (m) is the average square displacement during m time steps. Accordingly, p r(n) D pr(m) ; (13) which shows that the random walk of n steps consists of p D n/m copies of the one-step walks arranged in space in such a way pthat the average linear size of this arrangement is q D n/m times larger than the size of the m-step walk (Fig. 4). Applying Eq. (5), we get df D 2 which does not depend on the embedding space. It is the same in one-dimensional, two-dimensional, and three-dimensional space. Note that Brownian trajectory is self-similar only in a statistical sense: each of the n concatenated copies of a m-step trajectory are different from each other, but their average square displacements are the same. Our eye can easily catch this statistical self-similarity. A random walk is a cloudy object of an elongated shape, its inertia ellipsoid is characterized by the average ratios of the squares of its axis: 12.1/2.71/1 [18]. The Brownian trajectory itself has a relevance in biology, since it may describe the changing of the electrical potential of a neuron [46], spreading of a colony [68], the foraging trajectory of a bacteria or an animal as well as the motion of proteins and other molecules in the cell. The self-avoiding walk (SAW) which has a smaller fractal dimension (df,SAW D 4/3 in d D 2 and df,SAW 1:7 in
Fractals in Biology
Fractals in Biology, Figure 5 A self-avoiding walk (a) of n D 104 in comparison with a random walk (b) of the same number of steps, both in d D 3. Both walks are produced by molecular dynamic simulations of the bead-on-a-string model of polymers. The hard core diameter of monomers in the SAW is equal to the bond length `, while for the RW it is zero. Comparing their fractal dimensions df,SAW 1:7, and df,RW D 2, one can predict that the average size (radius of inertia) of SAW must be n1/df,SAW 1/df,RW 2:3 times larger than that of RW. Indeed the average radius of inertia of SAW and RW are 98` and 40`, respectively. The fact that their complex shapes resemble leaving creatures was noticed by P. G. de Gennes in a cartoon published in his book “Scaling Concepts in Polymer Physics” [34]
d D 3) is a model of a polymer (Fig. 5) in a good solvent, such as for example a random coil conformation of a protein. Thus a SAW provides another example of a fractal object which has certain relevance in molecular biology. The fractal properties of SAWs are well established in the works of Nobel Laureates Flory and de Gennes [34,44,63]. Diffusion Limited Aggregation and Bacterial Colonies The properties of a Brownian trajectory explain the ramified structure of the DLA cluster. Since the Brownian trajectory is not straight, it is very difficult for it to penetrate deep into the fiords of a DLA cluster. With a much higher probability it hits the tips of the branches. Thus DLA cluster (Fig. 6a) has a tree-like structure with usually 5 main branches in 2 dimensions. The analytical determination of its fractal dimension is one of the most challenging questions in modern mathematics. In computer simulations it is defined by measuring the number of aggregated particles versus the radius of gyration of the aggregate (Fig. 6c). The fractal dimension thus found is approximately 1.71
Fractals in Biology, Figure 6 a A DLA cluster of n D 214 particles produced by the aggregation of the random walks on the square lattice (top). A comparison of a small BD cluster (b) and a DLA cluster (c). The color in b and c indicates the deposition time of a particle. The slopes of the loglog graphs of the mass of the growing aggregates versus their gyration radius give the values of their fractal dimensions in the limit of large mass
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Fractals in Biology, Figure 7 A typical DLA-like colony grown in a Petri dish with a highly viscous substrate (top). A change in morphology from DLA-like colony growth to a swimming chiral pattern presumably due to a cell morphotype transition (bottom) (from [15])
Fractals in Biology
in two dimensions and 2.50 in three dimensions [34]. It seems that as the aggregate grows larger the fractal dimension slightly increases. Note that if the deposition is made by particles moving along straight lines (ballistic deposition) the aggregate changes its morphology and becomes almost compact and circular with small fluctuation on the boundaries and small holes in the interior (Fig. 6b). The fractal dimension of the ballistic aggregate coincides with the dimension of embedding space. Ballistic deposition (BD) belongs to the same universality class as the Eden model, the first model to describe the growth of cancer. In this model, each cell on the surface of the cluster can produce an offspring with equal probability. This cell does not diffuse but occupies one of the empty spaces neighboring to the parent cell [13,83]. DLA is supposed to be a good model for growth of bacterial colonies in the regime when nutrients are coming via diffusion in a viscous media from the exterior. Under different conditions bacteria colonies observe various transition in their morphology. If the nutrient supply is greater, the colonies increase their fractal dimension and start to resemble BD. An interesting phenomenon is observed upon changing the viscosity of the substrate [15]. If the viscosity is small it is profitable for bacteria to swim in order to get to the regions with high nutrient concentrations. If the viscosity is large it is more profitable to grow in a static colony which is supplied by the diffusion of the nutrient from the outside. It seems that the way how the colony grows is inherited in the bacteria gene expression. When a bacteria grown at high viscosity is planted into a low viscosity substrate, its descendants continue to grow in a DLA pattern until a transition in a gene-expression happens in a sufficiently large number of neighboring bacteria which change their morphotype to a swimming one (Fig. 7). All the descendants of these bacteria start to swim and thus quickly take over the slower growing morphotype. Conversely, when a swimming bacteria is planted into a viscous media its descendants continue to swim until a reverse transition of a morphotype happens and the descendants of the bacteria with this new morphotype start a DLA-like growing colony. The morphotype transition can be also induced by fungi. Thus, it is likely that although bacteria are unicellular organisms, they exchange chemical signals similarly to the cells in the multicellular organisms, which undergo a complex process of cell differentiation during organism development (morphogenesis). There is evidence that the tree roots also follow the DLA pattern, growing in the direction of diffusing nutrients [43]. Coral reefs [80] whose life depends on the diffu-
sion of oxygen and nutrients also may to a certain degree follow DLA or BD patterns. Another interesting conjecture is that neuronal dendrites grow in vivo obeying the same mechanism [29]. In this case, they follow signaling chemicals released by other cells. It was also conjectured that even fingers of vertebrate organisms may branch in a DLA controlled fashion, resembling the pattern of viscous fingering which are observed when a liquid of low viscosity is pushed into a liquid of higher viscosity. Measuring Fractal Dimension of Real Biological Fractals There were attempts to measure fractal dimension of growing neurons [29] and other biological objects from photographs using a box-counting method and a circle method developed for empirical determination of fractal dimension. The circle method is the simplified version of the radial distribution function analysis described in the definition section. It consists of placing a center of a circle or a sphere of a variable radius R at each point of an object and counting the average number of other points of this object M(R) found inside these circles or spheres. The slope of ln M(R) vs. ln R gives an estimate of the fractal dimension. The box counting method consists of placing a square grid of a given spacing ` on a photograph and counting number of boxes n(`) needed to cover all the points belonging to a fractal set under investigation. Each box containing at least one point of a fractal set is likely to contain on average N(`) D `d f other points of this set, and n(`)N(`) D N, where N is the total number of the points. Thus for a fractal image n(`) N/`d f `d f . Accordingly the fractal dimension of the image can be estimated by the slope of a graph of ln n(`) vs. ln `. The problem is that real biological objects do not have many orders of self-similarity, and also, as we see above, only a skeleton or a perimeter of a real object has fractal properties. The box counting method usually produces curvy lines on a log–log paper with at best one decade of approximately constant slope (Fig. 8). What is most disappointing is that almost any image analyzed in this fashion produces a graph of similar quality. For example, an Einstein cartoon presented in [93] has the same fractal dimension as some of the growing neurons. Therefore, experimental determination of the fractal dimension from the photographs is no longer in use in scientific publications. However, in the 1980s and early 1990s when the computer scanning of images became popular and the fractals were at the highest point of their career, these exercises were frequent.
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Fractals in Biology, Figure 8 An image of a neuron and its box-counting analysis (a) together with the analogous analysis of an Einstein cartoon (b). The log–log plot of the box-counting data has a shape of a curve with the changing slope. Bars represent the local slope of this graph, which may be interpreted as some effective fractal dimension. However, in both graphs the slopes change dramatically from 1 to almost 1.7 and it is obvious that any comparison to the DLA growth mechanism based on these graphs is invalid (from [93])
For example, R. Voss [137] analyzed fractal dimensions of Chinese graphics of various historical periods. He found that the fractal dimension of the drawings fluctuated from century to century and was about 1.3 at the time of the highest achievements of this technique. It was argued that the perimeters of many natural objects such as perimeters of the percolation clusters and random walks have similar fractal dimension; hence the works of the artists who were able to reproduce this feature were the most pleasant for our eyes. Recently, the box counting method was used to analyze complex networks such as protein interaction networks (PIN). In this case the size of the box was defined as the maximal number of edges needed to connect any two nodes belonging to the box. The box-counting method appears to be a powerful tool for analyzing self-similarity of the network structure [120,121].
Percolation and Forest Fires It was also conjectured that biological habitats of certain species may have fractal properties. However it is not clear whether we have a true self-similarity or just an apparent mosaic pattern which is due to complex topography and geology of the area. If there is no reasonable theoretical model of an ecological process, which displays true fractal properties the assertions of the fractality of a habitat are pointless. One such model, which produces fractal clusters is percolation [26,127]. Suppose a lightning hits a random tree in the forest. If the forest is sufficiently dry, and the inflammable trees are close enough the fire will spread from a tree to a tree and can burn the entire forest. If the inflammable trees form a connected cluster, defined as such that its any two trees are connected with a path along which the fire can spread, all the trees in this cluster will
Fractals in Biology
be burnt down. Percolation theory guarantees that there is a critical density pc of inflammable trees below which all the connected clusters are finite and the fire will naturally extinguish burning only an infinitesimally small portion of the forest. In contrast, above this threshold there exists a giant cluster which constitutes a finite portion of the forest so that the fire started by a random lightning will on average destroy a finite portion of the forest. Exactly at the critical threshold, the giant cluster is a fractal with a fractal dimension df D 91/48 1:89. The probability to burn a finite cluster of mass S follows a power law P(S) S d/d f . Above and below the percolation threshold, this distribution is truncated by an exponential factor which specifies that clusters of linear dimensions exceeding a characteristic size are exponentially rare. The structure of the clusters which do not exceed this characteristic scale is also fractal, so if their mass S is plotted versus their linear size R (e. g. radius of inertia) it follows a power law S Rdf In the natural environment the thunderstorms happen regularly and they produce fractal patterns of burned down patches. However, if the density of trees is small, the patches are finite and the fires are confined. As the density the trees reaches the critical threshold, the next thunderstorm is likely to destroy the giant cluster of the forest, which will produce a bare patch of a fractal shape spreading over the entire forest. No major forest fire will happen until the new forest covers this patch creating a new giant cluster, because the remaining disconnected groves are of finite size. There is an evidence that the forest is the system which drives itself to a critical point [30,78,111] (Fig. 9). Suppose that each year there is certain number of lightning strokes per unit area nl . The average number of trees in a cluster is hSi D ajp pc j , where D 43/18 is one of the critical exponents describing percolation, a is a constant, and p is the number of inflammable trees per unit area. Thus the number of trees destroyed annually per unit area is nl ajpc p] . On the other hand, since the trees are growing, the number of trees per unit area increases annually by nt , which is the parameter of the ecosystem. At equilibrium, nl ajpc p] D nt . Accordingly, the equilibrium density of trees must reach pe D pc (anl /nt )1/ . For very low nl , pe will be almost equal to pc , and the chance that in the next forest fire a giant cluster which spans the entire ecosystem will be burned is very high. It can be shown that for a given chance c of hitting such a cluster in a random lightning stroke, the density of trees must reach p D pc f (c)L1/ , where L is the linear size of the forest, D 4/3 is the correlation critical exponent and f (c) > 0 is a logarithmically growing function of c.
Accordingly, if nt /nl > bL / , where b is some constant, the chance of getting a devastating forest fire is close to 100%. We have here a paradoxical situation: the more frequent are the forest fires, the least dangerous they are. This implies that the fire fighters should not extinguish small forest fires which will be contained by themselves. Rather they should annually cut a certain fraction of trees to decrease nt . As we see, the forest fires can be regarded as a self-organized critical (SOC) system which drive themselves towards criticality. As in many SOC systems, here there are two processes one is by several orders of magnitudes faster than the other. In this case they are the tree growing process and the lightning striking process. The model reaches the critical point if a tuning parameter nt /nl ! 1 and in addition nt ! 0 and L ! 1, which are quite reasonable assumptions. In a regular critical system a tuning parameter (e. g. temperature) must be in the vicinity of a specific finite value. In a SOC system the tuning parameter must be just large enough. There is an evidence, that the forest fires follow a power law distribution [109]. One can also speculate that the areas burned down by the previous fires shape fractal habitats for light loving and fire-resistant trees such as pines. The attempts to measure a fractal dimension of such habitats from aerial photographs are dubious due to the limitations discussed above and also because by adjusting a color threshold one can produce fractal-like clusters in almost any image. This artifact is itself a trivial consequence of the percolation theory. The frequently reported Zipf’s law for the sizes of colonies of various species including the areas and populations of the cities [146] usually arises not from fractality of the habitats but from preferential attachment growth in which the old colonies grow in proportion to their present population, while the new colonies may form with a small probability. The preferential attachment model is a very simple mechanism which can create a power-law distribution of colony sizes P(S) S 2 , where is a small correction which is proportional to the probability of formation of new colonies [25]. Other examples of power laws in biology [67], such as distributions of clusters in the metabolic networks, the distribution of families of proteins, etc. are also most likely come from the preferential attachment or also, in some cases, can arise as artifacts of specifically selected similarity threshold, which brings a network under consideration to a critical point of the percolation theory. The epidemic spreading can also be described by a percolation model. In this case the contagious disease spreads from a person to a person as the forest fire spreads from
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Fractals in Biology, Figure 9 A sequence of frames of a forest fire model. Each tree occupies a site on 500 500 square lattice. At each time step a tree (a colored site) is planted at a randomly chosen empty site (black). Each 10,000 time steps a lightning strikes a randomly chosen tree and the forest fire eliminates a connected cluster of trees. The frames are separated by 60,000 time steps. The color code indicates the age of the trees from blue (young) to red (old). The initial state of the system is an empty lattice. As the concentration of trees reaches percolation threshold (frame 3) a small finite cluster is burned. However it does not sufficiently decrease the concentration of trees and it continues to build up until a devastating forest fire occurs between frames 3 and 4, with only few green groves left. Between frames 4 and 5 several lightnings hit these groves and they are burned down, while surrounding patch of the old fire continues to be populated by the new trees. Between frame 5 and 6 a new devastating forest fire occurs. At the end of the movie huge intermittent forest fires produce gigantic patches of dense and rare groves of various ages
a tree to a tree. Analogously to the forest fire model, a person who caught the disease dies or recovers and becomes immune to the disease, so he or she cannot catch it again. This model is called susceptible-infective-removed (SIR) [62]. The main difference is that the epidemics spread not on a two-dimensional plane but on the network describing contacts among people [61,86]. This network is usually supposed to be scale free, i. e. the number of contacts different people have (degree) is distributed according to a an inverse power law [2,71]. As the epidemic spreads the number of connections in the susceptible population depletes [115] so the susceptible population comes to a percolation threshold after which the epidemic stops.
This model explains for example why the Black Death epidemic stopped in the late 14th century after killing about one third of the total European population. Critical Point and Long-Range Correlations Percolation critical threshold [26,127] is an example of critical phenomena the most well known of which is the liquid-gas critical point [124]. As one heats a sample of any liquid occupying a certain fraction of a closed rigid container, part of the liquid evaporates and the pressure in the container increases so that the sample of liquid remains at equilibrium with its vapor. However, at certain tempera-
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ture the visible boundary between the liquid at the bottom and the gas on the top becomes fuzzy and eventually the system becomes completely nontransparent as milk. This phenomena is called critical opalescence and the temperature, pressure, and density at which it happens is called a critical point of this liquid. For water, the critical point is Tc D 374 Cı , Pc D 220 atm and c D 330 kg/m3 . As the temperature goes above the critical point the system becomes transparent again, but the phase boundary disappears: liquid and gas cannot coexist above the critical temperature. They form a single phase, supercritical fluid, which has certain properties of both gas (high compressibility) and liquid (high density and slow diffusivity). At the critical point the system consists of regions of high density (liquid like) and low density (gas like) of all sizes from the size of a single molecule to several microns across which are larger than the wave length of visible light 0:5 m. These giant density fluctuations scatter visible light causing the critical opalescence. The characteristic linear size of the density fluctuations is called the correlation length which diverges at critical temperature as jT Tc j . The shapes of these density fluctuations are self-similar, fractal-like. The system can be represented as a superposition of a uniform set with the density corresponding to the average density of the system and a fractal set with the fractal dimension df . The density correlation function h(r) decreases for r ! 1 as r exp(r/), where d 2 C D d df and d D 1; 2; 3; : : : is the dimension of space in which the critical behavior is observed. The exponential cutoff sets the upper limit of the fractal density fluctuations to be equal to the correlation length. The lower limit of the fractal behavior of the density fluctuations is one molecule. As the system approaches the critical point, the range of fractal behavior increases and can reach at least three orders of magnitude in the narrow vicinity of the critical point. The intensity of light scattered by density fluctuations at certain angle can be expressed via the Fourier transform of the density R correlation function S(f) h(r) exp(ir f)dr, where the integral is taken over d-dimensional space. This experimentally observed quantity S(f ) is called the structure factor. It is a powerful tool to study the fractal properties of matter since it can uncover the presence of a fractal set even if it is superposed with a set of uniform density. If the density correlation function has a power law behavior h(r) r , the structure factor also has a power law behavior S( f ) f d . For the true fractals with df < d, the average density over the entire embedding space is zero, so the correlation function coincides with the average density of points of the set at certain distance from a given point, h(r) rdC1 dM(r)/dr r d f d , where M(r) r d f is
the mass of the fractal within the radius r from a given point. Thus, for a fractal set of points with fractal dimension df , the structure factor has a simple power law form S( f ) f d f . Therefore whenever S( f ) f ˇ , ˇ < d is identified with the fractal dimension and the system is said to have fractal density fluctuations even if the average density of the system is not zero. There are several methods of detecting spatial correlations in ecology [45,108] including box-counting [101]. In addition one can study the density correlation function h(r) of certain species on the surface of the Earth defined as h(r) D N(r)/2 rr N/A, where N(r) is the average number of the representatives of a species within a circular rim of radius r and width r from a given representative, N is the total population, and A is the total area. For certain species, the correlation function may follow about one order of magnitude of a fractal (power-law) behavior [9,54,101]. One can speculate that there is some effective attraction between the individuals such as cooperation (mimicking the van der Walls forces between molecules) and a tendency to spread over the larger territory (mimicking the thermal motion of the particles). The interplay of these two tendencies may produce a fractal behavior like in the vicinity of the critical point. However, there is no reason of why the ecological system although complex and chaotic [81] should drive itself to a critical point. Whether or not there is a fractal behavior, the density correlation function is a useful way to describe the spatial distribution of the population. It can be applied not only in ecology but also in physiology and anatomy to describe the distribution of cells, for example, neurons in the cortex [22]. Lévy Flight Foraging A different set of mathematical models which produce fractal patterns and may be relevant in ecology is the Lévy flight and Lévy walk models [118,147]. The Lévy flight model is a generalization of a random walk, in which the distribution of the steps (flights) of length ` follows a power law P(`) ` with < 3. Such distributions do not have a finite variance. The probability density of the landing points of the Lévy flight converges not to a Gaussian as for a normal random walk with a finite step variance but to a Lévy stable distribution with the parameter ˛ 1. The landing points of a Lévy flight form fractal dust similar to a Cantor set with the fractal dimension df D ˛. It was conjectured that certain animals may follow Lévy flights during foraging [103,116,132,133]. It is a mathematical theorem [23,24] that in case when targets
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are scarce but there is a high probability that a new target could be discovered in the vicinity of the previously found target, the optimal strategy for a forager is to perform a Lévy flight with D 2 C , where is a small correction which depends on the density of the target sites, the radius of sight of the forager and the probability to find a new target in the vicinity of the old one. In this type of “inverse square” foraging strategy a balance is reached between finding new rich random target areas and returning back to the previously visited area which although depleted, may still provide some food necessary for survival. The original report [132] of the distribution of the flight times of wandering albatross was in agreement with the theory. However, recent work [39] using much longer flight time records and more reliable analysis showed that the distribution of flight times is better described by a Poisson distribution corresponding to regular random walk rather then by a power law. Several other reports of the Lévy flight foraging are also found dubious. The theory [23] however predicts that the inverse square law of foraging is optimal only in case of scarce sources distribution. Thus if the harvest is good and the food is plenty there is no reason to discover the new target locations and the regular random walk strategy becomes the most efficient. Subsequent observations show that power laws exist in some other marine predator search behavior [119]. In order to find a definitive answer, the study must be repeated for the course of many successive years characterized by different harvests. Once the miniature electronic devices for tracing animals are becoming cheaper and more efficient, a new branch of electronic ecology is emerging with the goal of quantifying migratory patterns and foraging strategies of various species in various environments. Whether or not the foraging patterns are fractal, this novel approach will help to establish better conservational policy with scientifically sound borders of wild-life reservations. Recent observations indicate that human mobility patterns might also possess Lévy flight properties [49].
pose that there is a population N t of a certain species enclosed in a finite habitat (e. g. island) at time t. Here t is an integer index which may denote year. Suppose that at time t C 1 the population becomes
Dynamic Fractals
Fractals and Time Series
It is not necessary that a fractal is a real geometrical object embedded in the regular three-dimensional space. Fractals can be found in the properties of the time series describing the behavior of the biological objects. One classical example of a biological phenomenon which under certain condition may display fractal properties is the famous logistic map [38,82,95,123] based on the ideas of a great British economist and demographer Robert Malthus. Sup-
As we can see in the previous section, even simple systems characterized by nonlinear feedbacks may display complex temporal behavior, which often becomes chaotic and sometimes to fractal. Obviously, such features must be present in the behavior of the nervous system, in particular in the human brain which is probably the most complex system known to contemporary science. Nevertheless, the source of fractal behavior can be sometimes trivial with-
N tC1 D bN t dN t2 ; where b is the natural birth rate and d is the death rate caused by the competition for limited resources. In the most primitive model, the animals are treated as particles randomly distributed over area A annihilating at each time step if the distance between them is less than r. In this case d D r2 /A. The normalized population x t D dN t /b obeys a recursive relation with a single parameter b: x tC1 D bx t (1 x t ) : The behavior of the population is quite different for different b. For b 1 the population dies out as t ! 1. For 1 < b b0 D 3 the population converges to a stable size. If b n1 < b b n , the population repetitively visits 2n values 0 < x1 ; : : : ; x2n < 1 called attractors as t ! 1. The bifurcation points 3 D b0 < b1 < b2 < < b n < b1 3:569945672 converge to a critical value b1 , at which the set of population sizes becomes a fractal with fractal dimension df 0:52 [50]. This fractal set resembles a Cantor set confined between 0 and 1. For b1 < b < 4, the behavior is extremely complex. At certain values b chaos emerges and the behavior of the population become unpredictable, i. e. exponentially sensitive to the initial conditions. At some intervals of parameter b, the predictable behavior with a finite attractor set is restored. For b > 4, the population inevitably dies out. Although the set of attractors becomes truly fractal only at certain values of the birth rate, and the particular value of the fractal dimension does not have any biological meaning, the logistic map has a paradigmatic value in the studies of population dynamics with an essentially Malthusian take home massage: excessive birth rate leads to disastrous consequences such as famines coming at unpredictable times, and in the case of Homo sapiens to devastating wars and revolutions.
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out evidence of any cognitive ability. An a example of such a trivial fractal behavior is the random firing of a neuron [46] which integrates through its dendritic synapses inhibitory and excitatory signals from its neighbors. The action potential of such a neuron can be viewed as performing a one-dimensional random walk going down if an inhibitory signal comes from a synapse or going up if an excitatory signal comes form a different synapse. As soon as the action potential reaches a firing threshold the neuron fires, its action potential drops to an original value and the random walk starts again. Thus the time intervals between the firing spikes of such a neuron are distributed as the returning times of a one-dimensional random walk to an origin. It is well known [42,58,104,141], that the probability density P(t) of the random walk returns scales as (t) with D 3/2. Accordingly, the spikes on the time axis form a fractal dust with the fractal dimension df D 1 D 1/2. A useful way to study the correlations in the time series is to compute its autocorrelation function and its Fourier transform which is called power spectrum, analogous to the structure factor for the spatial correlation analysis. Due to the property of the Fourier transform to convert a convolution into a product, the power spectrum is also equal to the square of the Fourier transform of the original time series. Accordingly it has a simple physical meaning telling how much energy is carried in a certain frequency range. In case of a completely uncorrelated signal, the power spectrum is completely flat, which means that all frequencies carry the same energy as in white light which is the mixture of all the rainbow colors of different frequencies. Accordingly a signal which has a flat power spectrum is called white noise. If the autocorrelation function C(t) decays for t ! 1 as C(t) t , where 0 < < 1, the power spectrum S(f ) of this time series diverges as f ˇ , with ˇ D 1 . Thus the LRPLC in the time series can be detected by studying the power spectrum. There are alternative ways of detecting LRPLC in time series, such as Hurst analysis [41], detrended fluctuation analysis (DFA) [19,98] and wavelet analysis [4]. These methods are useful for studying short time series for which the power spectrum is too noisy. They measure the Hurst exponent ˛ D (1 C ˇ)/2 D 1 /2. It can be shown [123] that for a time series which is equal to zero everywhere except at points t1 ; t2 ; : : : ; t n ; : : : , at which it is equal to unity and these points form a fractal set with fractal dimension df , the time autocorrelation function C(t) decreases for t ! 1 as C(t) t , where D 1 df . Therefore the power spectrum S(f ) of this time series diverges for f ! 0 as f ˇ , with ˇ D 1 D df . Accordingly, for the random
walk model of neuron firing, the power spectrum is characterized by ˇ D 1/2. In the more general case, the distribution of the intervals t n D t n t n1 can decay as a power law P(t) (t) , with 1 < < 3. For < 2, the set t1 ; t2 ; : : : ; t n is a fractal, and the power spectrum decreases as power law S( f ) f d f D f C1 . For 2 < < 3, the set t1 ; t2 ; : : : ; t n has a finite density, with df D D D 1. However, the power spectrum and the correlation function maintain their power law behavior for < 3. This behavior indicates that although the time series itself is uniform, the temporal fluctuations remain fractal. In this case, the exponent ˇ characterizing the low frequency limit is given by ˇ D 3 . The maximal value of ˇ D 1 is achieved when D 2. This type of signal is called 1/ f noise or “red” noise. If 3, ˇ D 0 in the limit of low frequencies and we again recover white noise. The physical meaning of 1/ f noise is that the temporal correlations are of infinite range. For the majority of the processes in nature, temporal correlations decay exponentially C(t) exp(t/), where the characteristic memory span is called relaxation or correlation time. For a time series with a finite correlation time , the power spectrum has a Lorentzian shape, namely it stays constant for f < 1/ and decreases as 1/ f 2 for f > 1/. The signal in which S( f ) 1/ f 2 is called brown noise, because it describes the time behavior of the one-dimensional Brownian motion. Thus for the majority of the natural processes, the time spectrum has a crossover from a white noise at low frequencies to a brown noise at high frequencies. The relatively unusual case when S( f ) f ˇ , where 0 < ˇ < 1 is called fractal noise because as we see above it describes the behavior of the fractal time series. 1/ f -noise is the special type of the fractal noise corresponding to the maximal value of ˇ, which can be achieved in the limit of low frequencies. R. Voss and J. Clarke [135,136] had analyzed the music written by different composers and found that it follows 1/ f noise over at least three orders of magnitude. It means that music does not have a characteristic time-scale. There is an evidence that physiological process such as heart-beat, gate, breath and sleeping patterns as well as certain types of human activity such as sending e-mails has certain fractal features [6,16,20,27,47,55,57, 72,73,97,99,100,113,126,129]. A. Goldberger [99,126] suggested that music is pleasant for us because it mimics the fractal features of our physiology. It has to be pointed out that in all these physiological time series there is no clear power law behavior expanding over many orders of magnitude. There is also no simple clear explanation of the origins of fractality. One possible mechanism could be due
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the distribution of the return times of a random walk, which has been used to explain the sleeping patterns and response to the e-mails. SOC and Biological Evolution Self-organized criticality (SOC) [8,92] describes the behavior of the systems far form equilibrium, the general feature of which is a slow increase of strain which is interrupted by an avalanche-like stress release. These avalanches are distributed in a power law fashion. The power spectrum of an activity at a given spatial site is described by a fractal noise. One of the most successful application of SOC is the explanation of the power-law distribution of the magnitudes of the earthquakes (Gutenberg–Richter’s law) [89,130]. The simple physical models of self-organized criticality are invasion percolation [26,127], sand-pile model [8], and Bak–Sneppen model of biological evolution [7]. In the one-dimensional Bak–Sneppen model, an ecosystem is represented by a linear chain of the pray-predator relationships in with each species is represented by a site on a straight line surrounded by its predator (a site to the right) and a pray (a site to the left). Each site is characterized by its fitness f which at the beginning is uniformly distributed between 0 and 1. At each time step, a site with the lowest fitness becomes extinct and is replaced by a mutated species with a new fitness randomly taken from a uniform distribution between 0 and 1. The fitnesses of its two neighbors (predator and prey) are also changed at random. After certain equilibration time, the fitness of almost all the species except a few becomes larger than a certain critical value fc . These few active species with low fitness which can spontaneously mutate form a fractal set on the pray-predator line. The activity of each site can be represented by a time series of mutations shown as spikes corresponding to the times of individual mutations at this site. The power spectrum of this time series indicates the presence of the fractal noise. At a steady state the minimal fitness value which spontaneously mutates fluctuates below fc with a small probability P( ) comes into the interval between fc and fc . The distribution of the first return times t to a given -vicinity of fc follows a power law with an -dependent exponential cut-off. Since each time step corresponds to a mutation, the time interval for which f stays below fc corresponds to an avalanche of mutations caused by a mutation of a very stable species with f > fc . Accordingly one can speculate, that evolution goes as a punctuated equilibrium so that an extinction of a stable species causes a gigantic extinction of many other species which hitherto have been well adopted. The problem with this SOC model is the def-
inition of a time step. In order to develop a realistic model of evolution one needs to assume that the real time needed for a spontaneous mutation of species with different fitness dramatically increases with f , going for example as exp( f A), where A is a large value. Unfortunately, it is impossible to verify the predictions of this model because paleontological records do not provide us with a sufficient statistics. Fractal Features of DNA Sequences DNA molecules are probably the largest molecules in nature [139]. Each strand of DNA in large human chromosomes consist of about 108 monomers or base-pairs (bp) which are adenine (A), cytosine (C), guanine (G), and thymine (T). The length of this molecule if stretched would reach several centimeters. The geometrical packing of DNA in a cell resembles a self-similar structure with at least 6 levels of packing: a turn of the double helix (10 bp), a nucleosome (200 bp), a unit of 30 nm fiber (6 nucleosomes), a loop domain ( 100 units of 30 nm fiber), a turn of a metaphase chromosome ( 100 loop domains), a metaphase chromosome ( 100 turns). The packing principle is quite similar to the organization of the information in the library: letters form lines, lines form pages, pages form books, books are placed on the shelves, shelves form bookcases, bookcases form rows, and rows are placed in different rooms. This structure however is not a rigorous fractal, because packing of the units on different levels follows different organization principles. The DNA sequence treated as a sequence of letters also has certain fractal properties [4,5,19,69,70,96]. This sequence can be transformed into a numerical sequence by several mapping rules: for example A rule, in which A is replaced by 1 and C, T, G are replaced by 0, or SW rule in which strongly bonded bp (C and G) are replaced by 1 and weakly bonded bp (A and T) are replaced by 1. Purine-Pyrimidine (RY) mapping rule (A,G: +1; C,T:1) and KM mapping rule (A,C:+1; G,T -1) are also possible. Power spectra of such sequences display large regions of approximate power law behavior in the range from f D 102 to f D 108 . For SW mapping rule we has almost perfect 1/ f noise in the region of low frequencies (Fig. 10). This is not surprising because chromosomes are organized in a large CG rich patches followed by AT rich patches called isochors which extend over millions of bp. The changing slope of the power spectra for different frequency ranges clearly indicates that DNA sequences are also not rigorous fractals but rather a mosaic structures with different organization principles on different length-scales [60,98]. A possible relation between the frac-
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Fractals in Biology, Figure 10 a Power spectra for seven different mapping rules computed for the Homo sapiens chromosome XIV, genomic contig NT_026437. The result is obtained by averaging 1330 power spectra computed by fast Fourier transform for non-overlapping segments of length N D 216 D 65536. b Power spectra for SW, RY, and KM mapping rules for the same contig extended to the low frequency region characterizing extremely long range correlations. The extension is obtained by extracting low frequencies from the power spectra computed by FFT with N D 224 16 106 bp. Three distinct correlation regimes can be identified. High frequency regime (f < 0:003) is characterized by small sharp peaks. Medium frequency regime (0:5 105 < f < 0:003) is characterized by approximate power-law behavior for RY and SW mapping rules with exponent ˇM D 0:57. Low frequency regime (f < 0:5 105 ) is characterized by ˇ D 1:00 for SW rule. The high frequency regime for RY rule can be approximated by ˇH D 0:16 in agreement with the data of Fig. 11. c RY Power spectra for the entire genome of E. coli (bacteria), S. cerevisae (yeast) chromosome IV, H. sapiens (human) chromosome XIV and the largest contig (NT_032977.6) on the chromosome I; and C. elegans (worm) chromosome X. It can be clearly seen that the high frequency peaks for the two different human chromosomes are exactly the same, while they are totally different from the high frequency peaks for other organisms. These high frequency peaks are associated with the interspersed repeats. One can also notice the presence of enormous peaks for f D 1/3 in E. coli and yeast, indicating that their genomes do not have introns, so that the lengths of coding segments are very large. The C. elegans data can be very well approximated by power law correlations S(f) f 0:28 for 104 < f < 102 . d Log–log plot of the RY power spectrum for E. coli with subtracted white noise level versus jf 1/3j. It shows a typical behavior for a signal with finite correlation length, indicating that the distribution of the coding segments in E. coli has finite average square length of approximately 3 103 bp
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Fractals in Biology, Figure 11 RY Power spectra averaged over all eukaryotic sequences longer than 512 bp, obtained by FFT with window size 512. Upper curve is average over 29,453 coding sequences; lower curve is average over 33,301 noncoding sequences. For clarity, the power spectra are shifted vertically by arbitrary quantities. The straight lines are least squares fits for second decade (Region M). The values of ˇM for coding and noncoding DNA obtained from the slopes of the fits are 0.03 and 0.21, respectively (from [21])
tal correlations of DNA sequences and packing of DNA molecules was suggested in [52,53]. An intriguing property of the DNA of multicellular organisms is that an overwhelming portion of it (97% in case of humans) is not used for coding proteins. It is interesting that the percent of non-coding DNA increases with the complexity of an organism. Bacteria practically do not have non-coding DNA, and Yeast has only 30% of it. The coding sequences form genes ( 104 bp) which carry information for one protein. The genes of multicellular organisms are broken by many noncoding intervening sequences (introns). Only exons which are short ( 102 bp) coding sequences located between introns ( 103 bp) are eventually translated into a protein. The genes themselves are separated by very long intergenic sequences 105 bp. Thus the coding structure of DNA resembles a Cantor set. The purpose and properties of coding DNA are well understood. Each three consequent bp form a codon, which is translated into one amino acid of the protein sequence. Accordingly, the power spectrum computed for the coding DNA has a characteristic peak at f D 1/3 corresponding to the inverse codon length (Figs. 11 and 12). Coding DNA is highly conserved and the power spectra of coding DNA of different organisms are very similar and in case of mammals are indistinguishable (Fig. 12). The properties of noncoding DNA are very different. Non coding DNA contains a lot of useful information in-
Fractals in Biology, Figure 12 Comparison of the correlation properties of coding and non-coding DNA of different mammals. Shown RY power spectra averaged over all complimentary DNA sequences of Homo sapiens (HS-C) and Mus musculus (mouse) (MM-C). The complimentary DNA sequences are obtained from messenger RNA by reverse transcriptase and thus lack non-coding elements. They are characterized by huge peaks at f D 1/3, corresponding to the inverse codon length 3 bp. The power spectra for human and mouse are almost indistinguishable. Also shown RY power spectra of large continuously sequenced segments of chromosomes (contigs) of about 107 bp long for mouse (MM), human (HS) and chimpanzee (PT, Pan troglodytes). Their power spectra have different high frequency peaks absent in the coding DNA power spectra: a peak at f D 1/2, corresponding to simple repeats and several large peaks in the range from f D 1/3 to f D 1/100 corresponding to interspersed repeats. Notice that the magnitudes of these peaks are similar for humans and chimpanzees (although for humans they are slightly larger, especially the peak at 80 bp corresponding to the long interspersed repeats) and much larger than those of mouse. This means that mouse has much smaller number of the interspersed repeats than primates. On the other hand, mouse has much larger fraction of dimeric simple repeats indicated by the peak at f D 1/2
cluding protein binding sites controlling gene transcription and expression and other regulatory sequences. However its overwhelming fraction lacks any known purpose. This “junk” DNA is full of simple repeats such as CACACACA. . . as well as the interspersed repeats or retroposons which are virus-like sequences inserting themselves in great number of copies into the intergenic DNA. It is a great challenge of molecular biology and genetics to understand the meaning of non-coding DNA and even to learn how to manipulate it. It would be very interesting to create transgenic animals without non-coding DNA and test if their phenotype will differ from that of the wild type species. The non-coding DNA even for very closely
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related species can significantly differ (Fig. 12). The length of simple repeats varies even for close relatives. That is why simple repeats are used in forensic studies. The power spectra of non-coding DNA significantly differ from those of coding DNA. Non-coding DNA does not have a peak at f D 1/3. The presence of simple repeats make non-coding DNA more correlated than coding DNA on a scale from 10 to 100 bp [21] (Fig. 11). This difference observed in 1992 [70,96] lead to the hypothesis that noncoding DNA is governed by some mutation-duplication stochastic process which creates long-range (fractal) correlations, while the coding DNA lacks long-range correlations because it is highly conserved. Almost any mutation which happens in coding DNA alter the sequence of the corresponding protein and thus may negatively affect its function and lead to a non-viable or less fit organism. Researchers have proposed using the difference in the long-range correlations to find the coding sequences in the sea of non-coding DNA [91]. However this method appears to be unreliable and today the non-coding sequences are found with much greater accuracy by the bioinformatics methods based on sequence similarity to the known proteins. Bioinformatics has developed powerful methods for comparing DNA of different species. Even a few hundred bp stretch of the mitochondrial DNA of a Neanderthal man can tell that Neanderthals diverged from humans about 500 000 years ago [65]. The power spectra and other correlation methods such as DFA or wavelet analysis does not have such an accuracy. Never the less, power spectra of the large stretches of DNA carry important information on the evolutionary processes in the DNA. They are similar for different chromosomes of the same organism, but differ even for closely related species (Fig. 12). In this sense they can play a role similar to the role played by X-ray or Raman spectra for chemical substances. A quick look at them can tell a human and a mouse and even a human and a monkey apart. Especially interesting is the difference in peak heights produced by different interspersed repeats. The height of these peaks is proportional to the number of the copies of the interspersed repeats. According to this simple criterion, the main difference between humans and chimps is the insertion of hundreds of thousands of extra copies of interspersed repeats into human DNA [59]. Future Directions All the above examples show that there are no rigorous fractals in biology. First of all, there are always a lower and an upper cutoff of the fractal behavior. For example, a polymer in a good solvent which is probably the most
rigorous of all real fractals in biology has the lower cutoff corresponding to the persistence length comprised of a few monomers and the upper cutoff corresponding to the length of the entire polymer or to the size of the compartment in which it is confined. For the hierarchical structures such as trees and lungs, in addition to the obvious lower and upper cutoffs, the branching pattern changes from one generation to the next. In the DNA, the packing principles employ different mechanisms on the different levels of packing. In bacterial colonies, the lower cutoff is due to some tendency of bacteria to clump together analogous to surface tension, and the upper cutoff is due to the finite concentration of the nutrients, which originally are uniformly distributed on the Petri dish. In the ideal DLA case, the concentration of the diffusing particles is infinitesimally small, so at any given moment of time there is only one particle in the vicinity of the aggregate. The temporal physiological series are also not exactly self-similar, but are strongly affected by the daily schedule with a characteristic frequency of 24 h and shorter overtones. Some of these signals can be described with help of multifractals. Fractals in ecology are limited by the topographical features of the land which may be also fractal due to complex geological processes, so it is very difficult to distinguish whether certain features are caused by biology or geology. The measurements of the fractal dimension are hampered by the lack of statistics, the noise in the image or signal, and by the crossovers due to intrinsic features of the system which is differently organized on different length scales. So very often the mosaic organization of the system with patches of several fixed length scales can be mistakenly identified with the fractal behavior. Thus the use of fractal dimension or Hurst exponent for diagnostics or distinguishing some parts of the system from one another has a limited value. After all, the fractal dimension is only one number which is usually obtained from a slope of a least square linear fit of a log–log graph over a subjectively identified range of values. Other features of the power spectrum, such as peaks at certain characteristic frequencies may have more biological information than fractal dimension. Moreover, the presence of certain fractal behavior may originate from simple physical principles, while the deviation from it may indicate the presence of a nontrivial biological phenomenon. On the other hand, fractal geometry is an important concept which can be used for qualitative understanding of the mechanisms behind certain biological processes. The use of similar organization principles on different length scales is a remarkable feature, which is certainly em-
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ployed by nature to design the shape and behavior of living organisms. Though fractals themselves may have a limited value in biology, the theory of complex systems in which they often emerge continues to be a leading approach in understanding life. One of the most impelling challenges of modern interdisciplinary science which involves biology, chemistry, physics, mathematics, computer science, and bioinformatics is to build a comprehensive theory of morphogenesis. Such a great mind as de Gennes turned to this subject in his late years [35,36,37]. In these studies the theory of complex networks [2,12,48,67,90] which describe interactions of biomolecules with many complex positive and negative feedbacks will take a leading part. Challenges of the same magnitude face researches in neuroscience, behavioral science, and ecology. Only complex interdisciplinary approach involving the specialists in the theory of complex systems may lead to new breakthroughs in this field.
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Fractals in Geology and Geophysics DONALD L. TURCOTTE Department of Geology, University of California, Davis, USA Article Outline Glossary Definition of the Subject Introduction Drainage Networks Fragmentation Earthquakes Volcanic Eruptions Landslides Floods Self-Affine Fractals Topography Earth’s Magnetic Field Future Directions Bibliography Glossary Fractal A collection of objects that have a power-law dependence of number on size. Fractal dimension The power-law exponent in a fractal distribution. Definition of the Subject The scale invariance of geological phenomena is one of the first concepts taught to a student of geology. When a photograph of a geological feature is taken, it is essential to include an object that defines the scale, for example, a coin or a person. It was in this context that Mandelbrot [7] introduced the concept of fractals. The length of a rocky coastline is obtained using a measuring rod with a specified length. Because of scale invariance, the length of the coastline increases as the length of the measuring rod decreases according to a power law. It is not possible to obtain a specific value for the length of a coastline due to small indentations down to a scale of millimeters or less. A fractal distribution requires that the number of objects N with a linear size greater than r has an inverse power-law dependence on r so that C (1) rD where C is a constant and the power D is the fractal dimension. This power-law scaling is the only distribution that is ND
scale invariant. However, the power-law dependence cannot be used to define a statistical distribution because the integral of the distribution diverges to infinity either for large values or small values of r. Thus fractal distributions never appear in compilations of statistical distributions. A variety of statistical distributions have power-law behavior either at large scales or small scales, but not both. An example is the Pareto distribution. Many geological phenomena are scale invariant. Examples include the frequency-size distributions of fragments, faults, earthquakes, volcanic eruptions, and landslides. Stream networks and landforms exhibit scale invariance. In terms of these applications there must always be upper and lower cutoffs to the applicability of a fractal distribution. As a specific application consider earthquakes on the Earth. The number of earthquakes has a power-law dependence on the size of the rupture over a wide range of sizes. But the largest earthquake cannot exceed the size of the Earth, say 104 km. Also, the smallest earthquake cannot be smaller than the grain size of rocks, say 1 mm. But this range of scales is 1010 . Actual earthquakes appear to satisfy fractal scaling over the range 1 m to 103 km. An example of fractal scaling is the number-area distribution of lakes [10], this example is illustrated in Fig. 1. Excellent agreement with the fractal relation given in Eq. (1) is obtained taking D D 1:90. The linear dimension r is taken to be the square root of the area A and the power-law (fractal) scaling extends from r D 100 m to r D 300 km. Introduction Fractal scaling evolved primarily as an empirical means of correlating data. A number of examples are given below. More recently a theoretical basis has evolved for the applicability of fractal distributions. The foundation of this basis is the concept of self-organized criticality. A number of simple computational models have been shown to yield fractal distributions. Examples include the sand-pile model, the forest-fire model, and the slider-block model. Drainage Networks Drainage networks are a universal feature of landscapes on the Earth. Small streams merge to form larger streams, large streams merge to form rivers, and so forth. Strahler [16] quantified stream networks by introducing an ordering system. When two like-order streams of order i merge they form a stream of order i C 1. Thus two i D 1 streams merge to form a i D 2 stream, two i D 2 streams merge to from a i D 3 stream and so forth. A bifurcation
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self-similar, side-branching topology was developed. Applications to drainage networks have been summarized by Peckham [11] and Pelletier [13]. Fragmentation
Fractals in Geology and Geophysics, Figure 1 Dependence of the cumulative number of lakes N with areas greater than A as a function of A. Also shown is the linear dimension r which is taken to be the square root of A. The straight-line correlation is with Eq. (1) taking the fractal dimension D D 1:90
ratio Rb is defined by Rb D
Ni N iC1
(2)
where N i is the number of streams of order i. A length order ratio Rr is defined by Rr D
r iC1 ri
(3)
where ri is the mean length of streams of order i. Empirically both Rb and Rr are found to be nearly constant for a range of stream orders in a drainage basin. From Eq. (1) the fractal dimension of a drainage basin DD
ln(N i /N iC1 ) ln Rb D ln(r iC1 /r i ) ln Rr
(4)
Typically Rb D 4:6, Rr D 2:2, and the corresponding fractal dimension is D D 1:9. This scale invariant scaling of drainage networks was recognized some 20 years before the concept of fractals was introduced in 1967. A major advance in the quantification of stream networks was made by Tokunaga [17]. This author was the first to recognize the importance of side branching, that is some i D 1 streams intersect i D 2, i D 3, and all higher-order streams. Similarly, i D 2 streams intersect i D 3 and higher- order streams and so forth. A fully
An important application of power-law (fractal) scaling is to fragmentation. In many examples the frequency-mass distributions of fragments are fractal. Explosive fragmentation of rocks (for example in mining) give fractal distributions. At the largest scale the frequency size distribution of the tectonic plates of plate tectonics are reasonably well approximated by a power-law distribution. Fault gouge is generated by the grinding process due to earthquakes on a fault. The frequency-mass distribution of the gouge fragments is fractal. Grinding (comminution) processes are common in tectonics. Thus it is not surprising that fractal distributions are ubiquitous in geology. As a specific example consider the frequency-mass distribution of asteroids. Direct measurements give a fractal distribution. Since asteroids are responsible for the impact craters on the moon, it is not surprising that the frequencyarea distribution of lunar craters is also fractal. Using evidence from the moon and a fractal extrapolation it is estimated that on average, a 1m diameter meteorite impacts the earth every year, that a 100 m diameter meteorite impacts every 10,000 years, and that a 10 km diameter meteorite impacts the earth every 100,000,000 years. The classic impact crater is Meteor Crater in Arizona, it is over 1 km wide and 200 m deep. Meteor Crater formed about 50,000 years ago and it is estimated that the impacting meteorite had a diameter of 30 m. The largest impact to occur in the 20th century was the June 30, 1908 Tunguska event in central Siberia. The impact was observed globally and destroyed over 1000 km2 of forest. It is believed that this event was the result of a 30 m diameter meteorite that exploded in the atmosphere. One of the major global extinctions occurred at the Cretaceous/Tertiary boundary 65 million years ago. Some 65% of the existing species were destroyed including dinosaurs. This extinction is attributed to a massive impact at the Chicxulub site on the Yucatan Peninsula, Mexico. It is estimated that the impacting meteorite had a 10 km diameter. In addition to the damage done by impacts there is evidence that impacts on the oceans have created massive tsunamis. The fractal power-law scaling can be used to quantify the risk of future impacts. Earthquakes Earthquakes universally satisfy several scaling laws. The most famous of these is Gutenberg–Richter frequency-
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magnitude scaling. The magnitude M of an earthquake is an empirical measure of the size of an earthquake. If the magnitude is increased by one unit it is observed that the cumulative number of earthquakes greater than the specified magnitude is reduced by a factor of 10. For the entire earth, on average, there is 1 magnitude 8 earthquake per year, 10 magnitude 7 earthquakes per year, and 100 magnitude 6 earthquakes per year. When magnitude is converted to rupture area a fractal relation is obtained. The numbers of earthquakes that occur in a specified region and time interval have a power-law dependence on the rupture area. The validity of this fractal scaling has important implications for probabilistic seismic risk assessment. The number of small earthquakes that occur in a region can be extrapolated to estimate the risk of larger earthquakes [1]. As an example consider southern California. On average there are 30 magnitude 4 or larger earthquakes per year. Using the fractal scaling it is estimated that the expected intervals between magnitude 6 earthquakes will be 3 years, between magnitude 7 earthquakes will be 30 years, and between magnitude 8 earthquakes will be 300 years. The fractal scaling of earthquakes illustrate a useful aspect of fractal distributions. The fractal distribution requires two parameters. The first parameter, the fractal dimension D (known as the b-value in seismology), gives the dependence of number on size (magnitude). For earthquakes the fractal dimension is almost constant independent of the tectonic setting. The second parameter gives the level of activity. For example, this can be the number of earthquakes greater than a specified magnitude in a region. This level of activity varies widely and is an accepted measure of seismic risk. The level is essentially zero in states like Minnesota and is a maximum in California. Volcanic Eruptions There is good evidence that the frequency-volume statistics of volcanic eruptions are also fractal [9]. Although it is difficult to quantify the volumes of magma and ash associated with older eruptions, the observations suggest that an eruption with a volume of 1 km3 would be expected each 10 years, 10 km3 each 100 years, and 100 km3 each 1000 years. For example, the 1991 Mount Pinatubo, Philippines eruption had an estimated volume of about 5 km3 . The most violent eruption in the last 200 years was the 1815 Tambora, Indonesia eruption with an estimated volume of 150 km3 . This eruption influenced the global climate in 1816 which was known as the year without a summer. It is estimated that the Long Valley, California eruption with an age of about 760,000 years had a volume of about
600 km3 and the Yellowstone eruptions of about 600,000 years ago had a volume of about 2000 km3 . Although the validity of the power-law (fractal) extrapolation of volcanic eruption volumes to long periods in the past can be questioned, the extrapolation does give some indication of the risk of future eruptions to global climate. There is no doubt that the large eruptions that are known to have occurred on time scales of 105 to 106 years would have a catastrophic impact on global agricultural production. Landslides Landslides are a complex natural phenomenon that constitutes a serious natural hazard in many countries. Landslides also play a major role in the evolution of landforms. Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt, or a large storm. The landslide event can include a single landslide or many thousands. The frequency-area distribution of a landslide event quantifies the number of landslides that occur at different sizes. It is generally accepted that the number of large landslides with area A has a power-law dependence on A with an exponent in the range 1.3 to 1.5 [5]. Unlike earthquakes, a complete statistical distribution can be defined for landslides. A universal fit to an inversegamma distribution has been found for a number of event inventories. This distribution has a power-law (fractal) behavior for large landslides and an exponential cut-off for small landslides. The most probable landslides have areas of about 40 m2 . Very few small landslides are generated. As a specific example we consider the 11,111 landslides generated by the magnitude 6.7 Northridge (California) earthquake on January 17, 1994. The total area of the landslides was 23.8 km2 and the area of the largest landslide was 0.26 km2 . The inventory of landslide areas had a good power-law dependence on area for areas greater than 103 m2 (103 km2 ). The number of landslides generated by earthquakes have a strong dependence on earthquake magnitude. Typically earthquakes with magnitudes M less than 4 do not generate any landslides [6]. Floods Floods are a major hazard to many cities and estimates of flood hazards have serious economic implications. The standard measure of the flood hazard is the 100-year flood. This is quantified as the river discharge Q100 expected during a 100 year period. Since there is seldom a long enough history to establish Q100 directly, it is necessary to extrapolate smaller floods that occur more often.
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One extrapolation approach is to assume that flood discharges are fractal (power-law) [3,19]. This scale invariant distribution can be expressed in terms of the ratio F of the peak discharge over a 10 year interval to the peak discharge over a 1 year interval, F D Q10 /Q1 . With self-similarity the parameter F is also the ratio of the 100 year peak discharge to the 10 year peak discharge, F D Q100 /Q10 . Values of F have a strong dependence on climate. In temperate climates such as the northeastern and northwestern US values are typically in the range F D 2–3. In arid and tropical climates such as the southwestern and southeastern US values are typically in the range F D 4–6. The applicability of fractal concepts to flood forecasting is certainly controversial. In 1982, the US government adopted the log-Pearson type 3 (LP3) distribution for the legal definition of the flood hazard [20]. The LP3 is a thintailed distribution relative to the thicker tailed power-law (fractal) distribution. Thus the forecast 100 year flood using LP3 is considerably smaller than the forecast using the fractal approach. This difference is illustrated by considering the great 1993 Mississippi River flood. Considering data at the Keukuk, Iowa gauging station [4] this flood was found to be a typical 100 year flood using the power-law (fractal) analysis and a 1000 to 10,000 year flood using the federal LP3 formulation. Concepts of self-similarity argue for the applicability of fractal concepts for flood-frequency forecasting. This applicability also has important implications for erosion. Erosion will be dominated by the very largest floods.
Adjacent values are not correlated with each other. In this case the spectrum is flat and the power spectral density coefficients are not a function of frequency, ˇ D 0. The classic example of a self-affine fractal is a Brownian walk. A Brownian walk is obtained by taking the running sum of a Gaussian white noise. In this case we have ˇ D 2. Another important self-affine time series is a red (or pink) noise with power spectral density coefficients proportional to 1/ f , that is ˇ D 1. We will see that the variability in the Earth’s magnetic field is well approximated by a 1/ f noise. Self-affine fractal time series in the range ˇ D 0 to 1 are known as fractional Gaussian noises. These noises are stationary and the standard deviation is a constant independent of the length of the time series. Self-affine time series with ˇ larger than 1 are known as fractional Brownian walk. These motions are not stationary and the standard deviation increases as a power of the length of the time series, there is a drift. For a Brownian walk the standard deviation increases with the square root of the length of the time series.
Self-Affine Fractals
Spectral expansions of global topography have been carried out, an example [15] is given in Fig. 2. Excellent agreement with the fractal relation given in Eq. (6) is obtained taking ˇ D 2, topography is well approximated by a Brownian walk. It has also shown that this fractal behavior of topography is found for the moon, Venus, and Mars [18].
Mandelbrot and Van Ness [8] extended the concept of fractals to time series. Examples of time series in geology and geophysics include global temperature, the strength of the Earth’s magnetic field, and the discharge rate in a river. After periodicities and trends have been removed, the remaining values are the stochastic (noise) component of the time series. The standard approach to quantifying the noise component is to carry out a Fourier transform on the time series [2]. The power-spectral density coefficients Si are proportional to the squares of the Fourier coefficients. The time series is a self-affine fractal if the power-spectral density coefficients have an inverse power-law dependence on frequency f i , that is Si D
C ˇ
(5)
fi
where C is a constant and ˇ is the power-law exponent. For a Gaussian white noise the values in the time series are selected randomly from a Gaussian distribution.
Topography The height of topography along linear tracks can be considered to be a continuous time series. In this case we consider the wave number ki (1/wave length) instead of frequency. Topography is a self-affine fractal if Si D
C k ˇi
(6)
Earth’s Magnetic Field Paleomagnetic studies have given the strength and polarity of the Earth’s magnetic field as a function of time over millions of years. These studies have also shown that the field has experienced a sequence of reversals. Spectral studies of the absolute amplitude of the field have been shown that it is a self-affine fractal [12,14]. The power-spectral density is proportional to one over the frequency, it is a 1/ f noise. When the fluctuations of the 1/ f noise take the magnitude to zero the polarity of the field reverses. The predicted distribution of polarity intervals is fractal and is in good agreement with the observed polarity intervals.
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Fractals in Geology and Geophysics, Figure 2 Power spectral density S as a function of wave number k for a spherical harmonic expansion of the Earth’s topography (degree l). The straight-line correlation is with Eq. (6) taking ˇ D 2, a Brownian walk
Future Directions There is no question that fractals are a useful empirical tool. They provide a rational means for the extrapolation and interpolation of observations. The wide applicability of power-law (fractal) distributions is generally accepted, but does this applicability have a more fundamental basis? Fractality appears to be fundamentally related to chaotic behavior and to numerical simulations exhibiting self-organized criticality. The entire area of fractals, chaos, selforganized criticality, and complexity remains extremely active, and it is impossible to predict with certainty what the future holds. Bibliography
4. Malamud BD, Turcotte DL, Barton CC (1996) The 1993 Mississippi river flood: A one hundred or a one thousand year event? Env Eng Geosci 2:479 5. Malamud BD, Turcotte DL, Guzzetti F, Reichenbach P (2004) Landslide inventories and their statistical properties. Earth Surf Process Landf 29:687 6. Malamud BD, Turcotte DL, Guzzetti F, Reichenbach P (2004) Landslides, earthquakes, and erosion. Earth Planet Sci Lett 229:45 7. Mandelbrot BB (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156:636 8. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422 9. McClelland L et al (1989) Global Volcanism 1975-1985. Prentice-Hall, Englewood Cliffs 10. Meybeck M (1995) Global distribution of lakes. In: Lerman A, Imboden DM, Gat JR (eds) Physics and Chemistry of Lakes, 2nd edn. Springer, Berlin, pp 1-35 11. Peckham SD (1989) New results for self-similar trees with applications to river networks. Water Resour Res 31:1023 12. Pelletier JD (1999) Paleointensity variations of Earth’s magnetic field and their relationship with polarity reversals. Phys Earth Planet Int 110:115 13. Pelletier JD (1999) Self-organization and scaling relationships of evolving river networks. J Geophys Res 104:7259 14. Pelletier JD, Turcotte DL (1999) Self-affine time series: II. Applications and models. Adv Geophys 40:91 15. Rapp RH (1989) The decay of the spectrum of the gravitational potential and the topography of the Earth. Geophys J Int 99:449 16. Strahler AN (1957) Quantitative analysis of watershed geomorphology. Trans Am Geophys Un 38:913 17. Tokunaga E (1978) Consideration on the composition of drainage networks and their evolution. Geogr Rep Tokyo Metro Univ 13:1 18. Turcotte DL (1987) A fractal interpretation of topography and geoid spectra on the earth, moon, Venus, and Mars. J Geophys Res 92:E597 19. Turcotte DL (1994) Fractal theory and the estimation of extreme floods. J Res Natl Inst Stand Technol 99:377 20. US Water Resources Council (1982) Guidelines for Determining Flood Flow Frequency. Bulletin 17B. US Geological Survey, Reston
Primary Literature 1. Kossobokov VG, Keilis-Borok VI, Turcotte DL, Malamud BD (2000) Implications of a statistical physics approach for earthquake hazard assessment and forecasting. Pure Appl Geophys 157:2323 2. Malamud BD, Turcotte DL (1999) Self-affine time series: I. Generation and analyses. Adv Geophys 40:1 3. Malamud BD, Turcotte DL (2006) The applicability of powerlaw frequency statistics to floods. J Hydrol 332:168
Books and Reviews Feder J (1988) Fractals. Plenum Press, New York Korvin G (1992) Fractal Models in the Earth Sciences. Elsevier, Amsterdam Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco Turcotte DL (1997) Fractals and Chaos in Geology and Geophysics, 2nd edn. Cambridge University Press, Cambridge
Fractals in the Quantum Theory of Spacetime
Fractals in the Quantum Theory of Spacetime LAURENT N OTTALE CNRS, Paris Observatory and Paris Diderot University, Paris, France Article Outline Glossary Definition of the Subject Introduction Foundations of Scale Relativity Theory Scale Laws From Fractal Space to Nonrelativistic Quantum Mechanics From Fractal Space-Time to Relativistic Quantum Mechanics Gauge Fields as Manifestations of Fractal Geometry Future Directions Bibliography Glossary Fractality In the context of the present article, the geometric property of being structured over all (or many) scales, involving explicit scale dependence which may go up to scale divergence. Spacetime Inter-relational level of description of the set of all positions and instants (events) and of their transformations. The events are defined with respect to a given reference system (i. e., in a relative way), but a spacetime is characterized by invariant relations which are valid in all reference systems, such as, e. g., the metric invariant. In the generalization to a fractal space-time, the events become explicitly dependent on resolution. Relativity The property of physical quantities according to which they can be defined only in terms of relationships, not in an abolute way. These quantities depend on the state of the reference system, itself defined in a relative way, i. e., with respect to other coordinate systems. Covariance Invariance of the form of equations under general coordinate transformations. Geodesics Curves in a space (more generally in a spacetime) which minimize the proper time. In a geometric spacetime theory, the motion equation is given by a geodesic equation. Quantum Mechanics Fundamental axiomatic theory of elementary particle, nuclear, atomic, molecular, etc. physical phenomena, according to which the state of
a physical system is described by a wave function whose square modulus yields the probability density of the variables, and which is solution of a Schrödinger equation constructed from a correspondence principle (among other postulates). Definition of the Subject The question of the foundation of quantum mechanics from first principles remains one of the main open problems of modern physics. In its current form, it is an axiomatic theory of an algebraic nature founded upon a set of postulates, rules and derived principles. This is to be compared with Einstein’s theory of gravitation, which is founded on the principle of relativity and, as such, is of an essentially geometric nature. In its framework, gravitation is understood as a very manifestation of the curvature of a Riemannian space-time. It is therefore relevant to question the nature of the quantum space-time and to ask for a possible refoundation of the quantum theory upon geometric first principles. In this context, it has been suggested that the quantum laws and properties could actually be manifestations of a fractal and nondifferentiable geometry of space-time [52,69,71], coming under the principle of scale relativity [53,54]. This principle extends, to scale transformations of the reference system, the theories of relativity (which have been, up to now, applied to transformations of position, orientation and motion). Such an approach allows one to recover the main tools and equations of standard quantum mechanics, but also to suggest generalizations, in particular toward high energies, since it leads to the conclusion that the Planck length scale could be a minimum scale in nature, unreachable and invariant under dilations [53]. But it has another radical consequence. Namely, it allows the possibility of a new form of macroscopic quantum-type behavior for a large class of complex systems, namely those whose behavior is characterized by Newtonian dynamics, fractal stochastic fluctuations over a large range of scales, and small scale irreversibility. In this case the equation of motion may take the form of a Schrödinger equation, which yields peaks of probability density according to the symmetry, field and limit conditions. These peaks may be interpreted as a tendency for the system to form structures [57], in terms of a macroscopic constant which is no longer ¯, therefore possibly leading to a new theory of self-organization. It is remarkable that, under such a relativistic view, the question of complexity may be posed in an original way. Namely, in a fully relativistic theory there is no intrinsic complexity, since the various physical properties of an ‘ob-
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ject’ are expected to vanish in the proper system of coordinates linked to the object. Therefore, in such a framework the apparent complexity of a system comes from the complexity of the change of reference frames from the proper frame to the observer (or measurement apparatus) reference frame. This does not mean that the complexity can always be reduced, since this change can itself be infinitely complex, as it is the case in the situation described here of a fractal and nondifferentiable space-time. Introduction There have been many attempts during the 20th century at understanding the quantum behavior in terms of differentiable manifolds. The failure of these attempts indicates that a possible ‘quantum geometry’ should be of a completely new nature. Moreover, following the lessons of Einstein’s construction of a geometric theory of gravitation, it seems clear that any geometric property to be attributed to space-time itself, and not only to particular objects or systems embedded in space-time, must necessarily be universal. Fortunately, the founders of quantum theory have brought to light a universal and fundamental behavior of the quantum realm, in opposition to the classical world; namely, the explicit dependence of the measurement results on the apparatus resolution described by the Heisenberg uncertainty relations. This leads one to contemplate the possibility that the space-time underlying the quantum mechanical laws can be explicitly dependent on the scale of observation [52,69,71]. Now the concept of a scale-dependent geometry (at the level of objects and media) has already been introduced and developed by Benoit Mandelbrot, who coined the word ‘fractal’ in 1975 to describe it. But here we consider a complementary program that uses fractal geometry, not only for describing ‘objects’ (that remain embedded in an Euclidean space), but also for intrinsically describing the geometry of space-time itself. A preliminary work toward such a goal may consist of introducing the fractal geometry in Einstein’s equations of general relativity at the level of the source terms. This would amount to giving a better description of the density of matter in the Universe accounting for its hierarchical organization and fractality over many scales (although possibly not all scales), then to solve Einstein’s field equations for such a scale dependent momentum-energy tensor. A full implementation of this approach remains a challenge to cosmology. But a more direct connection of the fractal geometry with fundamental physics comes from its use in describing not only the distribution of matter in space, but also the ge-
ometry of space-time itself. Such a goal may be considered as the continuation of Einstein’s program of generalization of the geometric description of space-time. In the new fractal space-time theory, [27,52,54,69,71], the essence of quantum physics is a manifestation of the nondifferentiable and fractal geometry of space-time. Another line of thought leading to the same suggestion comes, not from relativity and space-time theories, but from quantum mechanics itself. Indeed, it has been discovered by Feynman [30] that the typical quantum mechanical paths (i. e., those that contribute in a main way to the path integral) are nondifferentiable and fractal. Namely, Feynman has proved that, although a mean velocity can be defined for them, no mean-square velocity exists at any point, since it is given by hv 2 i / ıt 1 . One now recognizes in this expression the behavior of a curve of fractal dimension DF D 2 [1]. Based on these premises, the reverse proposal, according to which the laws of quantum physics find their very origin in the fractal geometry of space-time, has been developed along three different and complementary approaches. Ord and co-workers [50,71,72,73], extending the Feynman chessboard model, have worked in terms of probabilistic models in the framework of the statistical mechanics of binary random walks. El Naschie has suggested to give up not only the differentiability, but also the continuity of space-time. This leads him to work in terms of a ‘Cantorian’ spacetime [27,28,29], and to therefore use in a preferential way the mathematical tool of number theory (see a more detailed review of these two approaches in Ref. [45]). The scale relativity approach [52,54,56,63,69] which is the subject of the present article, is, on the contrary, founded on a fundamentally continuous geometry of space-time which therefore includes the differentiable and nondifferentiable cases, constrained by the principle of relativity applied to both motion and scale. Other applications of fractals to the quantum theory of space-time have been proposed in the framework of a possible quantum gravity theory. They are of another nature than those considered in the present article, since they are applicable only in the framework of the quantum theory instead of deriving it from the geometry, and they concern only very small scales on the magnitude of the Planck scale. We send the interested reader to Kröger’s review paper on “Fractal geometry in quantum mechanics, field theory and spin systems”, and to references therein [45]. In the present article, we summarize the steps by which one recovers, in the scale relativity and fractal space-time framework, the main tools and postulates of quantum me-
Fractals in the Quantum Theory of Spacetime
chanics and of gauge field theories. A more detailed account can be found in Refs. [22,23,54,56,63,67,68], including possible applications of the theory to various sciences. Foundations of Scale Relativity Theory The theory of scale relativity is based on giving up the hypothesis of manifold differentiability. In this framework, the coordinate transformations are continuous but can be nondifferentiable. This has several consequences [54], leading to the following preliminary steps of construction of the theory: (1) One can prove the following theorem [5,17,18,54, 56]: a continuous and nondifferentiable curve is fractal in a general meaning, namely, its length is explicitly dependent on a scale variable ", i. e., L D L("), and it diverges, L ! 1, when " ! 0. This theorem can be readily extended to a continuous and nondifferentiable manifold, which is therefore fractal not as an hypothesis, but as a consequence of giving up an hypothesis (that of differentiability). (2) The fractality of space-time [52,54,69,71] involves the scale dependence of the reference frames. One therefore adds a new variable " which characterizes the ‘state of scale’ to the usual variables defining the coordinate system. In particular, the coordinates themselves become functions of these scale variables, i. e., X D X("). (3) The scale variables " can never be defined in an absolute way, but only in a relative way. Namely, only their ratio D "0 /" has a physical meaning. In experimental situations, these scale variables amount to the resolution of the measurement apparatus (it may be defined as standard errors, intervals, pixel size, etc.). In a theoretical analysis, they are the space and time differential elements themselves. This universal behavior extends to the principle of relativity in such a way that it also applies to the transformations (dilations and contractions) of these resolution variables [52,53,54]. Scale Laws Fractal Coordinate and Differential Dilation Operator Consider a variable length measured on a fractal curve, and (more generally) a non-differentiable (fractal) curvilinear coordinate L(s; "), that depends on some parameter s which characterizes the position on the curve (it may be, e. g., a time coordinate), and on the resolution ". Such a coordinate generalizes the concept of curvilinear coordinates introduced for curved Riemannian space-times in Einstein’s general relativity [54] to nondifferentiable and fractal space-times.
Such a scale-dependent fractal length L(s; ") remains finite and differentiable when " ¤ 0; namely, one can define a slope for any resolution ", being aware that this slope is itself a scale-dependent fractal function. It is only at the limit " ! 0 that the length is infinite and the slope undefined, i. e., that nondifferentiability manifests itself. Therefore the laws of dependence of this length upon position and scale may be written in terms of a double differential calculus, i. e., it can be the solution of differential equations involving the derivatives of L with respect to both s and ". As a preliminary step, one needs to establish the relevant form of the scale variables and the way they intervene in scale differential equations. For this purpose, let us apply an infinitesimal dilation d to the resolution, which is therefore transformed as " ! "0 D "(1 C d ). The dependence on position is omitted at this stage in order to simplify the notation. By applying this transformation to a fractal coordinate L, one obtains, to the first order in the differential element, L("0 ) D L(" C " d ) D L(") C
@L(") " d @"
˜ L(") ; D (1 C Dd )
(1)
˜ is, by definition, the dilation operator. where D Since d"/" D d ln ", the identification of the two last members of Eq. (1) yields ˜ D" @ D @ : D @" @ ln "
(2)
This form of the infinitesimal dilation operator shows that the natural variable for the resolution is ln ", and that the expected new differential equations will indeed involve quantities such as @L(s; ")/@ ln ". This theoretical result agrees and explains the current knowledge according to which most measurement devices (of light, sound, etc.), including their physiological counterparts (eye, ear, etc.) respond according to the logarithm of the intensity (e. g., magnitudes, decibels, etc.). Self-similar Fractals as Solutions of a First Order Scale Differential Equation Let us start by writing the simplest possible differential equation of scale, and then solve it. We shall subsequently verify that the solutions obtained comply with the principle of relativity. As we shall see, this very simple approach already yields a fundamental result: it gives a foundation and an understanding from first principles for self-similar fractal laws, which have been shown by Mandelbrot and many others to be a general description of a large
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Fractals in the Quantum Theory of Spacetime, Figure 1 Scale dependence of the length (left) and of the effective fractal dimension DF D F C 1 (right) in the case of “inertial” scale laws (which are solutions of the simplest, first order scale differential equation). Toward the small scale one gets a scale-invariant law with constant fractal dimension, while the explicit scale-dependence is lost at scales larger than a transition scale
number of natural phenomena, in particular biological ones (see, e. g., [48,49,70], other volumes of these series and references therein). In addition, the obtained laws, which combine fractal and scale-independent behaviours, are the equivalent for scales of what inertial laws are for motion [49]. Since they serve as a fundamental basis of description for all the subsequent theoretical constructions, we shall now describe their derivation in detail. The simplest differential equation of explicit scale dependence which one can write is of first order and states that the variation of L under an infinitesimal scale transformation d ln " depends only on L itself. Basing ourselves on the previous derivation of the form of the dilation operator, we thus write @L(s; ") D ˇ(L) : @ ln "
(3)
The function ˇ is a priori unknown. However, still looking for the simplest form of such an equation, we expand ˇ(L) in powers of L, namely we write ˇ(L) D a C bL C : : : . Disregarding for the moment the s dependence, we obtain, to the first order, the following linear equation in which a and b are constants: dL D a C bL : d ln "
(4)
In order to find the solution of this equation, let us change the names of the constants as F D b and L0 D a/F , so that a C bL D F (L L0 ). We obtain the equation dL L L0
D F d ln " :
(5)
Its solution is (see Fig. 1)
F ; L(") D L0 1 C "
(6)
where is an integration constant. This solution corresponds to a length measured on a fractal curve up to a given point. One can now generalize it to a variable length that also depends on the position characterized by the parameter s. One obtains
F L(s; ") D L0 (s) 1 C (s) ; (7) " in which, in the most general case, the exponent F may itself be a variable depending on the position. The same kind of result is obtained for the projections on a given axis of such a fractal length [54]. Let X(s; ") be one of these projections, it reads
F : (8) X(s; ") D x(s) 1 C x (s) " In this case x (s) becomes a highly fluctuating function which may be described by a stochastic variable, as can be seen in Fig. 2. The important point here and for what follows is that the solution obtained is the sum of two terms (a classicallike, “differentiable part” and a nondifferentiable “fractal part”) which is explicitly scale-dependent and tends to infinity when " ! 0 [22,54]. By differentiating these two parts in the above projection, we obtain the differential formulation of this essential result, dX D dx C d ;
(9)
Fractals in the Quantum Theory of Spacetime
that this result has been derived from a theoretical analysis based on first principles, instead of being postulated or deduced from a fit of observational data. It should be noted that in the above expressions, the resolution is a length interval, " D ıX defined along the fractal curve (or one of its projected coordinate). But one may also travel on the curve and measure its length on constant time intervals, then change the time scale. In this case the resolution " is a time interval, " D ıt. Since they are related by the fundamental relation (see Fig. 2) ıX D F ıt ; the fractal length depends on the time resolution as 11/D F T : X(s; ıt) D X0 (s) ıt Fractals in the Quantum Theory of Spacetime, Figure 2 A fractal function. An example of such a fractal function is given by the projections of a fractal curve on Cartesian coordinates, as a function of a continuous and monotonous parameter (here the time t) which marks the position on the curve. The figure also exhibits the relation between space and time differential elements for such a fractal function, and compares the differentiable part ıx and nondifferentiable part ı of the space elementary displacement ıX D ıx C ı. While the differentiable coordinate variation ıx D hıXi is of the same order as the time differential ıt, the fractal fluctuation becomes much larger than ıt when ıt T, where T is a transition time scale, and it depends on the fractal dimension DF as ı / ıt1/DF . Therefore the two contributions to the full differential displacement are related by the fractal law ı DF / ıx, since ıx and ıt are differential elements of the same order
where dx is a classical differential element, while d is a differential element of fractional order (see Fig. 2, in which the parameter s that characterizes the position on the fractal curve has been taken to be the time t). This relation plays a fundamental role in the subsequent developments of the theory. Consider the case when F is constant. In the asymptotic small scale regime, " , one obtains a power-law dependence on resolution which reads F L(s; ") D L0 (s) : (10) " In this expression we recognize the standard form of a selfsimilar fractal behavior with constant fractal dimension DF D 1 C F , which has already been found to yield a fair description of many physical and biological systems [49]. Here the topological dimension is DT D 1, since we deal with a length, but this can be easily generalized to surfaces (DT D 2), volumes (DT D 3), etc., according to the general relation DF D DT C F . The new feature here is
(11)
(12)
An example of the use of such a relation is Feynman’s result according to which the mean square value of the velocity of a quantum mechanical particle is proportional to ıt 1 (see p. 176 in [30]), which corresponds to a fractal dimension DF D 2, as later recovered by Abbott and Wise [1] by using a space resolution. More generally (in the usual case when " D ıX), following Mandelbrot, the scale exponent F D DF DT can be defined as the slope of the (ln "; ln L) curve, namely F D
d ln L : d ln(/")
(13)
For a self-similar fractal such as that described by the fractal part of the above solution, this definition yields a constant value which is the exponent in Eq. (10). However, one can anticipate on the following, and use this definition to compute an “effective” or “local” fractal dimension, now variable, from the complete solution that includes the differentiable and the nondifferentiable parts, and therefore a transition to effective scale independence. Differentiating the logarithm of Eq. (16) yields an effective exponent given by F eff D : (14) 1 C ("/)F The effective fractal dimension DF D 1 C F therefore jumps from the nonfractal value DF D DT D 1 to its constant asymptotic value at the transition scale (see right part of Fig. 1). Galilean Relativity of Scales We can now check that the fractal part of such a law is compatible with the principle of relativity extended to scale transformations of the resolutions (i. e., with the principle of scale relativity). It reads L D L0 (/")F
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(Eq. 10), and it is therefore a law involving two variables (ln L and F ) as a function of one parameter (") which, according to the relativistic view, characterizes the state of scale of the system (its relativity is apparent in the fact that we need another scale to define it by their ratio). More generally, all the following statements remain true for the complete scale law including the transition to scale-independence, by making the replacement of L by L L0 . Note that, to be complete, we anticipate on what follows and consider a priori F to be a variable, even if, in the simple law first considered here, it takes a constant value. Let us take the logarithm of Eq. (10). It yields ln(L/L0 ) D F ln(/"). The two quantities ln L and F then transform under a finite scale transformation " ! "0 D " as ln
L("0 ) L0
D ln
L(") L0
F ln ;
(15)
and to be complete, F0 D F :
(16)
These transformations have exactly the same mathematical structure as the Galilean group of motion transformation (applied here to scale rather than motion), which reads x 0 D x t v;
t0 D t :
(17)
This is confirmed by the dilation composition law, " ! "0 ! "00 , which reads "00 "0 "00 (18) D ln C ln 0 ; " " " and is therefore similar to the law of composition of velocities between three reference systems K, K 0 and K 00 , ln
V 00 (K 00 /K) D V(K 0 /K) C V 0 (K 00 /K 0 ) :
(19)
Since the Galileo group of motion transformations is known to be the simplest group that implements the principle of relativity, the same is true for scale transformations. It is important to realize that this is more than a simple analogy: the same physical problem is set in both cases, and is therefore solved under similar mathematical structures (since the logarithm transforms what would have been a multiplicative group into an additive group). Indeed, in both cases, it is equivalent to finding the transformation law of a position variable (X for motion in a Cartesian system of coordinates, ln L for scales in a fractal system of coordinates) under a change of the state of the coordinate system (change of velocity V for motion and of resolution ln for scale), knowing that these state variables are defined only in a relative way. Namely, V is the
relative velocity between the reference systems K and K 0 , and is the relative scale: note that " and "0 have indeed disappeared in the transformation law, only their ratio remains. This remark establishes the status of resolutions as (relative) “scale velocities” and of the scale exponent F as a “scale time”. Recall finally that since the Galilean group of motion is only a limiting case of the more general Lorentz group, a similar generalization is expected in the case of scale transformations, which we shall briefly consider in Sect. “Special Scale-Relativity”. Breaking of Scale Invariance The standard self-similar fractal laws can be derived from the scale relativity approach. However, it is important to note that Eq. (16) provides us with another fundamental result, as shown in Fig. 1. Namely, it also contains a spontaneous breaking of the scale symmetry. Indeed, it is characterized by the existence of a transition from a fractal to a non-fractal behavior at scales larger than some transition scale . The existence of such a breaking of scale invariance is also a fundamental feature of many natural systems, which remains, in most cases, misunderstood. The advantage of the way it is derived here is that it appears as a natural, spontaneous, but only effective symmetry breaking, since it does not affect the underlying scale symmetry. Indeed, the obtained solution is the sum of two terms, the scale-independent contribution (differentiable part), and the explicitly scale-dependent and divergent contribution (fractal part). At large scales the scaling part becomes dominated by the classical part, but it is still underlying even though it is hidden. There is therefore an apparent symmetry breaking (see Fig. 1), though the underlying scale symmetry actually remains unbroken. The origin of this transition is, once again, to be found in relativity (namely, in the relativity of position and motion). Indeed, if one starts from a strictly scale-invariant law without any transition, L D L0 (/")F , then adds a translation in standard position space (L ! L C L1 ), one obtains F
F 1 L0 D L1 C L0 D L1 1 C : (20) " " Therefore one recovers the broken solution (that corresponds to the constant a ¤ 0 in the initial scale differential equation). This solution is now asymptotically scaledependent (in a scale-invariant way) only at small scales, and becomes independent of scale at large scales, beyond some relative transition 1 which is partly determined by the translation itself.
Fractals in the Quantum Theory of Spacetime
Multiple Scale Transitions Multiple transitions can be obtained by a simple generalization of the above result [58]. Still considering a perturbative approach and taking the Taylor expansion of the differential equation dL/d ln " D ˇ(L), but now to the second order of the expansion, one obtains the equation dL D a C bL C c L2 C : d ln "
(21)
One of its solutions, which generalizes that of Eq. (5), describes a scaling behavior which is broken toward both the small and large scales, as observed in most real fractal systems, 1 C (0 /")F L D L0 : (22) 1 C (1 /")F Due to the non-linearity of the ˇ function, there are now two transition scales in such a law. Indeed, when " < 1 < 0 , one has (0 /") 1 and (1 /") 1, so that L D L0 (0 /1 )F cst, independent of scale; when 1 < " < 0 , one has (0 /") 1 but (1 /") 1, so that the denominator disappears, and one recovers the previous pure scaling law L D L0 (0 /")F ; when 1 < 0 < ", one has (0 /") 1 and (1 /") 1, so that L D L0 D cst, independent of scale. Scale Relativity Versus Scale Invariance Let us briefly be more specific about the way in which the scale relativity viewpoint differs from scaling or simple scale invariance. In the standard concept of scale invariance, one considers scale transformations of the coordinate, X ! X0 D q X ;
(23)
then one looks for the effect of such a transformation on some function f (X). It is scaling when f (qX) D q˛ f (X) :
(24)
The scale relativity approach involves a more profound level of description, since the coordinate X is now explicitly resolution-dependent, i. e. X D X("). Therefore we now look for a scale transformation of the resolution, " ! "0 D " ;
(25)
which implies a scale transformation of the position variable that takes in the self-similar case the form X( ") D F X(") :
(26)
But now the scale factor on the variable has a physical meaning which goes beyond a trivial change of units. It corresponds to a coordinate measured at two different resolutions on a fractal curve of fractal dimension DF D 1 C F , and one can obtain a scaling function of a fractal coordinate, f ( F X) D ˛F f (X) :
(27)
In other words, there are now three levels of transformation in the scale relativity framework (the resolution, the variable, and its function) instead of only two in the usual conception of scaling. Generalized Scale Laws Discrete Scale Invariance, Complex Dimension, and Log-Periodic Behavior Fluctuations with respect to pure scale invariance are potentially important, namely the log-periodic correction to power laws that is provided, e. g., by complex exponents or complex fractal dimensions. It has been shown that such a behavior provides a very satisfactory and possibly predictive model of the time evolution of many critical systems, including earthquakes and market crashes ([79] and references therein). More recently, it has been applied to the analysis of major event chronology of the evolutionary tree of life [14,65,66], of human development [11] and of the main economic crisis of western and precolumbian civilizations [36,37,40,65]. One can recover log-periodic corrections to self-similar power laws through the requirement of covariance (i. e., of form invariance of equations) applied to scale differential equations [58]. Consider a scale-dependent function L("). In the applications to temporal evolution quoted above, the scale variable is identified with the time interval jt t c j, where t c is the date of a crisis. Assume that L satisfies a first order differential equation, dL L D 0 ; d ln "
(28)
whose solution is a pure power law L(") / " (cf Sect. “Self-Similar Fractals as Solutions of a First Order Scale Differential Equation”). Now looking for corrections to this law, one remarks that simply incorporating a complex value of the exponent would lead to large log-periodic fluctuations rather than to a controllable correction to the power law. So let us assume that the right-hand side of Eq. (28) actually differs from zero: dL L D : d ln "
(29)
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Fractals in the Quantum Theory of Spacetime
Fractals in the Quantum Theory of Spacetime, Figure 3 Scale dependence of the length L of a fractal curve and of the effective “scale time” F D DF DT (fractal dimension minus topological dimension) in the case of a log-periodic behavior with fractal/non-fractal transition at scale , which reads L(") D L0 f1 C (/") exp[b cos(! ln("/))]g
We can now apply the scale covariance principle and require that the new function be a solution of an equation which keeps the same form as the initial equation, d 0 D 0 : d ln "
(30)
Setting 0 D C , we find that L must be a solution of a second-order equation: d2 L dL (2 C ) C ( C )L D 0 : (d ln ")2 d ln "
(31)
The solution reads L(") D a" (1 C b" ), and finally, the choice of an imaginary exponent D i! yields a solution whose real part includes a log-periodic correction: L(") D a" [1 C b cos(! ln ")] :
(32)
As previously recalled in Sect. “Breaking of Scale Invariance,” adding a constant term (a translation) provides a transition to scale independence at large scales (see Fig. 3). Lagrangian Approach to Scale Laws In order to obtain physically relevant generalizations of the above simplest (scale-invariant) laws, a Lagrangian approach can be used in scale space analogously to using it to derive the laws of motion, leading to reversal of the definition and meaning of the variables [58]. This reversal is analogous to that achieved by Galileo concerning the laws of motion. Indeed, from the Aristotle viewpoint, “time is the measure of motion”. In the same way, the fractal dimension, in its standard (Mandelbrot’s)
acception, is defined from the topological measure of the fractal object (length of a curve, area of a surface, etc.) and resolution, namely (see Eq. 13) tD
x v
$
F D DF D T D
d ln L : d ln(/")
(33)
In the case mainly considered here, when L represents a length (i. e., more generally, a fractal coordinate), the topological dimension is DT D 1 so that F D DF 1. With Galileo, time becomes a primary variable, and the velocity is deduced from space and time, which are therefore treated on the same footing, in terms of a space-time (even though the Galilean space-time remains degenerate because of the implicitly assumed infinite velocity of light). In analogy, the scale exponent F D DF 1 becomes in this new representation a primary variable that plays, for scale laws, the same role as played by time in motion laws (it is called “djinn” in some publications which therefore introduce a five-dimensional ‘space-time-djinn’ combining the four fractal fluctuations and the scale time). Carrying on the analogy, in the same way that the velocity is the derivative of position with respect to time, v D dx/dt, we expect the derivative of ln L with respect to scale time F to be a “scale velocity”. Consider as reference the self-similar case, that reads ln L D F ln(/"). Deriving with respect to F , now considered as a variable, yields d ln L/dF D ln(/"), i. e., the logarithm of resolution. By extension, one assumes that this scale velocity provides a new general definition of resolution even in more general situations, namely, d ln L V D ln : (34) D " dF
Fractals in the Quantum Theory of Spacetime
One can now introduce a scale Lagrange function e L(ln L; V ; F ), from which a scale action is constructed: Z 2 e e SD (35) L(ln L; V ; F )dF : 1
The application of the action principle yields a scale Euler– Lagrange equation which reads L d @e @e L : D dF @V @ ln L
(36)
One can now verify that in the free case, i. e., in the ab˜ ln L D 0), one recovsence of any “scale force” (i. e., @L/@ ers the standard fractal laws derived hereabove. Indeed, in this case the Euler–Lagrange equation becomes ˜ @L/@V D const ) V D const:
(37)
which is the equivalent for scale of what inertia is for motion. Still in analogy with motion laws, the simplest possible form for the Lagrange function is a quadratic dependence on the scale velocity, (i. e., L˜ / V 2 ). The constancy of V D ln(/") means that it is independent of the scale time F . Equation (34) can therefore be integrated to give the usual power law behavior, L D L0 (/")F , as expected. But this reversed viewpoint also has several advantages which allow a full implementation of the principle of scale relativity: The scale time F is given the status of a fifth dimension and the logarithm of the resolution V D ln(/"), is given the status of a scale velocity (see Eq. 34). This is in accordance with its scale-relativistic definition, in which it characterizes the state of scale of the reference system, in the same way as the velocity v D dx/dt characterizes its state of motion. (ii) This allows one to generalize the formalism to the case of four independent space-time resolutions, V D ln( /" ) D d ln L /dF . (iii) Scale laws more general than the simplest self-similar ones can be derived from more general scale Lagrangians [57,58] involving “scale accelerations” I% D d2 ln L/dF2 D d ln(/")/dF , as we shall see in what follows. (i)
Note however that there is also a shortcoming in this approach. Contrary to the case of motion laws, in which time is always flowing toward the future (except possibly in elementary particle physics at very small time scales), the variation of the scale time may be non-monotonic, as exemplified by the previous case of log-periodicity. Therefore this Lagrangian approach is restricted to monotonous variations of the fractal dimension, or, more generally, to scale intervals on which it varies in a monotonous way.
Scale Dynamics The previous discussion indicates that the scale invariant behavior corresponds to freedom (i. e. scale force-free behavior) in the framework of a scale physics. However, in the same way as there are forces in nature that imply departure from inertial, rectilinear uniform motion, we expect most natural fractal systems to also present distortions in their scale behavior with respect to pure scale invariance. This implies taking non-linearity in the scale space into account. Such distorsions may be, as a first step, attributed to the effect of the dynamics of scale (“scale dynamics”), i. e., of a “scale field”, but it must be clear from the very beginning of the description that they are of geometric nature (in analogy with the Newtonian interpretation of gravitation as the result of a force, which has later been understood from Einstein’s general relativity theory as a manifestation of the curved geometry of space-time). In this case the Lagrange scale-equation takes the form of Newton’s equation of dynamics, FD
d2 ln L ; dF2
(38)
where is a “scale mass”, which measures how the system resists to the scale force, and where I% D d2 ln L/dF2 D d ln(/")/dF is the scale acceleration. In this framework one can therefore attempt to define generic, scale-dynamical behaviours which could be common to very different systems, as corresponding to a given form of the scale force. Constant Scale Force A typical example is the case of a constant scale force. Setting G D F/, the potential reads ' D G ln L, analogous to the potential of a constant force f in space, which is ' D f x, since the force is @'/@x D f . The scale differential equation is d2 ln L DG: dF2
(39)
It can be easily integrated. A first integration yields d ln L/dF D GF CV0 , where V0 is a constant. Then a second integration yields a parabolic solution (which is the equivalent for scale laws of parabolic motion in a constant field), 1 V D V0 C GF ; ln L D ln L0 C V0 F C G F2 ; (40) 2 where V D d ln L/dF D ln(/"). However the physical meaning of this result is not clear under this form. This is due to the fact that, while in the case of motion laws we search for the evolution of the system with time, in the case of scale laws we search for the
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Fractals in the Quantum Theory of Spacetime
Fractals in the Quantum Theory of Spacetime, Figure 4 Scale dependence of the length of a fractal curve ln L and of its effective fractal dimension (DF D DT CF , where DT is the topological dimension) in the case of a constant scale force, with an additional fractal to non-fractal transition
dependence of the system on resolution, which is the directly measured observable. Since the reference scale is arbitrary, the variables can be re-defined in such a way that V0 D 0, i. e., D 0 . Indeed, from Eq. (40) one gets F D (V V0 )/G D [ln(/") ln(/0 )]/G D ln(0 /")/G. Then one obtains 0 L 1 1 2 0 F D ln ; ln ln : (41) D G " L0 2G " The scale time F becomes a linear function of resolution (the same being true, as a consequence, of the fractal dimension DF D 1 C F ), and the (ln L; ln ") relation is now parabolic instead of linear (see Fig. 4 and compare to Fig. 1). Note that, as in previous cases, we have considered here only the small scale asymptotic behavior, and that we can once again easily generalize this result by including a transition to scale-independence at large scale. This is simply achieved by replacing L by (L L0 ) in every equations. There are several physical situations where, after careful examination of the data, the power-law models were clearly rejected since no constant slope could be defined in the (log L; log ") plane. In the several cases where a clear curvature appears in this plane, e. g., turbulence [26], sandpiles [9], fractured surfaces in solid mechanics [10], the physics could come under such a scale-dynamical description. In these cases it might be of interest to identify and study the scale force responsible for the scale distortion (i. e., for the deviation from standard scaling). Special Scale-Relativity Let us close this section about the derivation of scale laws of increasing complexity by coming back to the question of
finding the general laws of scale transformations that meet the principle of scale relativity [53]. It has been shown in Sect. “Galilean Relativity of Scales” that the standard selfsimilar fractal laws come under a Galilean group of scale transformations. However, the Galilean relativity group is known, for motion laws, to be only a degenerate form of the Lorentz group. It has been proved that a similar result holds for scale laws [53,54]. The problem of finding the linear transformation laws of fields in a scale transformation V D ln (" ! "0 ) amounts to finding four quantities, a(V ); b(V ); c(V ), and d(V ), such that ln
L0 L0
D a(V ) ln
F 0 D c(V ) ln
L L0 L L0
C b(V ) F ; (42) C d(V )F :
Set in this way, it immediately appears that the current ‘scale-invariant’ scale transformation law of the standard form (Eq. 8), given by a D 1; b D V ; c D 0 and d D 1, corresponds to a Galilean group. This is also clear from the law of composition of dilatations, " ! "0 ! "00 , which has a simple additive form, V 00 D V C V 0 :
(43)
However the general solution to the ‘special relativity problem’ (namely, find a; b; c and d from the principle of relativity) is the Lorentz group [47,53]. This result has led to the suggestion of replacing the standard law of dilatation, " ! "0 D %" by a new Lorentzian relation, namely, for " < 0 and "0 < 0 ln
"0 ln("/0 ) C ln % D : 0 1 C ln % ln("/0 )/ ln2 (/0 )
(44)
Fractals in the Quantum Theory of Spacetime
Fractals in the Quantum Theory of Spacetime, Figure 5 Scale dependence of the logarithm of the length and of the effective fractal dimension, DF D 1 C F , in the case of scale-relativistic Lorentzian scale laws including a transition to scale independence toward large scales. The constant C has been taken to be C D 4 2 39:478, which is a fundamental value for scale ratios in elementary particle physics in the scale relativity framework [53,54,63], while the effective fractal dimension jumps from DF D 1 to DF D 2 at the transition, then increases without any limit toward small scales
This relation introduces a fundamental length scale , which is naturally identified, towards the small scales, with the Planck length (currently 1:6160(11) 1035 m) [53], D lP D („G/c 3 )1/2 ;
(45)
and toward the large scales (for " > 0 and "0 > 0 ) with the scale of the cosmological constant, L D 1/2 (see Chap. 7.1 in [54]). As one can see from Eq. (44), if one starts from the scale " D and apply any dilatation or contraction %, one obtains again the scale "0 D , whatever the initial value of 0 . In other words, can be interpreted as a limiting lower (or upper) length-scale, which is impassable and invariant under dilatations and contractions. Concerning the length measured along a fractal coordinate which was previously scale-dependent as ln(L/L0 ) D 0 ln(0 /") for " < 0 , it becomes in the new framework, in the simplified case when one starts from the reference scale L0 (see Fig. 5) ln
L L0
0 ln(0 /") : D p 1 ln2 (0 /")/ ln2 (0 /)
(46)
The main new feature of scale relativity with respect to the previous fractal or scale-invariant approaches is that the scale exponent F and the fractal dimension DF D 1 C F , which were previously constant (DF D 2; F D 1), are now explicitly varying with scale (see Fig. 5), following the law (given once again in the simplified case when we start from the reference scale L0 ): 0 F (") D p : (47) 2 1 ln (0 /")/ ln2 (0 /) Under this form, the scale covariance is explicit, since one keeps a power law form for the length variation, L D
L0 (/")F (") , but now in terms of a variable fractal dimen-
sion. For a more complete development of special relativity, including its implications regarding new conservative quantities and applications in elementary particle physics and cosmology, see [53,54,56,63]. The question of the nature of space-time geometry at the Planck scale is a subject of intense work (see, e. g., [3,46] and references therein). This is a central question for practically all theoretical attempts, including noncommutative geometry [15,16], supersymmetry, and superstring theories [35,75] – for which the compactification scale is close to the Planck scale – and particularly for the theory of quantum gravity. Indeed, the development of loop quantum gravity by Rovelli and Smolin [76] led to the conclusion that the Planck scale could be a quantized minimal scale in Nature, involving also a quantization of surfaces and volumes [77]. Over the last years, there has also been significant research effort aimed at the development of a ‘Doubly-Special-Relativity’ [2] (see a review in [3]), according to which the laws of physics involve a fundamental velocity scale c and a fundamental minimum length scale Lp , identified with the Planck length. The concept of a new relativity in which the Planck length-scale would become a minimum invariant length is exactly the founding idea of the special scale relativity theory [53], which has been incorporated in other attempts of extended relativity theories [12,13]. But, despite the similarity of aim and analysis, the main difference between the ‘Doubly-Special-Relativity’ approach and the scale relativity one is that the question of defining an invariant lengthscale is considered in the scale relativity/fractal space-time theory as coming under a relativity of scales. Therefore
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the new group to be constructed is a multiplicative group, that becomes additive only when working with the logarithms of scale ratios, which are definitely the physically relevant scale variables, as one can show by applying the Gell-Mann-Levy method to the construction of the dilation operator (see Sect. “Fractal Coordinate and Differential Dilation Operator”).
relations and the explicit dependence of measurement results on the resolution of the measurement apparatus. Let us start from the result of the previous section, according to which the solution of a first order scale differential equation reads for DF D 2, after differentiation and reintroduction of the indices, p (48) dX D dx C d D v ds C c ds ;
From Fractal Space to Nonrelativistic Quantum Mechanics
where c is a length scale which must be introduced for dimensional reasons and which, as we shall see, generalizes the Compton length. The are dimensionless highly fluctuating functions. Due to their highly erratic character, we can replace them by stochastic variables such that h i D 0, h( 0 )2 i D 1 and h( k )2 i D 1 (k D1 to 3). The mean is taken here on a purely mathematic probability law which can be fully general, since the final result does not depend on its choice.
The first step in the construction of a theory of the quantum space-time from fractal and nondifferentiable geometry, which has been described in the previous sections, has consisted of finding the laws of explicit scale dependence at a given “point” or “instant” (under their new fractal definition). The next step, which will now be considered, amounts to writing the equation of motion in such a fractal space(time) in terms of a geodesic equation. As we shall see, after integration this equation takes the form of a Schrödinger equation (and of the Klein-Gordon and Dirac equations in the relativistic case). This result, first obtained in Ref. [54], has later been confirmed by many subsequent physical [22,25,56,57] and mathematical works, in particular by Cresson and Ben Adda [6,7,17,19] and Jumarie [41,42,43,44], including attempts of generalizations using the tool of the fractional integro-differential calculus [7,21,44]. In what follows, we consider only the simplest case of fractal laws, namely, those characterized by a constant fractal dimension. The various generalized scale laws considered in the previous section lead to new possible generalizations of quantum mechanics [56,63]. Critical Fractal Dimension 2 Moreover, we simplify again the description by considering only the case DF D 2. Indeed, the nondifferentiability and fractality of space implies that the paths are random walks of the Markovian type, which corresponds to such a fractal dimension. This choice is also justified by Feynman’s result [30], according to which the typical paths of quantum particles (those which contribute mainly to the path integral) are nondifferentiable and of fractal dimension DF D 2 [1]. The case DF ¤ 2, which yields generalizations to standard quantum mechanics has also been studied in detail (see [56,63] and references therein). This study shows that DF D 2 plays a critical role in the theory, since it suppresses the explicit scale dependence in the motion (Schrödinger) equation – but this dependence remains hidden and reappears through, e. g., the Heisenberg
Metric of a Fractal Space-Time Now one can also write the fractal fluctuations in terms p of the coordinate differentials, d D dx . The identification of this expression with that of Eq. (3) leads one to recover the Einstein-de Broglie length and time scales, x D
c „ ; D dx/ds px
D
c „ D : dt/ds E
(49)
Let us now assume that the large scale (classical) behavior is given by the Riemannian metric potentials g (x; y; z; t). The invariant proper time dS along a geodesic in terms of the complete differential elements dX D dx C d dS 2 D g dX dX D g (dx Cd )(dx Cd ): (50) Now replacing the d’s by their expression, one obtains a fractal metric [54,68]. Its two-dimensional and diagonal expression, neglecting the terms of the zero mean (in order to simplify its writing) reads F 2 2 c dt dS 2 Dg00 (x; t) 1 C 02 dt x g11 (x; t) 1 C 12 (51) dx 2 : dx We therefore obtain generalized fractal metric potentials which are divergent and explicitly dependent on the coordinate differential elements [52,54]. Another equivalent way to understand this metric consists in remarking that it is no longer only quadratic in the space-time differental elements, but that it also contains them in a linear way.
Fractals in the Quantum Theory of Spacetime
As a consequence, the curvature is also explicitly scaledependent and divergent when the scale intervals tend to zero. This property ensures the fundamentally non-Riemannian character of a fractal space-time, as well as the possibility to characterize it in an intrinsic way. Indeed, such a characterization, which is a necessary condition for defining a space in a genuine way, can be easily made by measuring the curvature at smaller and smaller scales. While the curvature vanishes by definition toward the small scales in Gauss-Riemann geometry, a fractal space can be characterized from the interior by the verification of the divergence toward small scales of curvature, and therefore of physical quantities like energy and momentum. Now the expression of this divergence is nothing but the Heisenberg relations themselves, which therefore acquire in this framework the status of a fundamental geometric test of the fractality of space-time [52,54,69]. Geodesics of a Fractal Space-Time The next step in such a geometric approach consists of the identifying the wave-particles with fractal space-time geodesics. Any measurement is interpreted as a selection of the geodesics bundle linked to the interaction with the measurement apparatus (that depends on its resolution) and/or to the information known about it (for example, the which-way-information in a two-slit experiment [56]. The three main consequences of nondifferentiability are: (i) The number of fractal geodesics is infinite. This leads one to adopt a generalized statistical fluid-like description where the velocity V (s) is replaced by a scaledependent velocity field V [X (s; ds); s; ds]. (ii) There is a breaking of the reflexion invariance of the differential element ds. Indeed, in terms of fractal functions f (s; ds), two derivatives are defined, X(s C ds; ds) X(s; ds) ; ds X(s; ds) X(s ds; ds) 0 (s; ds) D ; X ds
0 XC (s; ds) D
(52)
which transform into each other under the reflection (ds $ ds), and which have a priori no reason to be equal. This leads to a fundamental two-valuedness of the velocity field. (iii) The geodesics are themselves fractal curves of fractal dimension DF D 2 [30]. This means that one defines two divergent fractal velocity fields, VC [x(s; ds); s; ds] D vC [x(s); s] C wC [x(s; ds); s; ds] and V [x(s; ds); s; ds] D v [x(s); s] C w [x(s; ds); s; ds], which can be decomposed in terms of
differentiable parts vC and v , and of fractal parts wC and w . Note that, contrary to other attempts such as Nelson’s stochastic quantum mechanics which introduces forward and backward velocities [51] (and which has been later disproved [34,80]), the two velocities are here both forward, since they do not correspond to a reversal of the time coordinate, but of the time differential element now considered as an independent variable. More generally, we define two differentiable parts of derivatives dC /ds and d /ds, which, when they are applied to x , yield the differential parts of the velocity fields, D d x /ds. vC D dC x /ds and v Covariant Total Derivative Let us first consider the non-relativistic case. It corresponds to a three-dimensional fractal space, without fractal time, in which the invariant ds is therefore identified with the time differential element dt. One describes the elementary displacements dX k , where k D 1; 2; 3, on the geodesics of a nondifferentiable fractal space in terms of the sum of the two terms (omitting the indices for simplicity) dX˙ D d˙ x C d˙ , where dx represents the differentiable part and d the fractal (nondifferentiable) part, defined as p d˙ x D v˙ dt ; d˙ D ˙ 2D dt 1/2 : (53) Here ˙ are stochastic dimensionless variables such that 2 i D 1, and D is a parameter that generh ˙ i D 0 and h ˙ alizes the Compton scale (namely, D D „/2m in the case of standard quantum mechanics) up to the fundamental constant c/2. The two time derivatives are then combined in terms of a complex total time derivative operator [54], b d 1 dC d i dC d D C : (54) dt 2 dt dt 2 dt dt Applying this operator to the differentiable part of the position vector yields a complex velocity V D
b d vC C v vC v x(t) D V iU D i : (55) dt 2 2
In order to find the expression for the complex time derivative operator, let us first calculate the derivative of a scalar function f . Since the fractal dimension is 2, one needs to go to the second order of expansion. For one variable it reads @f @ f dX 1 @2 f dX 2 df D C C : dt @t @X dt 2 @X 2 dt
(56)
Generalizing this process to three dimensions is straightforward.
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Fractals in the Quantum Theory of Spacetime
Let us now take the stochastic mean of this expression, i. e., we take the mean on the stochastic variables ˙ which appear in the definition of the fractal fluctuation d˙ . By definition, since dX D dx C d and hdi D 0, we have hdXi D dx, so that the second term is reduced (in 3 dimensions) to v:r f . Now concerning the term dX 2 /dt, it is infinitesimal and therefore it would not be taken into account in the standard differentiable case. But in the nondifferentiable case considered here, the mean square fluctuation is non-vanishing and of order dt, namely, hd 2 i D 2Ddt, so that the last term of Eq. (56) amounts in three dimensions to a Laplacian operator. One obtains, respectively for the (+) and () processes, @ d˙ f (57) D C v˙ :r ˙ D f : dt @t Finally, by combining these two derivatives in terms of the complex derivative of Eq. (54), it reads [54] b @ d D C V :r i D: dt @t
b d @L @L D0: dt @V @x
(62)
From the homogeneity of space and Noether’s theorem, one defines a generalized complex momentum given by the same form as in classical mechanics as PD
@L : @V
(63)
If the action is now considered as a function of the upper limit of integration in Eq. (61), the variation of the action from a trajectory to another nearby trajectory yields a generalization of another well-known relation of classical mechanics, P D rS :
(64)
(58) Motion Equation
Under this form, this expression is not fully covariant [74], since it involves derivatives of the second order, so that its Leibniz rule is a linear combination of the first and second order Leibniz rules. By introducing the velocity operator [61] b D V i Dr; V
Generalized Euler–Lagrange equations that keep their standard form in terms of the new complex variables can be derived from this action [22,54], namely
(59)
As an example, consider the case of a single particle in an external scalar field with potential energy (but the method can be applied to any situation described by a Lagrange function). The Lagrange function, L D 12 mv 2 , is generalized as L(x; V ; t) D 12 mV 2 . The Euler– Lagrange equations then keep the form of Newton’s fundamental equation of dynamics F D m dv/dt, namely,
it may be given a fully covariant expression, b d @ b :r : D CV dt @t
(60)
Under this form it satisfies the first order Leibniz rule for partial derivatives. We shall now see that b d/dt plays the role of a “covariant derivative operator” (in analogy with the covariant derivative of general relativity). Namely, one may write in its terms the equation of physics in a nondifferentiable space under a strongly covariant form identical to the differentiable case. Complex Action and Momentum The steps of construction of classical mechanics can now be followed, but in terms of complex and scale dependent quantities. One defines a Lagrange function that keeps its usual form, L(x; V ; t), but which is now complex. Then one defines a generalized complex action Z t2 SD L(x; V ; t)dt : (61) t1
m
b d V D r ; dt
(65)
which is now written in terms of complex variables and complex operators. In the case when there is no external field ( D 0), the covariance is explicit, since Eq. (65) takes the free form of the equation of inertial motion, i. e., of a geodesic equation, b d V D0: dt
(66)
This is analogous to Einstein’s general relativity, where the equivalence principle leads to the covariant equation of the motion of a free particle in the form of an inertial motion (geodesic) equation Du /ds D 0, written in terms of the general-relativistic covariant derivative D, of the four-vector u and of the proper time differential ds. The covariance induced by the effects of the nondifferentiable geometry leads to an analogous transformation of the equation of motions, which, as we show below, become after integration the Schrödinger equation, which
Fractals in the Quantum Theory of Spacetime
can therefore be considered as the integral of a geodesic equation in a fractal space. In the one-particle case the complex momentum P reads P D mV ;
(67)
so that from Eq. (64) the complex velocity V appears as a gradient, namely the gradient of the complex action V D r S/m :
(68)
Wave Function Up to now the various concepts and variables used were of a classical type (space, geodesics, velocity fields), even if they were generalized to the fractal and nondifferentiable, explicitly scale-dependent case whose essence is fundamentally not classical. We shall now make essential changes of variable, that transform this apparently classical-like tool to quantum mechanical tools (without any hidden parameter or new degree of freedom). The complex wave function is introduced as simply another expression for the complex action S, by making the transformation D e i S/S 0 :
(69)
Note that, despite its apparent form, this expression involves a phase and a modulus since S is complex. The factor S0 has the dimension of an action (i. e., an angular momentum) and must be introduced because S is dimensioned while the phase should be dimensionless. When this formalism is applied to standard quantum mechanics, S0 is nothing but the fundamental constant ¯. As a consequence, since S D i S0 ln
;
(70)
one finds that the function is related to the complex velocity appearing in Eq. (68) as follows V D i
S0 r ln m
:
(71)
This expression is the fundamental relation that connects the two description tools while giving the meaning of the wave function in the new framework. Namely, it is defined here as a velocity potential for the velocity field of the infinite family of geodesics of the fractal space. Because of nondifferentiability, the set of geodesics that defines a ‘particle’ in this framework is fundamentally non-local. It can easily be generalized to a multiple particle situation (in particular to entangled states) which are described by
a single wave function , from which the various velocity fields of the subsets of the geodesic bundle are derived as V k D i (S0 /m k ) r k ln , where k is an index for each particle. The indistinguishability of identical particles naturally follows from the fact that the ‘particles’ are identified with the geodesics themselves, i. e., with an infinite ensemble of purely geometric curves. In this description there is no longer any point-mass with ‘internal’ properties which would follow a ‘trajectory’, since the various properties of the particle – energy, momentum, mass, spin, charge (see next sections) – can be derived from the geometric properties of the geodesic fluid itself. Correspondence Principle Since we have P D i S0 r ln tain the equality [54] P
D i S0 (r )/ , we ob-
D i„r
(72)
in the standard quantum mechanical case S0 D „, which establishes a correspondence between the classical momentum p, which is the real part of the complex momentum in the classical limit, and the operator i„r. This result is generalizable to other variables, in particular to the Hamiltonian. Indeed, a strongly covariant form of the Hamiltonian can be obtained by using the fully covariant form of the covariant derivative operator given by Eq. (60). With this tool, the expression of the relation between the complex action and the complex Lagrange function reads LD
b dS @S b :r S : D CV dt dt
(73)
Since P D r S and H D @S/@t, one obtains for the generalized complex Hamilton function the same form it has in classical mechanics, namely [63,67], b :P L : H DV
(74)
After expanding the velocity operator, one obtains H D V :P i Dr:P L, which includes an additional term [74], whose origin is now understood as an expression of nondifferentiability and strong covariance. Schrödinger Equation and Compton Relation The next step of the construction amounts to writing the fundamental equation of dynamics Eq. (65) in terms of the function . It takes the form iS0
b d (r ln ) D r : dt
(75)
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Fractals in the Quantum Theory of Spacetime
As we shall now see, this equation can be integrated in a general way in the form of a Schrödinger equation. Replacing b d/dt and V by their expressions yields
r˚ D iS0
@ r ln @t
i
D2
S0 (r ln :r)(r ln ) m C D(r ln ) : (76)
This equation may be simplified thanks to the identity [54], r
D 2(r ln :r)(r ln ) C (r ln ) : (77)
We recognize, in the right-hand side of Eq. (77), the two terms of Eq. (76), which were respectively a factor of S0 /m and D. This leads to the definition of the wave function as D e i S/2mD ;
(78)
which means that the arbitrary parameter S0 (which is identified with the constant ¯ in standard QM) is now linked to the fractal fluctuation parameter by the relation S0 D 2mD :
(79)
This relation (which can actually be proved instead of simply being set as a simplifying choice, see [62,67]) is actually a generalization of the Compton relation, since the geometric parameter D D hd 2 i/2dt can be written in terms of a length scale as D D c/2, so that, when S0 D „, it becomes D „/mc. But a geometric meaning is now given to the Compton length (and therefore to the inertial mass of the particle) in the fractal space-time framework. The fundamental equation of dynamics now reads r D 2imD
@ r ln @t
˚ i 2D(r ln :r)(r ln ) C D(r ln ) : (80)
Using the above remarkable identity and the fact that @/@t and r commute, it becomes
r @ D 2Dr i ln C D : (81) m @t The full equation becomes a gradient,
r
2D r m
and it can be easily integrated to finally obtain a generalized Schrödinger equation [54]
i @ /@t C D
D0
(82)
C iD
@ @t
2m
D0;
(83)
up to an arbitrary phase factor which may be set to zero by a suitable choice of the phase. One recovers the standard Schrödinger equation of quantum mechanics for the particular case when D D „/2m. Von Neumann’s and Born’s Postulates In the framework described here, “particles” are identified with the various geometric properties of fractal space(time) geodesics. In such an interpretation, a measurement (and more generally any knowledge about the system) amounts to selecting the sub-set of the geodesics family which only contains the geodesics with the geometric properties corresponding to the measurement result. Therefore, just after the measurement, the system is in the state given by the measurement result, which is precisely the von Neumann postulate of quantum mechanics. The Born postulate can also be inferred from the scalerelativity construction [22,62,67]. Indeed, the probability for the particle to be found at a given position must be proportional to the density of the geodesics fluid at this point. The velocity and the density of the fluid are expected to be solutions of a Euler and continuity system of four equations, with four unknowns, ( ; Vx ; Vy ; Vz ). Now, by separating the real and imaginary parts of the p Schrödinger equation, setting D P e i and using a mixed representation (P; V ), where V D fVx ; Vy ; Vz g, one precisely obtains such a standard system of fluid dynamics equations, namely, p ! @ 2 P ; C V r V D r 2D p @t P (84) @P C div(PV ) D 0 : @t This allows one to unequivocally identify P D j j2 with the probability density of the geodesics and therefore with the probability of presence of the ‘particle’. Moreover, p 2 P (85) Q D 2D p P can be interpreted as the new potential which is expected to emerge from the fractal geometry, in analogy with the identification of the gravitational field as a manifestation of the curved geometry in Einstein’s general relativity. This result is supported by numerical simulations, in which the
Fractals in the Quantum Theory of Spacetime
probability density is obtained directly from the distribution of geodesics without writing the Schrödinger equation [39,63]. Nondifferentiable Wave Function In more recent works, instead of taking only the differentiable part of the velocity field into account, one constructs the covariant derivative and the wave function in terms of the full velocity field, including its divergent nondifferentiable part of zero mean [59,62]. As we shall briefly see now, this still leads to the standard form of the Schrödinger equation. This means that, in the scale relativity framework, one expects the Schrödinger equation to have fractal and nondifferentiable solutions. This result agrees with a similar conclusion by Berry [8] and Hall [38], but it is considered here as a direct manifestation of the nondifferentiability of space itself. Consider the full complex velocity field, including its differentiable and nondifferentiable parts,
vC C v vC v i 2 2 wC C w wC w C i : 2 2
˜ DV CW D V
(86)
It is related to a full complex action S˜ and a full wave function ˜ as ˜ D V C W D r S˜/m D 2i Dr ln ˜ : V
(87)
Under the standard point of view, the complex fluctuation W is infinite and therefore r ln ˜ is undefined, so that
Eq. (87) would be meaningless. In the scale relativity approach, on the contrary, this equation keeps a mathematical and physical meaning, in terms of fractal functions, which are explicitly dependent on the scale interval dt and divergent when dt ! 0. After some calculations [59,62], one finds that the covariant derivative built from the total process (including the differentiable and nondifferentiable divergent terms) is finally b d @ D C V˜ :r i D : dt @t
(88)
The subsequent steps of the derivation of the Schrödinger equation are unchanged (now in terms of scale-dependent fractal functions), so that one obtains D2 ˜ C i D
@˜ ˜ D0; @t 2m
(89)
where ˜ can now be a nondifferentiable and fractal function. The research of such a behavior in laboratory experiments is an interesting new challenge for quantum physics. One may finally stress the fact that this result is obtained by accounting for all the terms, differentiable (dx) and nondifferentiable (d), in the description of the elementary displacements in a nondifferentiable space, and that it does not depend at all on the probability distribution of the stochastic variables d, about which no hypothesis is needed. This means that the description is fully general, and that the effect on motion of a nondifferentiable space(-time) amounts to a fundamental indeterminism, i. e., to a total loss of information about the past path which will in all cases lead to the QM description. From Fractal Space-Time to Relativistic Quantum Mechanics All these results can be generalized to relativistic quantum mechanics, which corresponds in the scale relativity framework to a full fractal space-time. This yields, as a first step, the Klein–Gordon equation [22,55,56]. Then an account of the new two-valuedness of the velocity allows one to suggest a geometric origin for the spin and to obtain the Dirac equation [22]. Indeed, the total derivative of a physical quantity also involves partial derivatives with respect to the space variables, @/@x . From the very definition of derivatives, the discrete symmetry under the reflection dx $ dx is also broken. Since, at this level of description, one should also account for parity as in the standard quantum theory, this leads to introduce a bi-quaternionic velocity field [22], in terms of which the Dirac bispinor wave function can be constructed. The successive steps that lead to the Dirac equation naturally generalize the Schrödinger case. One introduces a biquaternionic generalization of the covariant derivative that keeps the same form as in the complex case, namely, b d D V @ C i @ @ ; ds 2
(90)
where D 2D/c. The biquaternionic velocity field is related to the biquaternionic (i. e., bispinorial) wave function, by V D i
S0 m
1
@
:
(91)
This is the relativistic expression of the fundamental relation between the scale relativity tools and the quantum
587
588
Fractals in the Quantum Theory of Spacetime
mechanical tools of description. It gives a geometric interpretation to the wave function, which is, in this framework, a manifestation of the geodesic fluid and of its associated fractal velocity field. The covariance principle allows us to write the equation of motion under the form of a geodesic differential equation, b d V D0: ds
(92)
After some calculations, this equation can be integrated and factorized, and one finally derives the Dirac equation [22], @ mc 1@ D ˛ k k i ˇ c @t „ @x
:
(93)
Finally it is easy to recover the Pauli equation and Pauli spinors as nonrelativistic approximations of the Dirac equation and Dirac bispinors [23]. Gauge Fields as Manifestations of Fractal Geometry General Scale Transformations and Gauge Fields Finally, let us briefly recall the main steps of applying of the scale relativity principles to the foundation of gauge theories, in the Abelian [55,56] and non-Abelian [63,68] cases. This application is based on a general description of the internal fractal structures of the “particle” (identified with the geodesics of a nondifferentiable space-time) in terms of scale variables ˛ˇ (x; y; z; t) D %˛ˇ "˛ "ˇ whose true nature is tensorial, since it involves resolutions that may be different for the four space-time coordinates and may be correlated. This resolution tensor (similar to a covariance error matrix) generalizes the single resolution variable ". Moreover, one considers a more profound level of description in which the scale variables may now be functions of the coordinates. Namely, the internal structures of the geodesics may vary from place to place and during the time evolution, in agreement with the nonabsolute character of the scale space. This generalization amounts to the construction of a ‘general scale relativity’ theory. We assume, for simplicity of the writing, that the two tensorial indices can be gathered under one common index. We therefore write the scale variables under the simplified form ˛1 ˛2 D ˛ , ˛ D 1 to N D n(n C 1)/2, where n is the number of space-time dimensions (n D 3; N D 6 for fractal space, n D 4; N D 10 for fractal spacetime and n D 5; N D 15 in the special scale relativity case where one treats the djinn (scale-time F ) as a fifth dimension [53]).
Let us consider infinitesimal scale transformations. The transformation law on the ˛ can be written in a linear way as 0˛ D ˛ C ı˛ D (ı˛ˇ C ı ˛ˇ ) ˇ ;
(94)
where ı˛ˇ is the Kronecker symbol. Let us now assume that the ˛ ’s are functions of the standard space-time coordinates. This leads one to define a new scale-covariant derivative by writing the total variation of the resolution variables as the sum of the inertial variation, described by the covariant derivative, and of the new geometric contribution, namely,
d˛ D D˛ ˇ ı ˛ˇ D D˛ ˇ W˛ˇ dx :
(95)
This covariant derivative is similar to that of general relativity, i. e., it amounts to the subtraction of the new geometric part in order to only keep the inertial part (for which the motion equation will therefore take a geodesical, free-like form). This is different from the case of the previous quantum-covariant derivative, which includes the effects of nondifferentiability by adding new terms in the total derivative. In this new situation we are led to introduce “gauge field potentials” W˛ˇ that enter naturally in the geometrical framework of Eq. (95). These potentials are linked to the scale transformations as follows: ı ˛ˇ D W˛ˇ dx :
(96)
But one should keep in mind, when using this expression, that these potentials find their origin in a covariant derivative process and are therefore not gradients. Generalized Charges After having written the transformation law of the basic variables (the ˛ ’s), one now needs to describe how various physical quantities transform under these ˛ transformations. These new transformation laws are expected to depend on the nature of the objects they transform (e. g., vectors, tensors, spinors, etc.), which implies a jump to group representations. We anticipate the existence of charges (which are fully constructed herebelow) by generalizing the relation (91) to multiplets between the velocity field and the wave function. In this case the multivalued velocity becomes a biquaternionic matrix,
V jk D i
1 j @
k
:
(97)
The biquaternionic, and therefore non-commutative, nature of the wave function [15], which is equivalent to Dirac
Fractals in the Quantum Theory of Spacetime
bispinors, plays an essential role here. Indeed, the general structure of Yang–Mills theories and the correct construction of non-Abelian charges can be obtained thanks to this result [68]. The action also becomes a tensorial biquaternionic quantity, dS jk D dS jk x ; V jk ; ˛ ;
(98)
and, in the absence of a field (free particle) it is linked to the generalized velocity (and therefore to the spinor multiplet) by the relation
@ S jk D mc V jk D i„
1 j @
k
:
(99)
Now, in the presence of a field (i. e., when the secondorder effects of the fractal geometry appearing in the right hand side of Eq. (95) are included), using the complete expression for @ ˛ ,
@ ˛ D D ˛ W˛ˇ ˇ ;
(100)
one obtains a non-Abelian relation, @ S jk D D S jk ˇ
@S jk W : @˛ ˛ˇ
(101)
This finally leads to the definition of a general group of scale transformations whose generators are T ˛ˇ D ˇ @˛
In this contribution, we have recalled the main steps that lead to a new foundation of quantum mechanics and of gauge fields on the principle of relativity itself (once it includes scale transformations of the reference system), and on the generalized geometry of space-time which is naturally associated with such a principle, namely, nondifferentiability and fractality. For this purpose, two covariant derivatives have been constructed, which account for the nondifferentiable and fractal geometry of space-time, and which allow one to write the equations of motion as geodesics equations. After a change of variable, these equations finally take the form of the quantum mechanical and quantum field equations. Let us conclude by listing some original features of the scale relativity approach which could lead to experimental tests of the theory and/or to new experiments in the future [63,67]: (i) nondifferentiable and fractal solutions of the Schrödinger equation; (ii) zero particle interference in a Young slit experiment; (iii) possible breaking of the Born postulate for a possible effective kinetic energy operator b T ¤ („2 /2m) [67]; (iv) underlying quantum behavior in the classical domain, at scales far larger than the de Broglie scale [67]; (v) macroscopic systems described by a Schrödinger-type mechanics based on a generalized macroscopic parameter D ¤ „/2m (see Chap. 7.2 in [54] and [24,57]); (vi) applications to cosmology [60]; (vii) applications to life sciences and other sciences [4,64,65], etc.
(102)
(where we use the compact notation @˛ D @/@˛ ), yielding the generalized charges, @S jk g˜ ˛ˇ : t D ˇ c jk @˛
Future Directions
(103)
This unified group is submitted to a unitarity condition, must since, when it is applied to the wave functions, be conserved. Knowing that ˛; ˇ represent two indices each, this is a large group – at least SO(10) – that contains the electroweak theory [33,78,81] and the standard model U(1) SU(2) SU(3) and its simplest grand unified extension SU(5) [31,32] as a subset (see [53,54] for solutions in the special scale relativity framework to the problems encountered by SU(5) GUTs). As it is shown in more detail in Ref. [68], the various ingredients of Yang–Mills theories (gauge covariant derivative, gauge invariance, charges, potentials, fields, etc.) may subsequently be recovered in such a framework, but they now have a first principle and geometric scalerelativistic foundation.
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Fractals Meet Chaos
Fractals Meet Chaos TONY CRILLY Middlesex University, London, UK Article Outline Glossary Definition of the Subject Introduction Dynamical Systems Curves and Dimension Chaos Comes of Age The Advent of Fractals The Merger Future Directions Bibliography Glossary Dimension The traditional meaning of dimension in modern mathematics is “topological dimension” and is an extension of the classical Greek meaning. In modern concepts of this dimension can be defined in terms of a separable metric space. For the practicalities of Fractals and Chaos the notion of dimension can be limited to subsets of Euclidean n-space where n is an integer. The newly arrived “fractal dimension” is metrically based and can take on fractional values. Just as for topological dimension. As for topological dimension itself there is a profusion of different (but related) concepts of metrical dimension. These are widely used in the study of fractals, the ones of principal interest being: Hausdorff dimension (more fully, Hausdorff–Besicovitch dimension), Box dimension (often referred to as Minkowski– Bouligand dimension), Correlation dimension (due to A. Rényi, P. Grassberger and I. Procaccia). Other types of metric dimension are also possible. There is “divider dimension” (based on ideas of an English mathematician/meteorologist L. F. Richardson in the 1920s); the “Kaplan–Yorke dimension” (1979) derived from Lyapunov exponents, known also as the “Lyapunov dimension”; “packing dimension” introduced by Tricot (1982). In addition there is an overall general dimension due to A. Rényi (1970) which admits box dimension, correlation dimension and information dimension as special cases. With many
of the concepts of dimension there are upper and lower refinements, for example, the separate upper and lower box dimensions. Key references to the vast (and highly technical) subject of mathematical dimension include [31,32,33,60,73,92,93]. Hausdorff dimension (Hausdorff–Besicovitch dimension). In the study of fractals, the most sophisticated concept of dimension is Hausdorff dimension, developed in the 1920s. The following definition of Hausdorff dimension is given for a subset A of the real number line. This is readily generalized to subsets of the plane, Euclidean 3-space and Euclidean n-space, and more abstractly to separable metric spaces by taking neighborhoods as disks instead of intervals. Let fU i g be an r-covering of A, (a covering of A where the width of all intervals U i , satisfies w(U i ) r). The measure mr is defined by mr (A) D inf
1 X
! w(U i )
;
iD1
where the infimum (or greatest of the minimum values) is taken over all r-coverings of A. The Hausdorff dimension DH of A is: DH D lim mr (A) ; r!0
provided the limit exists. The subset E D f1/n : n D 1; 2; ; 3; : : :g of the unit interval has DH D 0 (the Hausdorff dimension of a countable set is always zero). The Hausdorff dimension is the basis of “fractal dimension” but because it takes into account intervals of unequal widths it may be difficult to calculate in practice. Box dimension (or Minkowski–Bouligand dimension, known also as capacity dimension, cover dimension, grid dimension). The box counting dimension is a more direct and practical method for computing dimension in the case of fractals. To define it, we again confine our attention to the real number line in the knowledge that box dimension is readily extended to subsets of more general spaces. As before, let fU i g be an r-covering of A, and let N r (A) be the least number of sets in such a covering. The box dimension DB of A is defined by: log N r (A) : r!0 log 1/r
DB D lim
The box dimension of the subset E D f1/n : n D 1; 2; 3; : : :g of the unit interval can be calculated to give DB D 0:5.
537
538
Fractals Meet Chaos
In general Hausdorff and Box dimensions are related to each other by the inequality DB DH , as happens in the above example. The relationship between DH and DB is investigated in [49]. For compact, selfsimilar fractal sets DB D DH but there are fractal sets for which DB > DH [76]. Though Hausdorff dimension and Box dimension have similar properties, Box dimension is only finitely additive, while Hausdorff dimension is countably additive. Correlation dimension For a set of n points A on the real number line, let P(r) be the probability that two different points of A chosen at random are closer than r apart. For a large number of points n, the graph of v D log P(r) against u D log r is approximated to its slope for small values of r, and theoretically, a straight line. The correlation dimension DC is defined as its slope for small values of r, that is, dv : r!0 du
DC D lim
The correlation dimension involves the separation of points into “boxes”, whereas the box dimension merely counts the boxes that cover A. If Pi is the probability of a point of A being in box i (approximately n i /n where ni is the number of points in box i and n is the totality of points in A) then an alternative definition of correlation dimension is P log Pi2 i : DC D lim r!0 log r Attractor A point set in phase space which “attracts” trajectories in its vicinity. More formally, a bounded set A in phase space is called an attractor for the solution x(t) of a differential equation if x(0) 2 A ) x(t) 2 A f or al l t. Thus, an attractor is invariant under the dynamics (trajectories which start in A remain in A). There is a neighborhood U A such that any trajectory starting in U is attracted to A (the trajectory gets closer and closer to A). If B A and if B satisfies the above two properties then B D A. An attractor is therefore the minimal set of points A which attracts all orbits starting at some point in a neighborhood of A. Orbit is a sequence of points fx i g D x0 ; x1 ; x2 ; : : : defined by an iteration x n D f n (x0 ). If n is a positive it is called a forwards orbit, and if n is negative a backwards orbit. If x0 D x n for some finite value n, the orbit is periodic. In this case, the smallest value of n for
which this is true is called the period of the orbit. For an invertible function f , a point x is homoclinic to a if lim f n (x) D lim f n (x) D a
as n ! 1 :
and in this case the orbit f f n (x0 )g is called a homoclinic orbit – the orbit which converges to the same saddle point a forwards or backwards. This term was introduced by Poincaré. The terminology “orbit” may be regarded as applied to the solution of a difference equation, in a similar way the solution of a differential equation x(t) is termed a trajectory. Orbit is the term used for discrete dynamical system and trajectory for the continuous time case. Basin of attraction If a is an attractive fixed point of a function f , its basin of attraction B(a) is the subset of points defined by B(a) D fx : f k (x) ! a ; as k ! 1g : It is the subset containing all the initial points of orbits attracted to a. The basins of attraction may have a complicated structure. An important example applies to the case where a is a point in the complex plane C. Julia set A set J f is the boundary between the basins of attraction of a function f . For example, in the case where z D ˙1 are attracting points (solutions of z2 1 D 0), the Julia set of the “Newton–Fourier” function f (z) D z((z2 1)/2z) is the set of complex numbers which lie along the imaginary axis x D 0 (as proved by Schröder and Cayley in the 1870s). The case of the Julia set involved with the solutions of z3 1 D 0 was beyond these pioneers and is fractal in nature. An alternative definition for a Julia set is the closure of the subset of the complex plane whose orbits of f tend to infinity. Definition of the Subject Though “Chaos” and “Fractals” are yoked together to form “Fractals and Chaos” they have had separate lines of development. And though the names are modern, the mathematical ideas which lie behind them have taken more than a century to gain the prominence they enjoy today. Chaos carries an applied connotation and is linked to differential equations which model physical phenomena. Fractals is directly linked to subsets of Euclidean space which have a fractional dimension, which may be obtained by the iteration of functions. This brief survey seeks to highlight some of the significant points in the history of both of these subjects. There are brief academic histories of the field. A history of
Fractals Meet Chaos
Chaos has been shown attention [4,46], while an account of the early history of the iteration of complex functions (up to Julia and Fatou) is given in [3]. A broad survey of the whole field of fractals is given in [18,50]. Accounts of Chaos and Fractals tend to concentrate on one side at the expense of the other. Indeed, it is only quite recently that the two subjects have been seen to have a substantial bearing on each other [72]. This account treats a “prehistory” and a “modern period”. In the prehistory period, before about 1960, topics which contributed to the modern theory are not so prominent. There is a lack of impetus to create a new field. In the modern period there is a greater sense of continuity, driven on by the popular interest in the subject. The modern theory coincided with a rapid increase in the power of computers unavailable to the early workers in the prehistory period. Scientists, mathematicians, and a wider public could now “see” the beautiful geometrical shapes displayed before them [25]. The whole theory of “fractals and chaos” necessarily involves nonlinearity. It is a mathematical theory based on the properties of processes which are assumed to be modeled by nonlinear differential equations and nonlinear functions. Chaos shows itself when solutions to these differential equations become unstable. The study of stability is rooted in the mathematics of the nineteenth century. Fractals are derived from the geometric study of curves and sets of points generally, and from abstract iterative schemes. The modern theory of fractals is the outcome of explorations by mathematicians and scientists in the 1960s and 1970s, though, as we shall see, it too has an extensive prehistory. Recently, there has been an explosion of published work in Fractals and Chaos. Just in Chaos theory alone a bibliography of six hundred articles and books compiled by 1982, grew to over seven thousand by the end of 1990 [97]. This is most likely an excessive underestimate. Fractals and Chaos has since grown into a wide-ranging and variegated theory which is rapidly developing. It has widespread applications in such areas as Astronomy the motions of planets and galaxies Biology population dynamics chemistry chemical reactions Economics time series, analysis of financial markets Engineering capsize of ships, analysis of road traffic flow Geography measurement of coastlines, growth of cities, weather forecasting Graphic art analysis of early Chinese Landscape Paintings, fractal geometry of music Medicine dynamics of the brain, psychology, heart rhythms.
There are many works which address the application of Chaos and Fractals to a broad sweep of subjects. In particular, see [13,19,27,59,63,64,87,96]. Introduction “Fractals and Chaos” is the popular name for the subject which burst onto the scientific scene in the 1970s and 1980s and drew together practical scientists and mathematicians. The subject reached the popular ear and perhaps most importantly, its eye. Computers were becoming very powerful and capable of producing remarkable visual images. Quite elementary mathematics was capable of creating beautiful pictures that the world had never seen before. The popular ear was captivated – momentarily at least – by the neologisms created for the theory. “Chaos” was one, but “fractals”, the “horseshoe”, and the superb “strange attractors” became an essential part of the scientific vocabulary. In addition, these were being passed around by those who dwelled far from the scientific front. It seemed all could take part, from scientific and mathematical researchers to lay scientists and amateur mathematicians. All could at least experience the excitement the new subject offered. Fractals and Chaos also carried implications for the philosophy of science. The widespread interest in the subject owes much to articles and books written for the popular market. J. Gleick’s Chaos was at the top of the heap in this respect and became a best seller. He advised his readership that chaos theory was one of the great discoveries of the twentieth century and quoted scientists who placed chaos theory alongside the revolutions of Relativity and Quantum Mechanics. Gleick claimed that “chaos [theory] eliminates the Laplacean fantasy of deterministic predictability”. In a somewhat speculative flourish, he reasoned: “Of the three theories, the revolution in chaos applies to the universe we see and touch, to objects at human scale. Everyday experience and real pictures of the world become legitimate targets for inquiry. There has long been a feeling, not always expressed openly, that theoretical physics has strayed far from human intuition about the world [38].” More properly chaos is “deterministic chaos”. The equations and functions used to model a dynamical system are stated exactly. The contradictory nature of chaos is that the mathematical solutions appeared to be random. On the one hand the situation is deterministic, but on the other there did not seem to be any order in the solutions. Chaos theory resolves this difficulty by conceptualizing the notion of orbits, trajectories phase space, attractors, and fractals. What appeared paradoxical seen through the
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old lenses offered explanation through a more embracing mathematical theory. In the scientific literature the modern Chaos is presented through “dynamical systems”, and this terminology gives us a clue to its antecedents. Dynamical systems conveys the idea of a physical system. It is clear that mechanical problems, for example, those involving motion are genuine dynamical systems because they evolve in space through time. Thus the pendulum swings backwards and forwards in time, planets trace out orbits, and the beat of a heart occurs in time. The differential equations which describe physical dynamical system give rise to chaos, but how do fractals enter the scene? In short, the trajectories in the phase space which describes the physical system through a process of spreading and folding pass through neighborhoods of the attractor set infinitely often, and on magnification reveal them running closer and closer together. This is the fine structure of the attractor. Measuring the metric dimension of this set results in a fraction and it is there that the connection with fractals is made. But this is running ahead of the story. Dynamical Systems The source of “Chaos” lies in the analysis of physical systems and goes back to the eighteenth century and the work of the Swiss mathematician Leonhard Euler. The most prolific mathematician of all time, Euler (whose birth tercentenary occurred in 2007) was amongst natural philosophers who made a study of differential equations in order to solve practical mechanical and astronomical problems. Chief among the problems is the problem of fluid flow which occurs in hydrodynamics. Sensitive Initial Conditions The key characteristic of “chaotic solutions” is their sensitivity to initial conditions: two sets of initial conditions close together can generate very different solution trajectories, which after a long time has elapsed will bear very little relation to each other. Twins growing up in the same household will have a similar life for the childhood years but their lives may diverge completely in the fullness of time. Another image used in conjunction with chaos is the so-called “butterfly effect” – the metaphor that the difference between a butterfly flapping its wings in the southern hemisphere (or not) is the difference between fine weather and hurricanes in Europe. The butterfly effect notion most likely got its name from the lecture E. Lorenz gave in Washington in 1972 entitled “Predictability: Does the Flap of a Butterfly’s wings in Brazil Set off a Tornado in
Texas?” [54]. An implication of chaos theory is that prediction in the long term is impossible for we can never know for certain whether the “causal” butterfly really did flap its wings. The sensitivity of a system to initial conditions, the hallmark of what makes a chaotic solution to a differential equation is derived from the stability of a system. Writing in 1873, the mathematical physicist James Clerk Maxwell alluded to this sensitivity in a letter to man of science Francis Galton: Much light may be thrown on some of these questions [of mechanical systems] by the consideration of stability and instability. When the state of things is such that an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable; but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the condition is said to be unstable [44]. H. Poincaré too, was well aware that small divergences in initial conditions could result in great differences in future outcomes, and said so in his discourse on Science and Method: “It may happen that slight differences in the initial conditions produce very great differences in the final phenomena; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible.” As an example of this he looked at the task of weather forecasting: Why have the meteorologists such difficulty in predicting the weather with any certainty? Why do the rains, the tempests seem to us to come by chance, so that many persons find it quite natural to pray for rain or shine, when they would think it ridiculous to pray for an eclipse? We see that great perturbations generally happen in regions where the atmosphere is in unstable equilibrium. . . . one tenth of a degree more or less at any point, and the cyclone bursts here and not there, and spreads its ravages over countries it would have spared. . . . Here again we find the same contrast between a very slight cause, unappreciable to the observer, and important effects, which are sometimes tremendous disasters [75]. So here is the answer to one conundrum – Poincaré is the true author of the butterfly effect! Weather forecasting is ultimately a problem of fluid flow in this case air flow.
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Fluid Motion and Turbulence The study of the motion of fluids predates Poincaré and Lorenz. Euler published “General Principles of the Motion of Fluids” in 1755 in which he wrote down a set of partial differential equations to describe the motion of non-viscous fluids. These were improved by the French engineer C. Navier when in 1821 he published a paper which took viscosity into account. From a different starting point in 1845 the British mathematical physicist G. G. Stokes derived the same equations in a paper entitled “On the Theories of the Internal Friction of Fluids in Motion.” These are the Navier–Stokes equations, a set of nonlinear partial differential equations which generally apply to fluid motion and which are studied by the mathematical physicist. In modern vector notation (in the case of unforced incompressible flow) they are @v 1 C v:rv D r p C r 2 v ; @t
where v is velocity, p is pressure, and and are density and viscosity constants. It is the nonlinearity of the Navier–Stokes equations which makes them intractable, the nonlinearity manifested by the “products of terms” like v:rv which occur in them. They have been studied intensively since the nineteenth century particularly in special forms obtained by making simplifying assumptions. L. F. Richardson, who enters the subject of Fractals and Chaos at different points made attempts to solve nonlinear differential equations by numerical methods. In the 1920s, Richardson adapted words from Gulliver’s Travels in one of his well-known refrains on turbulence: “big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity – in the molecular sense” [79]. This numerical work was hampered by the lack of computing power. The only computers available in the 1920s were human ones, and the traditions of using paid “human computers” was still in operation. Richardson visualized an orchestra of human computers harmoniously carrying out the vast array of calculations under the baton of a mathematician. There were glimmers of all this changing, and during the 1920s the appearance of electrical devices gave an impetus to the mathematical study of both numerical analysis and the study of nonlinear differential equations. John von Neumann for one saw the need for the electronic devices as an aid to mathematics. Notwithstanding this development, the problem of fluid flow posed intrinsic mathematical difficulties. In 1932, Horace Lamb addressed the British Association for the Advancement of Science, with a prophetic
statement dashed with his impish touch of humor: “I am an old man now, and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electro-dynamics, and the other is the turbulent motion of fluids. And about the former I am really rather optimistic.” Forty years on Werner Heisenberg continued in the same vein. He certainly knew about quantum theory, having invented it, but chose relativity as the competing theory with turbulence for his own lamb’s tale. On his death bed it is said he compared quantum theory with turbulence and is reputed to have singled out turbulence as of the greater difficulty. In 1941, A. Kolmogorov, the many-sided Russian mathematician published two papers on problems of turbulence caused by the jet engine and astronomy. Kolmogorov also made contributions to probability and topology though these two papers are the ones rated highly by fluid dynamicists and physicists. In 1946 he was appointed to head the Turbulence Laboratory of the Academic Institute of Theoretical Geophysics. With Kolmolgov, an influential school of mathematicians that included L. S. Pontryagin, A. A. Andronov, D. V. Anosov and V. I. Arnol’d became active in Russia in the field of dynamical systems [24]. Qualitative Differential Equations and Topology Poincaré’s qualitative study of differential equations pioneered the idea of viewing the solution of differential equations as curves rather than functions and replacing the local with the global. Poincaré’s viewpoint was revolutionary. As he explained it in Science and Method: “formerly an equation was considered solved only when its solution had been expressed by aid of a finite number of known functions; but that is possible scarcely once in a hundred times. What we always can do, or rather what we should always seek to do, is to solve the problem qualitatively [his italics] so to speak; that is to say; seek to know the general form of the curve [trajectory] which represents the unknown function.” For the case of the plane, m D 2, for instance, what do the solutions to differential equations look like across the whole plane, viewing them as trajectories x(t) starting at an initial point? This contrasts with the traditional view of solving differential equations whereby specific functions are sought which satisfy initial and boundary conditions. Poincaré’s attack on the three body problem (the motion of the moon, earth, sun, is an example) was stimulated by a prize offered in 1885 to commemorate the sixtieth birthday of King Oscar II of Sweden. The problem set was for an n-body problem but Poincaré’s essay on the re-
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stricted three-body problem was judged so significant that he was awarded the prize in January 1889. Before publication, Poincaré went over his working and found a significant error. It proved a profound error. According to Barrow-Green his description of doubly asymptotic trajectories is the first mathematical description of chaotic motion in a dynamical system [6,7]. Poincaré introduced the “Poincaré section” the surface section that transversely intersects the trajectory in phase space and defines a sequence of points on it. In the case of the damped pendulum, for example, a sequence of points is obtained as the spiral trajectory in phase space hits the section. This idea brings a physical dynamical systems problem in conjunction with topology, a theme developed further by Birkhoff in 1927 [1,10]. Poincaré also introduced the idea of phase space but his study mainly revolves around periodic trajectories and not the non periodic ones typical of chaos. Poincaré said of periodic orbits, that they were “the only gap through which may attempt to penetrate, into a place where up to now was reputed to be unreachable”. The concentration on periodic trajectories has a parallel in nearby pure mathematics – where irregular curves designated as “monsters” were ignored in favor of “normal curves”. G. D. Birkhoff learned from Poincaré. It was said that, apart from J. Hadamard, no other mathematician knew Poincaré’s work as well as Birkhoff did. Like his mentor, Birkhoff adopted phase space as his template, and emphasised periodic solutions and treated conservative systems and not dissipative systems (such as the damped pendulum which loses energy). Birkhoff spent the major part of his career, between 1912 and 1945 contributing to the theory of dynamical systems. His aim was to provide a qualitative theory which characterized equilibrium points in connection with their stability. A dynamical system was defined by a set of n differential equations dx i /dt D F i (x1 ; : : : ; x n ) defined locally. An interpretation of these could be a situation in chemistry where x1 (t); : : : ; x n (t) might be the concentrations of n chemical reactants at time t where the initial values x1 (0); : : : ; x n (0) are given, though applications were not Birkhoff’s main concern. In 1941, towards the end of his career, Birkhoff reappraised the field in “Some unsolved problems of theoretical dynamics” Later M. Morse discussed these and pointed out that Birkhoff listed the same problems he had considered in 1920. In 1920 Birkhoff had written about dynamical systems in terms of the “general analysis” propounded by E. H. Moore in Chicago where he had been a student. In the essay of 1941 modern topological language was used:
As was first realized about fifty years ago by the great French mathematician, Henri Poincaré, the study of dynamical systems (such as the solar system) leads directly to extraordinary diverse and important problems in point-set theory, topology and the theory of functions of real variables. The idea was to describe the phase space by an abstract topological space. In his talk of 1941, Birkhoff continued: The kind of abstract space which it seems best to employ is a compact metric space. The individual points represent “states of motion”, and each curve of motion represents a complete motion of the abstract dynamical system [11]. Using Birkhoff’s reappraisal, Morse set out future goals: “ ‘Conservative flows’ are to be studied both in the topological and the statistical sense, and abstract variational theory is to enter. There is no doubt about the challenge of the field, and the need for a powerful and varied attack [68].” Duffing, Van der Pol and Radar It is significant that chaos theory was first derived from practical problems. Poincaré was an early pioneer with a problem in astronomy but other applications shortly arrived. C. Duffing (1918) introduced a second order nonlinear differential equation which described a mechanical oscillating device. In a simple form of it (with zero forcing term): d2 x/dt 2 C ˛dx/dt C (ˇx 3 C x) D 0 ; Duffing’s equation exhibits chaotic solutions. B. van der Pol working at the Radio Scientific Research group at the Philips Laboratory at Eindhoven, described “irregular noise” in an electronic diode. Van der Pol’s equations of the form (with right hand side forcing term): d2 x/dt 2 C k(x 2 1)dx/dt C x D A cos ˝ t described “relaxational” oscillations or arrhythmic beats of an electrical circuit. Such “relaxational” oscillations are of the type which also occur in the beating of the heart. Richardson noted the transition from periodic solutions to van der Pol’s equation (1926) to unstable solutions. Both Duffing’s equation and Van der Pol’s equation play an important part in chaos theory. In 1938 the English mathematician Mary Cartwright answered a call for help from the British Department of Scientific and Industrial Research. A solution to the differential equations connected with the new radar technology
Fractals Meet Chaos
was wanted, and van der Pol’s equation was relevant. It was in the connection this equation that she collaborated with J. E. Littlewood in the 1940s and made discoveries which presaged modern Chaos. The mathematical difficulty was caused by the equation being nonlinear but otherwise the equation appear nondescript. Yet in mathematics, the nondescript can yield surprises. A corner of mathematics was discovered that Cartwright described as “a curious branch of mathematics developed by different people from different standpoints – straight mechanics, radio oscillations, pure mathematics and servo-mechanisms of automatic control theory [65]” The mathematician and physicist Freeman Dyson wrote of this work and its practical significance: Cartwright had been working with Littlewood on the solutions of the equation, which describe the output of a nonlinear radio amplifier when the input is a pure sine-wave. The whole development of radio in World War II depended on high power amplifiers, and it was a matter of life and death to have amplifiers that did what they were supposed to do. The soldiers were plagued with amplifiers that misbehaved, and blamed the manufacturers for their erratic behavior. Cartwright and Littlewood discovered that the manufacturers were not to blame. The equation itself was to blame. They discovered that as you raise the gain [the ratio of output to input] of the amplifier, the solutions of the equation become more and more irregular. At low power the solution has the same period as the input, but as the power increases you see solutions with double the period, and finally you have solutions that are not periodic at all [30].
Crinkly Curves In 1872, K. Weierstrass introduced the famous function defined by a convergent series: f (x) D
1 X
b k cos(a k x) (a > 1; 0 < b < 1; ab > 1);
kD0
which was continuous everywhere but differentiable nowhere. It was the original “crinkly curve” but in 1904, H. von Koch produced a much simpler one based only on elementary geometry. While the idea of a function being continuous but not differentiable could be traced back to A-M Ampère at the beginning of the nineteenth century, von Koch’s construction has similarities with examples produced by B. Bolzano. Von Koch’s curve has become a classic half-way between a “monster curves” and regularity – an example of a curve of infinite length which encloses a finite area, as well as being an iconic fractal. In retrospect G. Cantor’s middle third set (also discovered by H.J.S. Smith (1875) a professor at Oxford), is also a fractal. It is a totally disconnected and uncountable set, with the curiosity that after subtractions of the middlethird segments from the unit interval the ultimate Cantor set has same cardinality as the original unit interval. At the beginning of the twentieth century searching questions about the theory of curves were being asked. The very basic question “what is a curve” had been brought to life by G. Peano by his space filling curve which is defined in accordance with Jordan’s definition of a curve but fills out a “two-dimensional square.” Clearly the theory of dimension needed serious attention for how could an ostensibly two-dimensional “filled-in square” be a curve? The Iteration of Functions
The story is now familiar: there is the phenomenon of period doubling of solutions followed by chaotic solutions as the gain of the amplifier is raised still higher. A further contribution to the theory of this equation was made by N. Levinson of the Massachusetts Institute of Technology in the United States.
Curves and Dimension The subject of “Fractals” is a more recent development. First inklings of them appeared in “Analysis situs” in the latter part of the nineteenth century when the young subject of topology was gaining ground. Questions were being asked about the properties of sets of points in Euclidean spaces, the nature of curves, and the meaning of dimension itself.
A principal source of fractals is obtained by the iteration of functions, what is now called a discrete dynamical system, or a system of symbolic dynamics [15,23]. One of the earliest forays in this field was made by the English mathematician A. Cayley, but he was not the first as D. S. Alexander has pointed out. F.W. K. E. (Ernst) Schröder anticipated him, and may even have been the inspiration for Cayley’s attraction to the problem. Is there a link between the two mathematicians? Schröder studied at Heidelberg and where L. O. Hesse was his doctoral adviser. Hesse who contributed to algebra, geometry, and invariant theory and was known to Cayley. Nowadays Schröder is known for his contributions to logic but in 1871 he published “Ueber iterite Functionen” in the newly founded Mathematische Annalen. The principal objective of this journal was the publication of articles
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on invariant theory then a preoccupation of the English and German schools of mathematics, and Cayley was at the forefront of this research. So did Cayley know of Schröder’s work? Cayley had an appetite for all things (pure) mathematical and had acquired encyclopedic knowledge of the field, just about possible in the 1870s. In 1871 Cayley published an article (“Note on the theory of invariants”) in the same volume and issue of the Mathematische Annalen in which Schröder’s article appeared, and actually published three articles in the volume. In this period of his life, when he was in his fifties he allowed his interests full rein and became a veritable magpie of mathematics. He covered a wide range of topics, too many to list here, but his contributions were invariably short. Moreover he dropped invariant theory at this point and only resumed in 1878 when J. J. Sylvester reawakened his interest. There was time to rediscover the four color problem and perhaps Schröder’s work on the iteration of functions. It was a field where Schroeder had previously “encountered very few collaborators.” The custom of English mathematicians citing previous work of others began in the 1880s and Cayley was one of the first to do this. The fact that Cayley did not cite Schröder is not significant. In February 1879, Cayley wrote to Sir William Thomson (the later Lord Kelvin) about the Newton–Fourier method of finding the root of an equation, named after I. Newton (c.1669) and J. Fourier (1818) that dealt with the real variable version of the method. This achieved a degree of significance in Cayley’s mind, for a few days later he wrote to another colleague about it: I have a beautiful question which is bothering me – the extension of the Newton–Fourier method of approximation to imaginary values: it is very easy and pretty for a quadric equation, but I do not yet see how it comes out for a cubic. The general notion is that the plane must be divided into regions; such that starting with a point P in one of these say the A-region . . . [his ellipsis], the whole series of derived points P1 ; P2 ; P3 ; : : : up is P1 (which will be the point A) lies in this [planar] region; . . . and so for the B. and C. regions. But I do not yet see how to find the bounding curves [of these regions] [21]. So Cayley’s regions are the modern basins of attraction for the point A, the bounding curves now known as Julia sets. He tried out the idea before the Cambridge Philosophical Society, and by the beginning of March had done enough to send the problem for publication: In connexion herewith, throwing aside the restrictions as to reality, we have what I call the Newton–
Fourier Imaginary Problem, as follows. Take f (u) a given rational and integral function [a polynomial] of u, with real or imaginary coefficients; z, a given real or imaginary value, and from this derive z1 by the formula z1 D z
f (z) f 0 (z)
and thence z1 ; z2 ; z3 ; : : : each from the preceding one by the like formula. . . . The solution is easy and elegant in the case of a quadric equation: but the next succeeding case of the cubic equation appears to present considerable difficulty [17]. Cayley’s connection with German mathematicians was close in the 1870s, and later on in 1879 (and in 1880) he went on tours of Germany, and visited mathematicians. No doubt he took the problem with him, and it was one he returned to periodically. Both Cayley and Schröder solved this problem for the roots of z2 D 1 but their methods differ. Cayley’s is geometrical whereas Schröder’s was analytical. There are only two fixed points z D ˙1 and the boundary (Julia set) between the two basins of attraction of the root finding function is the “y-axis.” The algorithm for finding the complex roots of the cubic equation z3 D 1 has three stable fixed points at z D 1; z D exp(2 i/3); z D exp(2 i/3) and with the iteration z ! z (z3 1)/3z2 , three domains, basins of attraction exist with highly interlaced boundaries. The problem of determining the Julia set for the quadratic was straightforward but the cubic seemed impossible [42]. For good reason, with the aid of modern computing machinery the Julia set in the case of the cubic is an intricately laced trefoil. But some progress was made before computers entered the field. After WWI G. Julia and P. Fatou published lengthy papers on iteration and later C. L. Siegel studied the field [35,48,85]. Topological Dimension The concept of dimension has been an enduring study for millennia, and fractals has prolonged the centrality of this concept in mathematics. The ordinary meaning of dimensions one, two, three dimensions applied to line, plane, solid of the Greeks, was initially extended to n-dimensions. Since the 1870s mathematicians ascribed newer meanings to the term. At the beginning of the twentieth century, Topology was in its infancy, and emerging from “analysis situs.” An impetus was the Hausdorff’s definition of a topological space in terms of neighborhoods in 1914. The evolution of topological dimension was developed in the hands
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of such mathematicians as L. E. J. Brouwer, H. Poincaré, K. Menger, P. Urysohn, F. Hausdorff but in the new surroundings, the concepts of curve and dimension proved elusive. Poincaré made several attempts to define dimension. One was in terms of group theory, one he described as a “dynamical theory.” This was unsatisfactory for why should dimension depend on the idea of a group. He gave another definition that he described as a “statical theory.” Accordingly in papers of 1903, and 1912, a topological notion of n-dimensions (where n a natural number) was based on the notion of a cut, and Poincaré wrote: If to divide a continuum it suffices to consider as cuts a certain number of elements all distinguishable from one another, we say this continuum is of one dimension; if, on the contrary, to divide a continuum it is necessary to consider as cuts a system of elements themselves forming one or several continua, we shall say that this continuum is of several dimensions. If to divide a continuum C, cuts which form one or several continua of one dimension suffice, we shall say that C is a continuum of two dimensions; if cuts which form one or several continua of at most two dimensions suffice, we shall say that C is a continuum of three dimensions; and so on [74]. To illustrate the pitfalls of this game of cat and mouse where “definition” attempts to capture the right notion, we see this definition yielded some curious results – the dimension of a double cone ostensibly of two dimensions turns out to be of one dimension, since one can delete the zero dimensional point where the two ends of the cone meet. Curves were equally difficult to pin down. Menger used a physical metaphor to get at the pure notion of a curve: We can think of a curve as being represented by fine wires, surfaces as produced from thin metal sheets, bodies as if they were made of wood. Then we see that in order to separate a point in the surface from points in a neighborhood or from other surfaces, we have to cut the surfaces along continuous lines with a scissors. In order to extract a point in a body from its neighborhood we have to saw our way through whole surfaces. On the other hand in order to excise a point in a curve from its neighborhood irrespective of how twisted or tangled the curve may be, it suffices to pinch at discrete points with tweezers. This fact, that is independent of the particular form of curves or surfaces we consider, equips us with a strong conceptual description [22].
Menger set out his ideas about basic notions in mathematical papers and in the books Dimensionstheorie (1928) and Kurventheorie (1932). In Dimensionstheorie he gave an inductive definition of dimension, on the implicit understanding that dimension only made sense for n an integer ( 1): A space is called at most n-dimensional, if every point is contained in arbitrarily small neighborhoods with an most (n1)-dimensional boundaries. A space that is not at most (n 1)-dimensional we call at least n-dimensional. . . . A Space is called n-dimensional, if it is both at most n-dimensional and also at least n-dimensional, in other words, if every point is contained in arbitrarily small neighborhoods with at most (n 1)-dimensional boundaries, but at least one point is not contained in arbitrarily small boundaries with less than (n 1)-dimensional boundaries. . . . The empty set and only this is (–1)-dimensional (and at most (–1)-dimensional. A space that for no natural number n is n-dimensional we call infinite dimensional. Different notions of topological dimension defined in the 1920s and 1930s, were ind X (the small inductive dimension), Ind X (the large inductive dimension), and dim X (the Lebesgue covering dimension). Researchers investigated the various inequalities between these and the properties of abstract topological spaces which ensured all these notions coincided. By 1950, the theory of topological dimension was still in its infancy [20,47,66]. Metric Dimension For the purposes of fractals, it is the metric definitions of dimension which are fundamental. For instance, how can we define the dimension of the Cantor’s “middle third” set which takes into account its metric structure? What about the iconic fractal known as the von Koch curve snow flake curve (1904)? These sets pose interesting questions. Pictorially von Koch’s curve is made up of small 1 dimensional lines and we might be persuaded that it too should be 1 dimensional. But the real von Koch curve is defined as a limiting curve and for this there are differences between it and a line. Between any two points on a 1 dimensional line there is a finite distance but between any two points on the Koch curve the distance is infinite. This suggests its dimensionality is greater than 1 while at the same time it does not fill out a 2 dimensional region. The Sierpinski curve (or gasket) is another example, but perhaps more spectacular is the “sponge curve”. In Kurventheorie (1932) Menger gave what is now known as
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the Menger Sponge, the set obtained from the unit cube by successively deleting sub-blocks in the same way as Cantor’s middle third set is obtained from the unit interval. A colorful approach to defining metric dimension is provided by L. F. Richardson. Richardson (1951) took on the practical measurement of coastlines from maps. To measure the coastline, Richardson used dividers set a disP tance l apart. The length of the coastline will be L D l after walking the dividers around the coast. Richardson was a practical man, and he found by plotting L against l, the purely empirical result that L / l ˛ for a constant ˛ that depends on the chosen coastline. So, for a given country we get an approximate length L D cl ˛ but the smaller the dividers are set apart the longer becomes the coastline! The phenomenon of the coastline length being “divider dependent” explains the discrepancy of 227 kilometers in the measurement by of the border between Spain and Portugal (987 km stated by Spain, and 1214 km stated by Portugal). The length of the Australian coastline turns out to be 9400 km if one divider space represents 2000 km, and is 14,420 km if the divider space represent 100 km. Richardson cautions “to speak simply of the “length” of a coast is therefore to make an unwarranted assumption. When a man says that he “walked 10 miles along the coast,” he usually means that he walked 10 miles [on a smooth curve] near the coast.” Richardson goes into incredible numerical data and his work is of an empirical nature, but it paid off, and he was able to combine two branches of science, the empirical and the mathematical. Moreover here we have a link with chaos: Richardson described the “sensitivity to initial conditions” where a small difference in the setting of the dividers can result in a large difference in the “length” of the coastline. The constant ˛ is a characteristic of the coastline or frontier whose value is dependent on its place in the range between smoothness and jaggedness. So if the frontier is a straight line, ˛ would be zero, and increases the more irregular the coast line. In the case of Australia, ˛ was found to be about 0.13 and for the very irregular west coast of Britain, ˛ was found to be about 0.25. The value 1 C ˛ anticipates the mathematical concept known as “divider dimension”. So, for example, the divider dimension of the west coast of Britain would be 1.25. The divider dimension idea is fruitful – it can be applied to Brownian motion, a mathematical theory set out by Norbert Wiener in the 1920s but it is not the main one when applied to fractal dimension. “Fractal dimension” means Hausdorff dimension, or as we already noted, the Hausdorff–Besicovitch dimension. Generally Hausdorff dimension is difficult to calculate, but for the self-similar sets and curves such as Can-
tor’s middle-third set, the Sierpinski curve, the von Koch curve, Menger’s sponge, it is equivalent to calculating the box dimension or Minkowski–Bouligand dimension (see also [61]). The middle third set has fractal dimension DH D log 2/ log 3 D 0:63 : : :, and the Sierpinski curve has DH D log 3/ log 2 D 1:58 : : :, the von Koch curve DH D log 4/ log 3 D 1:26 : : :, and Menger’s sponge embedded in three-dimensional Euclidean space has Hausdorff dimension DH D log 20/ log 3 D 2:72 : : :. Chaos Comes of Age The modern theory of Chaos occurred around the end of the 1950s. S. Smale, Y. Ueda, E. N. Lorenz made discoveries which ushered in the new age. The subject became extremely popular in the early 1970s and the “avant-garde” looked back to this period as the beginning of chaos theory. They contributed some of the most often cited papers in further developments. Physical Systems In 1961 Y. Ueda, a third year undergraduate student in Japan discovered a curious phenomenon in connection with the single “Van der Pol” type nonlinear equation d2 x/dt 2 C k( x 2 1)dx/dt C x 3 D ˇ cos t ; another example of an equation used to model an oscillator. With the parameter values set at k D 0:2; D 8; ˇ D 0:35 Ueda found the dynamics was “chaotic” – though of course he did not use that term. With an analogue computer, of a type then used to solve differential equations, the attractor in phase space appeared as a “shattered egg.” Solutions for many other values of the parameters revealed orderly behavior, but what was special about the values 0.2, 8, 0.35? Ueda had no idea he had stumbled on a major discovery. Forty years later he reminisced on the higher principles of the scientific quest: “but while I was toiling alone in my laboratory,” he said, “I was never trying to pursue such a grandiose dream as making a revolutionary new discovery, not did I ever anticipate writing a memoir about it. I was simply trying to find an answer to a persistent question, faithfully trying to follow the lead of my own perception of a problem [2].” In 1963, E. N. Lorenz published a landmark paper (with at least 5000 citations to date) but it was one published in a journal off the beaten track for mathematicians. Lorenz investigated a simple model for atmospheric convection along the lines of the Rayleigh–Bernard convection model. To see what Lorenz actually did we quote the abstract to his original paper:
Fractals Meet Chaos
Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamical flow. Solutions of these equations can be identified with trajectories in phase space. For those solutions with bounded solutions, it is found that non periodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are non periodic [53]. The oft quoted equations (with simplifying assumptions, and a truncated version of the Navier–Stokes equations) used to model Rayleigh–Bernard convection are the nonlinear equations [89]: dx/dt D (y x) dy/dt D rx y xz dz/dt D x y bz ; where is called the Prandtl number, r is the Rayleigh number and b is a geometrically determined parameter. From the qualitative viewpoint, the solution of these equations, (x1 (t); x2 (t); x3 (t)) with initial values (x1 (0); x2 (0); x3 (0)) traces out a trajectory in 3-dimensional phase space. Lorenz solved the equations numerically by a forward difference procedure based on the time honored Runge–Kutta method which set up an iterative scheme of difference equations. Lorenz discovered that the case where the parameters have specific values of D 10; r D 28; b D 8/3 gave chaotic trajectories which wind around the famous Lorenz attractor in the phase space – and by accident, he discovered the butterfly effect. The significance of Lorenz’s work was the discovery of Chaos in low dimensional systems – the equations described dynamical behavior of a kind seen by only a few – including Ueda in Japan. One can imagine his surprise and delight, for though working in Meteorology, Lorenz had been a graduate student of G. D. Birkhoff at Harvard. Once the chink in the mathematical fabric was made, the same kind of mathematical behavior of chaos was discovered in other systems of differential equations. The Lorenz attractor became the subject of intensive research. Once it was discovered by mathematicians – not many mathematicians read the meteorology journals – it excited much attention and helped to launch the chaos craze. But one outstanding problem remained: did the
Lorenz attractor actually exist or was its presence due to the accumulation of numerical errors in the approximate methods used to solve the differential equations. Computing power was in its infancy and Lorenz made his calculations on a fairly primitive string and sealing-wax Royal McBee LGP-30 machine. Some believed that the numerical evidence was sufficient for the existence of a Lorenz attractor but this did not satisfy everyone. This question of actual existence resisted all attempts at its solution because there were no mathematical tools to solve the equations explicitly. The problem was eventually cracked by W. Tucker in 1999 then a postgraduate student at the University of Uppsala [94]. Tucker’s proof that the Lorenz attractor actually exists involved a rigorous computer algorithm in conjunction with a rigorous set of bounds on the possible numerical errors which could occur. It is very technical [95]. Other sets of differential equations which exemplified the chaos phenomena, one even more basic than Lorenz’s is due to O. Rössler, equations which modeled chemical reactions: dx/dt D y z dy/dt D x C ˛ y dz/dt D ˛ z C xz : Rössler discovered chaos for the parameter values ˛ D 0:2 and D 5:7 [77]. This is the simplest system yet found, for it has only one nonlinear term xz compared with two in the case of Lorenz’s equations. Other examples of chaotic solutions are found in the dripping faucet experiment caried out by Shaw (1984) [84]. Strange Attractors An attractor is a set of points in phase space with the property that a trajectory with an initial point near it will be attracted to it – and if the trajectory touches the attractor it is trapped to stay within it. But what makes an attractor “strange”? The simple pendulum swings to and fro. This is a dissipative system as the loss of energy causes the bob to come to rest. Whatever the initial point stating point of the bob, the trajectory in phase space will spiral down to the origin. The attractor in this case is simply the point at the origin; the rest point where displacement is nil and velocity is nil. This point is said to attract all solutions of the differential equations which models the pendulum. This single point attractor is hardly strange. If the pendulum is configured to swing to and fro with a fixed amplitude, the attractor in phase space will be a circle. In this case the system conserves energy and the whole
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system is called a conservative system. If the pendulum is slightly perturbed it will return to the previous amplitude, and in this sense the circle or a limit cycle will have attracted the displaced trajectory. Neither is the circle strange. The case of the driven pendulum is different. In the driven pendulum the anchor point of the pendulum oscillates with a constant amplitude a and constant drive frequency f . The equation of motion of the angular displacement from the vertical with damping parameter q can be described as a second order nonlinear (because of the presence of the sin term in the differential equation): d2 /dt 2 C (1/q) d /dt C sin D a cos f t : Alternatively this motion can be described by three simultaneous equations: dw/dt D (1/q)w sin C a cos d /dt D w d/dt D f ; where is the phase of the drive term. The three variables (w; ; ) describe the motion of the driven pendulum in three–dimensional phase space and its shape will depend on the values of the parameters (a; f ; q). For some values, just as for the Lorenz attractor, the motion will be chaotic [5]. For an attractor to be strange, it should be fractal in structure. The notion of strange attractor is due to D. Ruelle and F. Takens in papers of 1971 [82]. In their work there is no mention of fractals, simply because fractals had not risen to prominence at that time. Strange attractors for Ruelle and Takens were infinite sets of points in phase space corresponding to points of a physical dynamic system which appeared to have a complicated evolution – they were just very weird sets of points. But the principle was gained. Dynamical systems, like the gusts in wind turbulence, are in principle modeled by deterministic differential equations but now their solution trajectories seemed random. In classical physics mechanical processes were supposed to be as uncomplicated as the pendulum where the attractor was a single point or a circular limit cycle. The seemingly random processes that now appeared offered the mathematician and physicist a considerable challenge. The name “strange attractor” caught on and quickly captured the scientific and popular imagination. Ruelle asked Takens if he had dreamed up the name, and he replied: “Did you ever ask God whether he created this damned universe? . . . I don’t remember anything . . . I often create without remembering it . . . ” and Ruelle wrote
the “creation of strange attractors thus seems to be surrounded by clouds and thunder. Anyway, the name is beautiful, and I well suited to these astonishing objects, of which we understand so little [80].” Ruelle wrote “These systems of curves [arising in the study of turbulence], these clouds of points, sometimes evoke galaxies or fireworks, other times quite weird and disturbing blossomings. There is a whole world of forms still to be explored, and harmonies still to be discovered [45,91].” Initially a strange attractor was a set which was extraordinary in some sense and there were naturally attempts to pin this down. In particular distinctions between “strange attractors” and “chaotic behavior” were drawn out [14]. Accordingly, a strange attractor is an attractor which is not (i) a finite set of points (ii) a closed curve (iii) a smooth or piecewise smooth surface or a volume bounded by such a surface. “Chaotic behavior” relates to the behavior of trajectories on the points of the attractor where nearby orbits around the attractor diverge with time. A strange nonchaotic attractor is one (with the above exclusions) where the trajectories are not chaotic, that is, there is an absence of sensitivity to initial conditions. Pure Dynamical Systems Chaos had its origins in real physical problems and the consequent differential equations, but the way had been set by Poincaré and Birkhoff for the entry of pure mathematicians. S. Smale gained his doctorate from the University of Michigan in 1956 supervised by the topologist R. Bott, stepped into this role. In seeking out fresh fields for research he surveyed Birkhoff’s Collected Papers and read the papers of Levinson on the van der Pol equation.. On a visit to Rio de Janeiro in late 1959, Smale focused on general dynamical systems. What better place to do research than on the beach: My work was mostly scribbling down ideas and trying to see how arguments could be sequenced. Also I would sketch crude diagrams of geometric objects flowing through space and try to link the pictures with formal deductions. Deeply involved in this kind of thinking and writing on a pad of paper, the distraction of the beach did not bother me. Moreover, one could take time off from the research to swim [88]. He got into hotter water to the north when it was brought to the attention of government officials that national research grants were being spent at the seaside. They seemed unaware that a mathematical researcher is always working. Smale certainly had a capacity for hard concentrated work.
Fractals Meet Chaos
Receiving a letter from N. Levinson, about the Van der Pol equation, he reported: I worked day and night to try to resolve the challenge to my beliefs that letter posed. It was necessary to translate Levinson’s analytic argument into my own geometric way of thinking. At least in my own case, understanding mathematics doesn’t come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. . . . In any case I eventually convinced myself that Levinson was correct, and that my conjecture was wrong. Chaos was already implicit in the analyzes of Cartwright and Littlewood! The paradox was resolved, I had guessed wrongly. But while learning that, I discovered the horseshoe [8]. The famous “horseshoe” mapping allowed him to construct a proof of the Poincaré conjecture in dimensions greater than or equal to five, and for this, he was awarded a Field’s medal in 1966. Iteration provides an example of a dynamical system, of a kind involving discrete time intervals. These were the systems suggested by Schröder and Cayley, and take on a pure mathematical guise with no physical experimenting to act as a guide. In the late 1950s, a research group in Toulouse led by I. Gumowski and his student C. Mira pursued an exploration of nonlinear systems from this point of view. Their approach was inductive, and started with the simplest example of nonlinear maps of the plane which were then iterated. Functions chosen for exploration were the family, x ! y F(x) y ! x C F(y F(x)) ; for various rational functions F(x). The pictures in the plane were noted for their spectacular “chaos esthétique.” The connection of chaos with the iteration of functions was pioneered by Gumowski and Mira, O. M. Sharkovsky (in 1964), Smale (in 1967) [86], and N. Metropolis, M. Stein, P. Stein (in 1973) [67]. M. Feigenbaum (in the late 1970s) studied the quadratic logistic map x ! x(1 x) and its iterates and discovered period doubling and the connection with physical phenomena. For the value D 3:5699456 : : : the orbit is non periodic and the attractor is the Cantor set so it is a strange attractor. Feigenbaum studied one dimensional iterative maps of a general type x ! f (x) for a general f and discovered properties independent of the form of the recursion function. The Feigenbaum number, the bifurcation rate ı with
value ı D 4:6692 : : : is the limit of the parameter intervals in which period doubling takes place before the onset of chaos. The intervals are shortened at each period doubling by the inverse of the Feigenbaum number, a universal value for many functions [36,37]. In the case of fluid flow, it was discovered that the transition from smooth flow to turbulence is linked with period doubling. The iteration of functions, a very simple process, was brought to the attention of the wider scientific public by R. May. In his oft quoted piece on the one dimension quadratic mapping, he stressed the importance it held for mathematical education: I would therefore urge that people be introduced to, say, equation x ! x(1 x) early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems. Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties [62]. The example of x ! x(1 x) provides another link with the mathematical past, which amply demonstrates that Fractals and Chaos is not merely a child of the sixties but is joined to the mainstream of mathematics which slowly evolves over centuries. The Schwarzian derivative y D f (x) defined by dx d3 y 3 S(y) D dy dx 3 2
dx d2 y dy dx 2
!2 ;
is connected with complex analysis and also introduced into invariant theory (where it was called a differentiant) of a hundred years previously. It was introduced by D. Singer in 1978 in the context of one dimensional dynamical systems [26]. While the quadratic map is one dimensional, we can go into two dimensions with pairs of nonlinear difference equations. A beautiful example is the one (x; y) ! (y x 2 ; a C bx) or in the equivalent form (x; y) ! (1Cyax 2 ; bx). In this case, for some values of the parameters ellipses are generated, but in others we gain strange attractors. This is the Hénon schemeb D 0:3, and a near 1.4 the attractor has an infinity of attracting points, and being a fractal set it is a strange attractor [81]. A Poincaré
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section of this attractor is a Cantor set, further evidence for the set being a strange attractor. The Hénon map is one of the earliest examples illustrating chaos. It has Jacobian (which measures area) b for all points in the plane, so if jbj < 1 the Hénon mapping of the plane contracts area by a constant factor b for any point in the plane [41]. The effect of iterating this mapping is to stretch and fold its image in the manner of the Smale horseshoe mapping. The Hénon attractor is the prototype strange attractor. It arises in the case of a diffeomorphism of the plane which stretches and folds an open set U with ¯ U. the property that f (U) The Advent of Fractals The term was coined by B. B. Mandelbrot in the 1970s and now Mandelbrot is regarded as the “father of fractals” [55,56]. His first forays were to study fractals which were invariant under linear transformations. His uncle (Szolem Mandelbrojt, a professor of Mathematics and Mechanics at the Collège de France and later at the French Académie des Sciences) encouraged him to read the original papers by G. Julia and P. Fatou as a young man. Euclid’s Elements Fractals are revolutionary because they challenge one of the sacred texts of mathematics. Euclid’s Elements had reigned over mathematics for well over two thousand years and enjoys the distinction of still be referred to by working mathematicians. Up to the nineteenth century it was the essential fare of mathematics taught in schools. In Britain the Elements exerted the same authority as the Bible which was the other book committed to memory by generations of school pupils. Towards the end of the nineteenth century its central position in the curriculum was challenged, and it was much for the same reasons that Mandelbrot challenged it in the 1980s. Praised for its logical structure, learning Euclid’s deductive proofs by heart was simply not the way to teach an understanding of geometry. In France it had been displaced at the beginning of the century but in Britain and many other countries the attack on its centrality came at the end of the century. Ironically one of the main upholders of the sovereignty of Euclid and a staunch defender of the sacred book was Cayley. He was just the man who a few years before had shown himself to be more in the Mandelbrot mode, who threw aside “the restrictions as to reality” in his investigation of primitive symbolic dynamic systems. And there is a second irony, for Cayley was a man who loved nature and likened the beauty of the natural world to the terrain of mathematics itself – and in prac-
tical terms used his mountaineering exploits as a way of discovering new ideas in the subject. A century later Mandelbrot’s criticism came but from a different direction but containing enough phrases which would have gained a nod of agreement from those who mounted their opposition all those years before. Mandelbrot’s opening paragraph launched the attack: Why is geometry often described as “cold” and “dry”? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are nor spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line [57]. Mandelbrot was setting out an ambitious claim, and behind it was the way he used for conducting mathematical research. He advocated a different methodology from the Euclidean style updated by the Bourbakian mantra of “definition, theorem, proof”. In fact he describes himself as an exile, driven from France by the Bourbaki school. Jean Dieudonné, a leading Bourbakian was at the opposite end of the spectrum in his mathematical style. Dieudonné’s objective, as set out in a graduate text Foundations of Modern Analysis (1960) was to “train the student in the use of the most fundamental mathematical tool of our time – the axiomatic method with which he [sic] will have had very little contact, if any at all, during his undergraduate years”. Echoing the attitude of the nineteenth century geometer Jacob Steiner, who eschewed diagrams of any kind as inimical to the property development of the subject, Dieudonné appealed only to axiomatic methods, and to make the point in his book, he deliberately avoided introducing any diagrams and appeal to “geometric intuition”[28]. What Mandelbrot advocated was the methodology of the physician’s and lawyer’s “casebook.” He noted “this term has no counterpart in science, and I suggest we appropriate it”. This was rather novel and presents a counterpoint to the image that the Bourbakian ideal of great theorems with nicely turned out proofs are what is required above all else. If mathematics is presented as a completed house built by a great mathematician, as the Bourkadians could suggest, questions are only to be found in the higher reaches of the subjects, that demand years of study to reach them. In his autobiography, Mandelbrot claimed “in pure mathematics, my main contribution has not been to provide proofs, but to ask new questions – usually very hard ones – suggested by physics and pictures”. His questions spring from elementary objects in mathematics, but ones
Fractals Meet Chaos
which start off in the world. He summarizes his long career in his autobiography: My whole career became one long, ardent pursuit of the concept of roughness. The roughness of clusters in the physics of disorder, of turbulent flows, of exotic noises, of chaotic dynamical systems, of the distribution of galaxies, of coastlines, of stock-prize charts, and of mathematical constructions [58]. The “monster” examples which previously existed as instances of known theorems (and therefore were of little interest in themselves from the novelty point of view) or put aside because they were not examples, were now brought into the limelight and put on to the dissecting table to be investigated. What is a Fractal? In his essay (1975) Mandelbrot said “I stopped short of giving a mathematical definition, because I felt this notion – like a good wine – demanded a bit of ageing before being ‘bottled’.” [71]. He knew of Hausdorff dimension and developed an intuitive understanding for it, but postponed a definition. A little later he adopted a working definition, and in his Fractal Geometry of Nature (1982) he gave his manifesto for fractals. What is Mandelbrot’s subsequent working definition of a fractal? It is simply a point set for which the Hausdorff dimension exceeds its topological dimension, DH > DT . Examples are Cantor middle third set. where DH D log 2/ log 3 D 0:63 : : : with DT D 0, and the von Koch Curve where DH D log 4/ log 3 D 1:26 : : : with DT D 1. Mandelbrot’s definition gives rise to a programme of research similar in nature to one carried out by Menger and Urysohn in the 1920s. Just as they asked “what is a curve?” and “what is dimension?” it could now be asked “what is a fractal exactly?” and “what is fractal dimension?” The whole game of topological dimension was to be played over for metric dimension. Mandelbrot’s definition of a fractal fired a first shot in answering one question but, just as there were exceptions to C. Jordan’s definitions of curve like Peano’s famous space filling curve of the 1890s, exceptions were found to Mandelbrot’s preliminary definition. It was a regret, for example, that the example of the “devil’s staircase” (characterized in terms of a continuous weight function defined on the Cantor middle third set) which looked like a fractal did not conform to the working definition of a fractal since in this case DH D DT D 1. There are, however, many examples of obvious fractals, in nature and in mathematics. The pathological curves of the late nineteenth century which suffered this fate of
not conforming to definitions of “normal curves” were brought back to life. The pathologies, such as those invented by Brouwer and Cantor had been put in a cupboard and locked away. An instance of this is Brouwer’s decomposition of the plane into three “countries” so that the boundary points touch each other (not just meeting at one point). Brouwer’s decomposition is not unlike the shape of a fractal. Cantor’s examples were equally pathological. Mandelbrot made his own discoveries, but when he opened the cupboard of “monster curves” he liked what he saw. Otto Rössler, with justification called Mandelbrot “Cantor’s time-displaced younger friend”. As Freeman Dyson wrote: “Now, as Mandelbrot points out, . . . , Nature has played a joke on the mathematicians. The nineteenth century mathematicians may have been lacking in imagination [in limiting themselves to Euclid and Newton], but nature has not. The same pathological structures [the gallery of monster curves] that the mathematicians invented to break loose from nineteenth century naturalism turn out to be inherent in familiar objects all around us [38].” The Mandelbrot Set Just how did Mandelbrot arrive at the famous Mandelbrot set? With the freedom of being funded by an IBM fellowship, he went further into the study of p c (z) D z2 C c and experimented. The basement of the Thomas J. Watson Research Center in the early 1980s held a brand new VAX main frame computer and a Tektronix cathode ray tube for viewing results. The quadratic map p c (z) D z2 C c is the simplest of all the nonlinear rational transformations. The key element is to consider z; c as complex numbers. The Mandelbrot set M in the plane is defined as M D fc 2 C : the Julia set J c is connectedg or, alternatively, in terms of the sequence fp kc (0)g M D fc 2 C : fp kc (0)g is boundedg : It was an exciting event when Mandelbrot first glimpsed the set, which was initially called an M-set. First observed on the Tektronix cathode ray tube, was the messy outline a fault in the primitive equipment? He and his colleagues tried again with a more powerful computer but found “the mess had failed to vanish” and even that the image showed signs of being more systematic. Mandebrot reported “we promptly took a much closer look. Many specks of dirt duly vanished after we zoomed in. But some
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specks failed to vanish; in fact, they proved to resolve into complex structures endowed with ‘sprouts’ very similar to those of the whole set M. Peter Moldave and I could not contain our excitement.” Yet concrete mathematical properties were hard to come by. One immediate result did follow quickly, when the M-set, now termed the Mandelbrot set was proved to be a connected set [29].
As an example, the fractal dimension of the original Lorenz attractor derived from the meteorology differential equations was found to be approximately 2.05 [34].
The Merger
Newer Concepts of Dimension
When chaos theory was being discussed in the 1970s, strange attractors were regarded as weird subsets of Euclidean space. To be sure their importance was recognized but their nature was only vaguely understood. Towards the end of the 1970s, prompted by the advent of fractals, attractors could be viewed in a new light. But Chaos which arose in physical situations did not give rise to ordinary self-similar fractals, like the Koch curve, but to more complicated sets. The outcome of Chaos is apparent randomness while the key properties of ordinary fractals is regularity.
Just as for metric free topological dimension itself, there are a myriad of different concepts of metrically based dimension. What was wanted were measures which could be used in practice. What “dimensions” would shine light on the onset of chaotic solutions to a set of differential equations ? One was the Lyapunov dimension drawing on the work of the Russian mathematician Alexander Lyapunov, and investigated by J. L. Kaplan and J. A. Yorke (1979). The Lyapunov dimension of an attractor A embedded in an Euclidean space of dimension n is defined by:
Measuring Strange Attractors Once strange attractors were brought to light, the next stage was to measure them. This perspective was put by practitioners in the 1980s: Nonlinear physics presents us with a perplexing variety of complicated fractal objects and strange sets. Notable examples include strange attractors for chaotic dynamical systems. . . . Naturally one wishes to characterize the objects and described the events occurring on them. For example, in dynamical systems theory one is often interested in a strange attractor . . . [43]. At the same time other researchers put a similar point of view, and an explanation for making the calculations. The way to characterize strange attractors was by the natural measure of a fractal – its fractal dimension – and this became the way to proceed: The determination of the fractal dimension d of strange attractors has become a standard diagnostic tool in the analysis of dynamical systems. The dimension roughly speaking measures the number of degrees of freedom that are relevant to the dynamics. Most work on dimension has concerned maps, such as the Hénon map, or systems given by a few coupled ordinary differential equations, such as the Lorenz and Rössler models. For any chaotic system described by differential equations, d must be
greater than 2, but d could be much larger for a system described by partial differential equations, such as the Navier–Stokes equations [12].
DL D k C
1 C 2 C C k ; j kC1 j
where 1 ; 2 ; : : : ; k are Lyapunov exponents and k is P the maximum integer for which kiD1 i 0. In the case where k D 1 which occurs for two-dimensional chaotic maps, for instance, DL D 1
1 2
and because in this case 2 < 0 the value of the Lyapunov dimension will be sandwiched between 1 and the topological dimension which is 2. In the case of the Hénon mapping of the plane, the Lyapunov dimension of the attractor is approximately 1.25. In a different direction, a generalized dimension Dq was introduced by A. Rényi in the 1950s [78]. Working in probability and information theory, he defined a spectrum of dimensions: P q log Pi 1 i D q D lim r!0 q 1 log r depending on the value of q. The special cases are worth noting: D0 D DB the box dimension D1 D “information dimension“ related to C. Shannon’s entropy [39,40] D2 D DC the correlation dimension.
Fractals Meet Chaos
The correlation dimension is a very practical measure and can be applied to experimental data which has been presented visually, as well as other shapes, such as photographs where the calculation involves counting points. It has been used for applications in hydrodynamics, business, lasers, astronomy, signal analysis. It can be calculated directly for time series where the “phase space” points are derived from lags or time-delays. In two dimensions this would be a sequence of points (x(n); x(n C l)) where l is the chosen lag. A successful computation depends on the number of points chosen. For example, it is possible to estimate the fractal dimension of the Hénon attractor to within 6% of its supposed value with 500 data points [70]. There is a sequence of inequalities between the various dimensions of a set A from the embedding space of topological dimension D E D n to the topological dimension DT of A [90]: DT : : : D2 D1 DH D0 n : Historically these questions of inequalities between the various types of dimension are of the same type as had been asked about topological dimension fifty years before. Just as it was shown that the “coincidence theorem” ind X D Ind X D dim X holds for well behaved spaces, such as compact metric spaces, so it is true that DL D DB D D1 for ordinary point sets such as single points and limit cycles. For the Lorenz attractor DL agrees with the value of DB and that of D2 [69]. But each of the equalities fails for some fractal sets; the study of fractal dimension of attractors is covered in [34,83]. Multifractals An ordinary fractal is one where the generalized Rényi dimension Dq is independent of q and returns a single value for the whole fractal. This occurs for the standard fractals such as the von Koch curve. A multifractal is a set in which Dq is dependent on q and a continuous spectrum of values results. While an ordinary self-similar fractal are useful for expository purposes (and are beautifully symmetric shapes) the attractors found in physics are not uniformly self-similar. The resulting object is called a multifractal because it is multi-dimensional. The values of Dq is called its spectrum of the multifractal. Practical examples usually require their strange attractors to be modeled by multifractals.
Future Directions The range of written material on Fractals and Chaos has exploded since the 1970s. A vast number of expository articles have been lodged in such journals as Nature and New Scientist and technical articles in Physica D and Nonlinearity [9]. Complexity Theory and Chaos Theory are relatively new sciences that can revolutionize the way we see our world. Stephen Hawking has said, “Complexity will be the science of the 21st century.” There is even a relationship between fractals with the yet unproved Riemann hypothesis [51]. Many problems remain. Here it is sufficient to mention one representative of the genre. Fluid dynamicists are broadly in agreement that fluids are accurately modeled by the nonlinear Navier–Stokes equations. These are based on Newtonian principles and are deterministic, but theoretically the solution of these equations is largely unknown territory. A proof of the global regularity of the solutions represents a formidable challenge. The current view is that there is a dichotomy between laminar flow (like flow in the upper atmosphere) being smooth while turbulent flow (like flow near the earth’s surface) is violent. Yet even this “turbulent” flow could be regular but of a complicated kind. A substantial advance in the theory will be rewarded with one of the Clay Institute’s million dollar prizes. One expert is not optimistic the prize will be claimed in the near future [52]. The implication of Chaos dependent as it is on the sensitivity of initial conditions, suggests that forecasting some physical processes is theoretically impossible. Long range weather forecasting falls into this mould since it is predicated on knowing the weather conditions exactly at some point in time. There will inevitably be inaccuracies so exactness appears to be an impossibility. No doubt “Chaos and (multi)Fractals” is here to stay. Rössler wrote of Chaos as the key to understanding Nature: “hairs and noodles and spaghettis and dough and taffy form an irresistible, disentanglable mess. The world of causality is thereby caricatured and, paradoxically, faithfully represented [2]”. Meanwhile, the challenge for scientists and mathematicians remains [16].
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26. Devaney RL (2003) An Introduction to Chaotic Dynamical Systems, 2nd edn. Westview Press, Boulder 27. Diacu F, Holmes P (1996) Celestial Encounters: the Origins of Chaos and Stability. Princeton University Press, Princeton 28. Dieudonné J (1960) Foundations of Modern Analysis. Academic Press, New York 29. Douady A, Hubbard J (1982) Iteration des polynomes quadratiques complexes. Comptes Rendus Acad Sci Paris Sér 1 Math 294:123–126 30. Dyson F (1997) ‘Nature’s Numbers’ by Ian Stewart. Math Intell 19(2):65–67 31. Edgar G (1993) Classics on Fractals. Addison-Wesley, Reading 32. Falconer K (1990) Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester 33. Falconer K (1997) Techniques in Fractal Geometry. Wiley, Chichester 34. Farmer JD, Ott E, Yorke JA (1983) The Dimension of Chaotic Attractors. Physica D 7:153–180 35. Fatou P (1919/20) Sur les equations fonctionelles. Bull Soc Math France 47:161–271; 48:33–94, 208–314 36. Feigenbaum MJ (1978) Quantitative Universality for a Class of Nonlinear Transformations. J Stat Phys 19:25–52 37. Feigenbaum MJ (1979) The Universal Metric Properties of Nonlinear Transformations. J Stat Phys 21:669–706 38. Gleick J (1988) Chaos. Sphere Books, London 39. Grassberger P (1983) Generalised dimensions of strange attractors. Phys Lett A 97:227–230 40. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208 41. Hénon M (1976) A Two dimensional Mapping with a Strange Attractor. Commun Math Phys 50:69–77 42. Haeseler F, Peitgen H-O, Saupe D (1984) Cayley’s Problem and Julia Sets. Math Intell 6:11–20 43. Halsey TC, Jensen MH, Kadanoff LP, Procaccia Shraiman I (1986) Fractal measures and their singularities: the characterization of strange sets. Phys Rev A 33:1141–1151 44. Harman PM (1998) The Natural Philosophy of James Clerk Maxwell. Cambridge University Press, Cambridge 45. Hofstadter DR (1985) Metamathematical Themas: questing for the essence of mind and pattern. Penguin, London 46. Holmes P (2005) Ninety plus years of nonlinear dynamics: More is different and less is more. Int J Bifurc Chaos Appl Sci Eng 15(9):2703–2716 47. Hurewicz W, Wallman H (1941) Dimension Theory. Princeton University Press, Princeton 48. Julia G (1918) Sur l’iteration des functions rationnelles. J Math Pures Appl 8:47–245 49. Kazim Z (2002) The Hausdorff and box dimension of fractals with disjoint projections in two dimensions. Glasgow Math J 44:117–123 50. Lapidus ML, van Frankenhuijsen M (2004) Fractal geometry and Applications: A Jubilee of Benoît Mandelbrot, Parts, 1, 2. American Mathematical Society, Providence 51. Lapidus ML, van Frankenhuysen (2000) Fractal geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions. Birkhäuser, Boston 52. Li YC (2007) On the True Nature of Turbulence. Math Intell 29(1):45–48 53. Lorenz EN (1963) Deterministic Nonperiodic Flow. J Atmospheric Sci 20:130–141
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54. Lorenz EN (1993) The Essence of Chaos. University of Washington Press, Seattle 55. Mandelbrot BB (1975) Les objets fractals, forme, hazard et dimension. Flammarion, Paris 56. Mandelbrot BB (1980) Fractal aspects of the iteration of z ! (1 z) for complex ; z. Ann New York Acad Sci 357:249–259 57. Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco 58. Mandelbrot BB (2002) A maverick’s apprenticeship. Imperial College Press, London 59. Mandelbrot BB, Hudson RL (2004) The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. Basic Books, New York 60. Mattila P (1995) Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge 61. Mauldin RD, Williams SC (1986) On the Hausdorff dimension of some graphs. Trans Am Math Soc 298:793–803 62. May RM (1976) Simple Mathematical models with very complicated dynamics. Nature 261:459–467 63. May RM (1987) Chaos and the dynamics of biological populations. Proc Royal Soc Ser A 413(1844):27–44 64. May RM (2001) Stability and Complexity in Model Ecosystems, 2nd edn, with new introduction. Princeton University Press, Princeton 65. McMurran S, Tattersall J (1999) Mary Cartwright 1900–1998. Notices Am Math Soc 46(2):214–220 66. Menger K (1943) What is dimension? Am Math Mon 50:2–7 67. Metropolis, Stein ML, Stein P (1973) On finite limit sets for transformations on the unit interval. J Comb Theory 15:25–44 68. Morse M (1946) George David Birkhoff and his mathematical work. Bull Am Math Soc 52(5, Part 1):357–391 69. Nese JM, Dutton JA, Wells R (1987) Calculated Attractor Dimensions for low-Order Spectral Models. J Atmospheric Sci 44(15):1950–1972 70. Ott E, Sauer T, Yorke JA (1994) Coping with Chaos. Wiley Interscience, New York 71. Peitgen H-O, Richter PH (1986) The Beauty of Fractals. Springer, Berlin 72. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New York 73. Pesin YB (1997) Dimension Theory in Dynamical Systems: contemporary views and applications. University of Chicago Press, Chicago 74. Poincaré H (1903) L’Espace et ses trois dimensions. Rev Métaphys Morale 11:281–301 75. Poincaré H, Halsted GB (tr) (1946) Science and Method. In: Cattell JM (ed) Foundations of Science. The Science Press, Lancaster 76. Przytycki F, Urbañski M (1989) On Hausdorff dimension of some fractal sets. Studia Mathematica 93:155–186 77. Rössler OE (1976) An Equation for Continuous Chaos. Phys Lett A 57:397–398 78. Rényi A (1970) Probability Theory. North Holland, Amsterdam 79. Richardson LF (1993) Collected Papers, 2 vols. Cambridge University Press, Cambridge 80. Ruelle D (1980) Strange Attractors. Math Intell 2:126–137 81. Ruelle D (2006) What is a Strange Attractor? Notices Am Math Soc 53(7):764–765 82. Ruelle D, Takens F (1971) On the nature of turbulence. Commun Math Phys 20:167–192; 23:343–344
83. Russell DA, Hanson JD, Ott E (1980) Dimension of strange attractors. Phys Rev Lett 45:1175–1178 84. Shaw R (1984) The Dripping Faucet as a Model Chaotic System. Aerial Press, Santa Cruz 85. Siegel CL (1942) Iteration of Analytic Functions. Ann Math 43:607–612 86. Smale S (1967) Differentiable Dynamical Systems. Bull Am Math Soc 73:747–817 87. Smale S (1980) The Mathematics of Time: essays on dynamical systems, economic processes, and related topics. Springer, New York 88. Smale S (1998) Chaos: Finding a horseshoe on the Beaches of Rio. Math Intell 20:39–44 89. Sparrow C (1982) The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York 90. Sprott JC (2003) Chaos and Time-Series Analysis. Oxford University Press, Oxford 91. Takens F (1980) Detecting Strange Attractors in Turbulence. In: Rand DA, Young L-S (eds) Dynamical Systems and Turbulence. Springer Lecture Notes in Mathematics, vol 898. Springer, New York, pp 366–381 92. Theiler J (1990) Estimating fractal dimensions. J Opt Soc Am A 7(6):1055–1073 93. Tricot C (1982) Two definitions of fractional dimension. Math Proc Cambridge Philos Soc 91:57–74 94. Tucker W (1999) The Lorenz Attractor exists. Comptes Rendus Acad Sci Paris Sér 1, Mathematique 328:1197–1202 95. Viana M (2000) What’s New on Lorenz Strange Attractors. Math Intell 22(3):6–19 96. Winfree AT (1983) Sudden Cardiac death- a problem for topology. Sci Am 248:118–131 97. Zhang S-Y (compiler) (1991) Bibliography on Chaos. World Scientific, Singapore
Books and Reviews Abraham RH, Shaw CD (1992) Dynamics, The Geometry of Behavior. Addison-Wesley, Redwood City Barnsley MF, Devaney R, Mandelbrot BB, Peitgen H-O, Saupe D, Voss R (1988) The Science of Fractal Images. Springer, Berlin Barnsley MF, Rising H (1993) Fractals Everywhere. Academic Press, Boston Çambel AB (1993) Applied Complexity Theory: A Paradigm for Complexity. Academic Press, San Diego Crilly AJ, Earnshaw RA, Jones H (eds) (1991) Fractals and Chaos. Springer, Berlin Cvitanovic P (1989) Universality in Chaos, 2nd edn. Adam Hilger, Bristol Elliott EW, Kiel LD (eds) (1997) Chaos Theory in the Social Sciences: foundations and applications. University of Michigan Press, Ann Arbor Gilmore R, Lefranc M (2002) The Topology of Chaos: Alice in Stretch and Squeezeland. Wiley Interscience, New York Glass L, MacKey MM (1988) From Clocks to Chaos. Princeton University Press, Princeton Holden A (ed) (1986) Chaos. Manchester University Press, Manchester Kellert SH (1993) In the Wake of Chaos. University of Chicago Press, Chicago Lauwerier H (1991) Fractals: Endlessly Repeated Geometrical Figures. Princeton University Press, Princeton
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Mullin T (ed) (1994) The Nature of Chaos. Oxford University Press, Oxford Ott E (2002) Chaos in Dynamical Systems. Cambridge University Press, Cambridge Parker B (1996) Chaos in the Cosmos: The Stunning Complexity of the Universe. Plenum, New York, London Peitgen H-O, Jürgens, Saupe D, Zahlten C (1990) Fractals: An animated discussion with Edward Lorenz and Benoit Mandelbrot. A VHS film in color (63 mins). Freeman, New York
Prigogine I, Stengers I (1985) Order out of Chaos: man’s new dialogue with Nature. Fontana, London Ruelle D (1993) Chance and Chaos. Penguin, London Schroeder M (1991) Fractals, Chaos, Power Laws. Freeman, New York Smith P (1998) Explaining Chaos. Cambridge University Press, Cambridge Thompson JMT, Stewart HB (1988) Nonlinear Dynamics and Chaos. Wiley, New York
Hamiltonian Perturbation Theory (and Transition to Chaos)
Hamiltonian Perturbation Theory (and Transition to Chaos) HENK W. BROER1 , HEINZ HANSSMANN2 Instituut voor Wiskunde en Informatica, Rijksuniversiteit Groningen, Groningen, The Netherlands 2 Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands
1
Article Outline Glossary Definition of the Subject Introduction One Degree of Freedom Perturbations of Periodic Orbits Invariant Curves of Planar Diffeomorphisms KAM Theory: An Overview Splitting of Separatrices Transition to Chaos and Turbulence Future Directions Bibliography Glossary Bifurcation In parametrized dynamical systems a bifurcation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative change corresponds to transition between dynamical regimes. In the generic theory a finite list of cases is obtained, containing elements like ‘saddle-node’, ‘period doubling’, ‘Hopf bifurcation’ and many others. Cantor set, Cantor dust, Cantor family, Cantor stratification Cantor dust is a separable locally compact space that is perfect, i. e. every point is in the closure of its complement, and totally disconnected. This determines Cantor dust up to homeomorphisms. The term Cantor set (originally reserved for the specific form of Cantor dust obtained by repeatedly deleting the middle third from a closed interval) designates topological spaces that locally have the structure Rn Cantor dust for some n 2 N. Cantor families are parametrized by such Cantor sets. On the real line R one can define Cantor dust of positive measure by excluding around each rational number p/q an interval of size 2
; q
> 0; > 2:
Similar Diophantine conditions define Cantor sets in Rn . Since these Cantor sets have positive measure
their Hausdorff dimension is n. Where the unperturbed system is stratified according to the co-dimension of occurring (bifurcating) tori, this leads to a Cantor stratification. Chaos An evolution of a dynamical system is chaotic if its future is badly predictable from its past. Examples of non-chaotic evolutions are periodic or multi-periodic. A system is called chaotic when many of its evolutions are. One criterion for chaoticity is the fact that one of the Lyapunov exponents is positive. Diophantine condition, Diophantine frequency vector A frequency vector ! 2 Rn is called Diophantine if there are constants > 0 and > n 1 with jhk; !ij
jkj
for all k 2 Z n nf0g :
The Diophantine frequency vectors satisfying this condition for fixed and form a Cantor set of half lines. As the Diophantine parameter tends to zero (while remains fixed), these half lines extend to the origin. The complement in any compact set of frequency vectors satisfying a Diophantine condition with fixed has a measure of order O( ) as # 0. Integrable system A Hamiltonian system with n degrees of freedom is (Liouville)-integrable if it has n functionally independent commuting integrals of motion. Locally this implies the existence of a torus action, a feature that can be generalized to dissipative systems. In particular a mapping is integrable if it can be interpolated to become the stroboscopic mapping of a flow. KAM theory Kolmogorov–Arnold–Moser theory is the perturbation theory of (Diophantine) quasi-periodic tori for nearly integrable Hamiltonian systems. In the format of quasi-periodic stability, the unperturbed and perturbed system, restricted to a Diophantine Cantor set, are smoothly conjugated in the sense of Whitney. This theory extends to the world of reversible, volumepreserving or general dissipative systems. In the latter KAM theory gives rise to families of quasi-periodic attractors. KAM theory also applies to torus bundles, in which case a global Whitney smooth conjugation can be proven to exist, that keeps track of the geometry. In an appropriate sense invariants like monodromy and Chern classes thus also can be defined in the nearly integrable case. Also compare with Kolmogorov– Arnold–Moser (KAM) Theory. Nearly integrable system In the setting of perturbation theory, a nearly integrable system is a perturbation of an integrable one. The latter then is an integrable approximation of the former. See an above item.
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Normal form truncation Consider a dynamical system in the neighborhood of an equilibrium point, a fixed or periodic point, or a quasi-periodic torus, reducible to Floquet form. Then Taylor expansions (and their analogues) can be changed gradually into normal forms, that usually reflect the dynamics better. Often these display a (formal) torus symmetry, such that the normal form truncation becomes an integrable approximation, thus yielding a perturbation theory setting. See above items. Also compare with Normal Forms in Perturbation Theory. Persistent property In the setting of perturbation theory, a property is persistent whenever it is inherited from the unperturbed to the perturbed system. Often the perturbation is taken in an appropriate topology on the space of systems, like the Whitney Ck -topology [72]. Perturbation problem In perturbation theory the unperturbed systems usually are transparent regarding their dynamics. Examples are integrable systems or normal form truncations. In a perturbation problem things are arranged in such a way that the original system is wellapproximated by such an unperturbed one. This arrangement usually involves both changes of variables and scalings. Resonance If the frequencies of an invariant torus with multi- or conditionally periodic flow are rationally dependent, this torus divides into invariant sub-tori. Such resonances hh; !i D 0, h 2 Z k , define hyperplanes in !-space and, by means of the frequency mapping, also in phase space. The smallest number jhj D jh1 jC Cjh k j is the order of the resonance. Diophantine conditions describe a measure-theoretically large complement of a neighborhood of the (dense!) set of all resonances. Separatrices Consider a hyperbolic equilibrium, fixed or periodic point or invariant torus. If the stable and unstable manifolds of such hyperbolic elements are codimension one immersed manifolds, then they are called separatrices, since they separate domains of phase space, for instance, basins of attraction. Singularity theory A function H : Rn ! R has a critical point z 2 Rn where DH(z) vanishes. In local coordinates we may arrange z D 0 (and similarly that it is mapped to zero as well). Two germs K : (Rn ; 0) ! (R; 0) and N : (Rn ; 0) ! (R; 0) represent the same function H locally around z if and only if there is a diffeomorphism on Rn satisfying N D K ı: The corresponding equivalence class is called a singularity.
Structurally stable A system is structurally stable if it is topologically equivalent to all nearby systems, where ‘nearby’ is measured in an appropriate topology on the space of systems, like the Whitney Ck -topology [72]. A family is structurally stable if for every nearby family there is a re-parametrization such that all corresponding systems are topologically equivalent. Definition of the Subject The fundamental problem of mechanics is to study Hamiltonian systems that are small perturbations of integrable systems. Also, perturbations that destroy the Hamiltonian character are important, be it to study the effect of a small amount of friction, or to further the theory of dissipative systems themselves which surprisingly often revolves around certain well-chosen Hamiltonian systems. Furthermore there are approaches like KAM theory that historically were first applied to Hamiltonian systems. Typically perturbation theory explains only part of the dynamics, and in the resulting gaps the orderly unperturbed motion is replaced by random or chaotic motion. Introduction We outline perturbation theory from a general point of view, illustrated by a few examples. The Perturbation Problem The aim of perturbation theory is to approximate a given dynamical system by a more familiar one, regarding the former as a perturbation of the latter. The problem then is to deduce certain dynamical properties from the unperturbed to the perturbed case. What is familiar may or may not be a matter of taste, at least it depends a lot on the dynamical properties of one’s interest. Still the most frequently used unperturbed systems are: Linear systems Integrable Hamiltonian systems Normal form truncations, compare with Normal Forms in Perturbation Theory and references therein Etc. To some extent the second category can be seen as a special case of the third. To avoid technicalities in this section we assume all systems to be sufficiently smooth, say of class C 1 or real analytic. Moreover in our considerations " will be a real parameter. The unperturbed case always corresponds to " D 0 and the perturbed one to " ¤ 0 or " > 0.
Hamiltonian Perturbation Theory (and Transition to Chaos)
Examples of Perturbation Problems To begin with consider the autonomous differential equation x¨ C "x˙ C
dV (x) D 0 ; dx
modeling an oscillator with small damping. Rewriting this equation of motion as a planar vector field x˙ D
y
y˙ D "y
dV (x) ; dx
we consider the energy H(x; y) D 12 y 2 C V(x). For " D 0 the system is Hamiltonian with Hamiltonian function H. ˙ Indeed, generally we have H(x; y) D "y 2 , implying that for " > 0 there is dissipation of energy. Evidently for " ¤ 0 the system is no longer Hamiltonian. The reader is invited to compare the phase portraits of the cases " D 0 and " > 0 for V (x) D cos x (the 1 pendulum) or V (x) D 12 x 2 C 24 bx 4 (Duffing). Another type of example is provided by the non-autonomous equation dV x¨ C (x) D " f (x; x˙ ; t) ; dx which can be regarded as the equation of motion of an oscillator with small external forcing. Again rewriting as a vector field, we obtain ˙t D
1
x˙ D
y
y˙ D
dV (x) C " f (x; y; t) ; dx
now on the generalized phase space R3 D ft; x; yg. In the case where the t-dependence is periodic, we can take S1 R2 for (generalized) phase space. Remark A small variation of the above driven system concerns a parametrically forced oscillator like x¨ C (! 2 C " cos t) sin x D 0 ; which happens to be entirely in the world of Hamiltonian systems. It may be useful to study the Poincaré or period mapping of such time periodic systems, which happens to be a mapping of the plane. We recall that in the Hamiltonian cases this mapping preserves area. For general reference in this direction see, e. g., [6,7,27,66].
There are lots of variations and generalizations. One example is the solar system, where the unperturbed case consists of a number of uncoupled two-body problems concerning the Sun and each of the planets, and where the interaction between the planets is considered as small [6,9, 107,108]. Remark One variation is a restriction to fewer bodies, for example only three. Examples of this are systems like Sun–Jupiter–Saturn, Earth–Moon–Sun or Earth– Moon–Satellite. Often Sun, Moon and planets are considered as point masses, in which case the dynamics usually are modeled as a Hamiltonian system. It is also possible to extend this approach taking tidal effects into account, which have a non-conservative nature. The Solar System is close to resonance, which makes application of KAM theory problematic. There exist, however, other integrable approximations that take resonance into account [3,63]. Quite another perturbation setting is local, e. g., near an equilibrium point. To fix thoughts consider x˙ D Ax C f (x) ;
x 2 Rn
with A 2 gl(n; R), f (0) D 0 and Dx f (0) D 0. By the scaling x D "x¯ we rewrite the system to x˙¯ D A¯x C "g(x¯ ) : So, here we take the linear part as an unperturbed system. Observe that for small " the perturbation is small on a compact neighborhood of x¯ D 0. This setting also has many variations. In fact, any normal form approximation may be treated in this way Normal Forms in Perturbation Theory. Then the normalized truncation forms the unperturbed part and the higher order terms the perturbation. Remark In the above we took the classical viewpoint which involves a perturbation parameter controlling the size of the perturbation. Often one can generalize this by considering a suitable topology (like the Whitney topologies) on the corresponding class of systems [72]. Also compare with Normal Forms in Perturbation Theory and Kolmogorov–Arnold–Moser (KAM) Theory. Questions of Persistence What are the kind of questions perturbation theory asks? A large class of questions concerns the persistence of cer-
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tain dynamical properties as known for the unperturbed case. To fix thoughts we give a few examples. To begin with consider equilibria and periodic orbits. So we put x˙ D f (x; ") ;
x 2 Rn ; " 2 R ;
(1)
for a map f : RnC1 ! Rn . Recall that equilibria are given by the equation f (x; ") D 0. The following theorem that continues equilibria in the unperturbed system for " ¤ 0, is a direct consequence of the implicit function theorem. Theorem 1 (Persistence of equilibria) f (x0 ; 0) D 0 and that
Suppose that
Dx f (x0 ; 0) has maximal rank : Then there exists a local arc " 7! x(") with x(0) D x0 such that f (x("); ") 0 : Periodic orbits can be approximated in a similar way. Indeed, let the system (1) for D 0 have a periodic orbit 0 . Let ˙ be a local transversal section of 0 and P0 : ˙ ! ˙ the corresponding Poincaré map. Then P0 has a fixed point x0 2 ˙ \ 0 . By transversality, for j"j small, a local Poincaré map P" : ˙ ! ˙ is well-defined for (1). Observe that fixed points x" of P" correspond to periodic orbits " of (1). We now have, again as another direct consequence of the implicit function theorem. Theorem 2 (Persistence of periodic orbits) In the above assume that P0 (x0 ) D x0 and Dx P0 (x0 ) has no eigenvalue 1 : Then there exists a local arc " 7! x(") with x(0) D x0 such that P" (x(")) x" : Remark Often the conditions of Theorem 2 are not easy to verify. Sometimes it is useful here to use Floquet Theory, see [97]. In fact, if T 0 is the period of 0 and ˝ 0 its Floquet matrix, then Dx P0 (x0 ) D exp(T0 ˝0 ). The format of the Theorems 1 and 2 with the perturbation parameter " directly allows for algorithmic approaches. One way to proceed is by perturbation series, leading to asymptotic formulae that in the real analytic setting have positive radius of convergence. In the latter case the names of Poincaré and Lindstedt are associated with the method, cf. [10]. Also numerical continuation programmes exist based on the Newton method.
The Theorems 1 and 2 can be seen as special cases of a a general theorem for normally hyperbolic invariant manifolds [73], Theorem 4.1. In all cases a contraction principle on a suitable Banach space of graphs leads to persistence of the invariant dynamical object. This method in particular yields existence and persistence of stable and unstable manifolds [53,54]. Another type of dynamics subject to perturbation theory is quasi-periodic. We emphasize that persistence of (Diophantine) quasi-periodic invariant tori occurs both in the conservative setting and in many others, like in the reversible and the general (dissipative) setting. In the latter case this leads to persistent occurrence of families of quasiperiodic attractors [125]. These results are in the domain of Kolmogorov–Arnold–Moser (KAM) theory. For details we refer to Sect. “KAM Theory: An Overview” below or to [24], Kolmogorov–Arnold–Moser (KAM) Theory, the former reference containing more than 400 references in this area. Remark Concerning the Solar System, KAM theory always has aimed at proving that it contains many quasi-periodic motions, in the sense of positive Liouville measure. This would imply that there is positive probability that a given initial condition lies on such a stable quasi-periodic motion [3,63], however, also see [85]. Another type of result in this direction compares the distance of certain individual solutions of the perturbed and the unperturbed system, with coinciding initial conditions over time scales that are long in terms of ". Compare with [24]. Apart from persistence properties related to invariant manifolds or individual solutions, the aim can also be to obtain a more global persistence result. As an example of this we mention the Hartman–Grobman Theorem, e. g., [7,116,123]. Here the setting once more is x˙ D Ax C f (x) ;
x 2 Rn ;
with A 2 gl(n; R), f (0) D 0 and Dx f (0) D 0. Now we assume A to be hyperbolic (i. e., with no purely imaginary eigenvalues). In that case the full system, near the origin, is topologically conjugated to the linear system x˙ D Ax. Therefore all global, qualitative properties of the unperturbed (linear) system are persistent under perturbation to the full system. For details on these notions see the above references, also compare with, e. g., [30].
Hamiltonian Perturbation Theory (and Transition to Chaos)
It is said that the hyperbolic linear system x˙ D Ax is (locally) structurally stable. This kind of thinking was introduced to the dynamical systems area by Thom [133], with a first, successful application to catastrophe theory. For further details, see [7,30,69,116]. General Dynamics We give a few remarks on the general dynamics in a neighborhood of Hamiltonian KAM tori. In particular this concerns so-called superexponential stickiness of the KAM tori and adiabatic stability of the action variables, involving the so-called Nekhoroshev estimate. To begin with, emphasize the following difference between the cases n D 2 and n 3 in the classical KAM theorem of Subsect. “Classical KAM Theory”. For n D 2 the level surfaces of the Hamiltonian are three-dimensional, while the Lagrangian tori have dimension two and hence codimension one in the energy hypersurfaces. This means that for open sets of initial conditions, the evolution curves are forever trapped in between KAM tori, as these tori foliate over nowhere dense sets of positive measure. This implies perpetual adiabatic stability of the action variables. In contrast, for n 3 the Lagrangian tori have codimension n1 > 1 in the energy hypersurfaces and evolution curves may escape. This actually occurs in the case of so-called Arnold diffusion. The literature on this subject is immense, and we here just quote [5,9,93,109], for many more references see [24]. Next we consider the motion in a neighborhood of the KAM tori, in the case where the systems are real analytic or at least Gevrey smooth. For a definition of Gevrey regularity see [136]. First we mention that, measured in terms of the distance to the KAM torus, nearby evolution curves generically stay nearby over a superexponentially long time [102,103]. This property often is referred to as superexponential stickiness of the KAM tori, see [24] for more references. Second, nearly integrable Hamiltonian systems, in terms of the perturbation size, generically exhibit exponentially long adiabatic stability of the action variables, see e. g. [15,88,89,90,93,103,109,110,113,120], Nekhoroshev Theory and many others, for more references see [24]. This property is referred to as the Nekhoroshev estimate or the Nekhoroshev theorem. For related work on perturbations of so-called superintegrable systems, also see [24] and references therein. Chaos In the previous subsection we discussed persistent and some non-persistent features of dynamical systems un-
der small perturbations. Here we discuss properties related to splitting of separatrices, caused by generic perturbations. A first example was met earlier, when comparing the pendulum with and without (small) damping. The unperturbed system is the undamped one and this is a Hamiltonian system. The perturbation however no longer is Hamiltonian. We see that the equilibria are persistent, as they should be according to Theorem 1, but that none of the periodic orbits survives the perturbation. Such qualitative changes go with perturbing away from the Hamiltonian setting. Similar examples concern the breaking of a certain symmetry by the perturbation. The latter often occurs in the case of normal form approximations. Then the normalized truncation is viewed as the unperturbed system, which is perturbed by the higher order terms. The truncation often displays a reasonable amount of symmetry (e. g., toroidal symmetry), which generically is forbidden for the class of systems under consideration, e. g. see [25]. To fix thoughts we reconsider the conservative example x¨ C (! 2 C " cos t) sin x D 0 of the previous section. The corresponding (time dependent, Hamiltonian [6]) vector field reads ˙t D 1 x˙ D
y
y˙ D (! 2 C " cos t) sin x : Let P!;" : R2 ! R2 be the corresponding (area-preserving) Poincaré map. Let us consider the unperturbed map P!;0 which is just the flow over time 2 of the free pendulum x¨ C! 2 sin x D 0. Such a map is called integrable, since it is the stroboscopic map of a two-dimensional vector field, hence displaying the R-symmetry of a flow. When perturbed to the nearly integrable case " ¤ 0, this symmetry generically is broken. We list a few of the generic properties for such maps [123]: The homoclinic and heteroclinic points occur at transversal intersections of the corresponding stable and unstable manifolds. The periodic points of period less than a given bound are isolated. This means generically that the separatrices split and that the resonant invariant circles filled with periodic points with the same (rational) rotation number fall apart. In any concrete example the issue remains whether or not
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Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 1 Chaos in the parametrically forced pendulum. Left: Poincaré map P!;" near the 1 : 2 resonance ! D Right: A dissipative analogue
it satisfies appropriate genericity conditions. One method to check this is due to Melnikov, compare [66,137], for more sophisticated tools see [65]. Often this leads to elliptic (Abelian) integrals. In nearly integrable systems chaos can occur. This fact is at the heart of the celebrated non-integrability of the three-body problem as addressed by Poincaré [12,59, 107,108,118]. A long standing open conjecture is that the clouds of points as visible in Fig. 1, left, densely fill sets of positive area, thereby leading to ergodicity [9]. In the case of dissipation, see Fig. 1, right, we conjecture the occurrence of a Hénon-like strange attractor [14,22,126]. Remark The persistent occurrence of periodic points of a given rotation number follows from the Poincaré–Birkhoff fixed point theorem [74,96,107], i. e., on topological grounds. The above arguments are not restricted to the conservative setting, although quite a number of unperturbed systems come from this world. Again see Fig. 1.
1 2
and for " > 0 not too small.
ishing eigenvalues may be unavoidable and it becomes possible that the corresponding dynamics persist under perturbations. Perturbations may also destroy the Hamiltonian character of the flow. This happens especially where the starting point is a dissipative planar system and e. g. a scaling leads for " D 0 to a limiting Hamiltonian flow. The perturbation problem then becomes twofold. Equilibria still persist by Theorem 1 and hyperbolic equilibria moreover persist as such, with the sum of eigenvalues of order O("). Also for elliptic eigenvalues the sum of eigenvalues is of order O(") after the perturbation, but here this number measures the dissipation whence the equilibrium becomes (weakly) attractive for negative values and (weakly) unstable for positive values. The one-parameter families of periodic orbits of a Hamiltonian system do not persist under dissipative perturbations, the very fact that they form families imposes the corresponding fixed point of the Poincaré mapping to have an eigenvalue one and Theorem 2 does not apply. Typically only finitely many periodic orbits survive a dissipative perturbation and it is already a difficult task to determine their number.
One Degree of Freedom Planar Hamiltonian systems are always integrable and the orbits are given by the level sets of the Hamiltonian function. This still leaves room for a perturbation theory. The recurrent dynamics consists of periodic orbits, equilibria and asymptotic trajectories forming the (un)stable manifolds of unstable equilibria. The equilibria organize the phase portrait, and generically all equilibria are elliptic (purely imaginary eigenvalues) or hyperbolic (real eigenvalues), i. e. there is no equilibrium with a vanishing eigenvalue. If the system depends on a parameter such van-
Hamiltonian Perturbations The Duffing oscillator has the Hamiltonian function H(x; y) D
1 2 1 1 y C bx 4 C x 2 2 24 2
(2)
where b is a constant distinguishing the two cases b D ˙1 and is a parameter. Under variation of the parameter the equations of motion x˙ D y
Hamiltonian Perturbation Theory (and Transition to Chaos)
1 y˙ D bx 3 x 6
Dissipative Perturbations
display a Hamiltonian pitchfork bifurcation, supercritical for positive b and subcritical in case b is negative. Correspondingly, the linearization at the equilibrium x D 0 of the anharmonic oscillator D 0 is given by the matrix 0 1 0 0 whence this equilibrium is parabolic. The typical way in which a parabolic equilibrium bifurcates is the center-saddle bifurcation. Here the Hamiltonian reads H(x; y) D
1 2 1 3 ay C bx C cx 2 6
(3)
where a; b; c 2 R are nonzero constants, for instance a D b D c D 1. Note that this is a completely different unfolding of the parabolic equilibrium at the origin. A closer look at the phase portraits and in particular at the Hamiltonian function of the Hamiltonian pitchfork bifurcation reveals the symmetry x 7! x of the Duffing oscillator. This suggests the addition of the non-symmetric term x. The resulting two-parameter family H; (x; y) D
1 2 1 1 y C bx 4 C x 2 C x 2 24 2
of Hamiltonian systems is indeed structurally stable. This implies not only that all equilibria of a Hamiltonian perturbation of the Duffing oscillator have a local flow equivalent to the local flow near a suitable equilibrium in this two-parameter family, but that every one-parameter family of Z2 -symmetric Hamiltonian systems that is a perturbation of (2) has equivalent dynamics. For more details see [36] and references therein. This approach applies mutatis mutandis to every nondegenerate planar singularity, cf. [69,130]. At an equilibrium all partial derivatives of the Hamiltonian vanish and the resulting singularity is called non-degenerate if it has finite multiplicity, which implies that it admits a versal unfolding H with finitely many parameters. The family of Hamiltonian systems defined by this versal unfolding contains all possible (local) dynamics that the initial equilibrium may be perturbed to. Imposing additional discrete symmetries is immediate, the necessary symmetric versal unfolding is obtained by averaging HG D
1 X H ı g jGj g2G
along the orbits of the symmetry group G.
In a generic dissipative system all equilibria are hyperbolic. Qualitatively, i. e. up to topological equivalence, the local dynamics is completely determined by the number of eigenvalues with positive real part. Those hyperbolic equilibria that can appear in Hamiltonian systems (the eigenvalues forming pairs ˙) do not play an important role. Rather, planar Hamiltonian systems become important as a tool to understand certain bifurcations triggered off by non-hyperbolic equilibria. Again this requires the system to depend on external parameters. The simplest example is the Hopf bifurcation, a co-dimension one bifurcation where an equilibrium loses stability as the pair of eigenvalues crosses the imaginary axis, say at ˙i. At the bifurcation the linearization is a Hamiltonian system with an elliptic equilibrium (the co-dimension one bifurcations where a single eigenvalue crosses the imaginary axis through 0 do not have a Hamiltonian linearization). This limiting Hamiltonian system has a oneparameter family of periodic orbits around the equilibrium, and the non-linear terms determine the fate of these periodic orbits. The normal form of order three reads x˙ D y 1 C b(x 2 C y 2 ) C x C a(x 2 C y 2 ) x˙ D x 1 C b(x 2 C y 2 ) C y C a(x 2 C y 2 ) and is Hamiltonian if and only if (; a) D (0; 0). The sign of the coefficient distinguishes between the supercritical case a > 0, in which there are no periodic orbits coexisting with the attractive equilibria (i. e. when < 0) and one attracting periodic orbit for each > 0 (coexisting with the unstable equilibrium), and the subcritical case a < 0, in which the family of periodic orbits is unstable and coexists with the attractive equilibria (with no periodic orbits for parameters > 0). As ! 0 the family of periodic orbits shrinks down to the origin, so also this Hamiltonian feature is preserved. Equilibria with a double eigenvalue 0 need two parameters to persistently occur in families of dissipative systems. The generic case is the Takens–Bogdanov bifurcation. Here the linear part is too degenerate to be helpful, but the nonlinear Hamiltonian system defined by (3) with a D 1 D c and b D 3 provides the periodic and heteroclinic orbit(s) that constitute the nontrivial part of the bifurcation diagram. Where discrete symmetries are present, e. g. for equilibria in dissipative systems originating from other generic bifurcations, the limiting Hamiltonian system exhibits that same discrete symmetry. For more details see [54,66,82] and references therein. The continuation of certain periodic orbits from an unperturbed Hamiltonian system under dissipative per-
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turbation can be based on Melnikov-like methods, again see [66,137]. As above, this often leads to Abelian integrals, for instance to count the number of periodic orbits that branch off. Reversible Perturbations A dynamical system that admits a reflection symmetry R mapping trajectories '(t; z0 ) to trajectories '(t; R(z0 )) is called reversible. In the planar case we may restrict to the reversing reflection R:
R2 (x; y)
! 7!
R2 (x; y) :
(4)
All Hamiltonian functions H D 12 y 2 CV (x) which have an interpretation “kinetic C potential energy” are reversible, and in general the class of reversible systems is positioned between the class of Hamiltonian systems and the class of dissipative systems. A guiding example is the perturbed Duffing oscillator (with the roles of x and y exchanged so that (4) remains the reversing symmetry) 1 x˙ D y 3 y C "x y 6 y˙ D x that combines the Hamiltonian character of the equilibrium at the origin with the dissipative character of the two other equilibria. Note that all orbits outside the homoclinic loop are periodic. There are two ways in which the reversing symmetry (4) imposes a Hamiltonian character on the dynamics. An equilibrium that lies on the symmetry line fy D 0g has a linearization that is itself a reversible system and consequently the eigenvalues are subject to the same constraints as in the Hamiltonian case. (For equilibria z0 that do not lie on the symmetry line the reflection R(z0 ) is also an equilibrium, and it is to the union of their eigenvalues that these constraints still apply.) Furthermore, every orbit that crosses fy D 0g more than once is automatically periodic, and these periodic orbits form one-parameter families. In particular, elliptic equilibria are still surrounded by periodic orbits. The dissipative character of a reversible system is most obvious for orbits that do not cross the symmetry line. Here R merely maps the orbit to a reflected counterpart. The above perturbed Duffing oscillator exemplifies that the character of an orbit crossing fy D 0g exactly once is undetermined. While the homoclinic orbit of the saddle at the origin has a Hamiltonian character, the heteroclinic orbits between the other two equilibria behave like in a dissipative system.
Perturbations of Periodic Orbits The perturbation of a one-degree-of-freedom system by a periodic forcing is a perturbation that changes the phase space. Treating the time variable t as a phase space variable leads to the extended phase space S1 R2 and equilibria of the unperturbed system become periodic orbits, inheriting the normal behavior. Furthermore introducing an action conjugate to the “angle” t yields a Hamiltonian system in two degrees of freedom. While the one-parameter families of periodic orbits merely provide the typical recurrent motion in one degree of freedom, they form special solutions in two or more degrees of freedom. Arcs of elliptic periodic orbits are particularly instructive. Note that these occur generically in both the Hamiltonian and the reversible context. Conservative Perturbations Along the family of elliptic periodic orbits a pair e˙i˝ of Floquet multipliers passes regularly through roots of unity. Generically this happens on a dense set of parameter values, but for fixed denominator q in e˙i˝ D e˙2ip/q the corresponding energy values are isolated. The most important of such resonances are those with small denominators q. For q D 1 generically a periodic center-saddle bifurcation takes place where an elliptic and a hyperbolic periodic orbit meet at a parabolic periodic orbit. No periodic orbit remains under further variation of a suitable parameter. The generic bifurcation for q D 2 is the period-doubling bifurcation where an elliptic periodic orbit turns hyperbolic (or vice versa) when passing through a parabolic periodic orbit with Floquet multipliers 1. Furthermore, a family of periodic orbits with twice the period emerges from the parabolic periodic orbit, inheriting the normal linear behavior from the initial periodic orbit. In case q D 3, and possibly also for q D 4, generically two arcs of hyperbolic periodic orbits emerge, both with three (resp. four) times the period. One of these extends for lower and the other for higher parameter values. The initial elliptic periodic orbit momentarily loses its stability due to these approaching unstable orbits. Denominators q 5 (and also the second possibility for q D 4) lead to a pair of subharmonic periodic orbits of q times the period emerging either for lower or for higher parameter values. This is (especially for large q) comparable to the behavior at Diophantine e˙i˝ where a family of invariant tori emerges, cf. Sect. “Invariant Curves of Planar Diffeomorphisms” below. For a single pair e˙i˝ of Floquet multipliers this behavior is traditionally studied for the (iso-energetic)
Hamiltonian Perturbation Theory (and Transition to Chaos)
Poincaré-mapping, cf. [92] and references therein. However, the above description remains true in higher dimensions, where additionally multiple pairs of Floquet multipliers may interact. An instructive example is the Lagrange top, the sleeping motion of which is gyroscopically stabilized after a periodic Hamiltonian Hopf bifurcation; see [56] for more details. Dissipative Perturbations There exists a large class of local bifurcations in the dissipative setting, that can be arranged in a perturbation theory setting, where the unperturbed system is Hamiltonian. The arrangement consists of changes of variables and rescaling. An early example of this is the Bogdanov–Takens bifurcation [131,132]. For other examples regarding nilpotent singularities, see [23,40] and references therein. To fix thoughts, consider families of planar maps and let the unperturbed Hamiltonian part contain a center (possibly surrounded by a homoclinic loop). The question then is which of these persist when adding the dissipative perturbation. Usually only a definite finite number persists. As in Subsect. “Chaos”, a Melnikov function can be invoked here, possibly again leading to elliptic (Abelian) integrals, Picard Fuchs equations, etc. For details see [61,124] and references therein. Invariant Curves of Planar Diffeomorphisms This section starts with general considerations on circle diffeomorphisms, in particular focusing on persistence properties of quasi-periodic dynamics. Our main references are [2,24,29,31,70,71,139,140]. For a definition of rotation number, see [58]. After this we turn to area preserving maps of an annulus where we discuss Moser’s twist map theorem [104], also see [24,29,31]. The section is concluded by a description of the holomorphic linearization of a fixed point in a planar map [7,101,141,142]. Our main perspective will be perturbative, where we consider circle maps near a rigid rotation. It turns out that generally parameters are needed for persistence of quasiperiodicity under perturbations. In the area preserving setting we consider perturbations of a pure twist map.
view this two-parameter family as a one-parameter family of maps P" : T 1 [0; 1] ! T 1 [0; 1]; (x; ˛) 7! (x C 2 ˛ C "a(x; ˛; "); ˛) of the cylinder. Note that the unperturbed system P0 is a family of rigid circle rotations, viewed as a cylinder map, where the individual map P˛;0 has rotation number ˛. The question now is what will be the fate of this rigid dynamics for 0 ¤ j"j 1. The classical way to address this question is to look for a conjugation ˚" , that makes the following diagram commute T 1 [0; 1] " ˚"
!
P"
T 1 [0; 1] " ˚"
T 1 [0; 1]
!
P0
T 1 [0; 1] ;
i. e., such that P" ı ˚" D ˚" ı P0 : Due to the format of P" we take ˚" as a skew map ˚" (x; ˛) D (x C "U(x; ˛; "); ˛ C " (˛; ")) ; which leads to the nonlinear equation U(x C 2 ˛; ˛; ") U(x; ˛; ") D 2 (˛; ") C a (x C "U(x; ˛; "); ˛ C " (˛; "); ") in the unknown maps U and . Expanding in powers of " and comparing at lowest order yields the linear equation U0 (x C 2 ˛; ˛) U0 (x; ˛) D 20 (˛) C a0 (x; ˛) which can be directly solved by Fourier series. Indeed, writing X a0k (˛)eikx ; a0 (x; ˛) D k2Z
U0 (x; ˛) D
P˛;" : T ! T ; 1
1
x 7! x C 2 ˛ C "a(x; ˛; ")
of circle maps of class C 1 . It turns out to be convenient to
U0k (˛)eikx
k2Z
we find 0 D 1/(2)a00 and
Circle Maps We start with the following general problem. Given a twoparameter family
X
U0k (˛) D
a0k (˛) 2ik˛ e 1
:
It follows that in general a formal solution exists if and only if ˛ 2 R n Q. Still, the accumulation of e2ik˛ 1 on 0 leads to the celebrated small divisors [9,108], also see [24,29,31,55].
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The classical solution considers the following Diophantine non-resonance conditions. Fixing > 2 and
> 0 consider ˛ 2 [0; 1] such that for all rationals p/q ˇ ˇ ˇ ˇ ˇ˛ p ˇ q : (5) ˇ qˇ This subset of such ˛s is denoted by [0; 1]; and is wellknown to be nowhere dense but of large measure as > 0 gets small [115]. Note that Diophantine numbers are irrational. Theorem 3 (Circle Map Theorem) For sufficiently small and for the perturbation "a sufficiently small in the C 1 -topology, there exists a C 1 transformation ˚" : T 1 [0; 1] ! T 1 [0; 1], conjugating the restriction P0 j[0;1]; to a subsystem of P" . Theorem 3 in the present structural stability formulation (compare with Fig. 2) is a special case of the results in [29,31]. We here speak of quasi-periodic stability. For earlier versions see [2,9]. Remark Rotation numbers are preserved by the map ˚" and irrational rotation numbers correspond to quasi-periodicity. Theorem 3 thus ensures that typically quasi-periodicity occurs with positive measure in the parameter space. Note that since Cantor sets are perfect, quasi-periodicity typically has a non-isolated occurrence.
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 2 Skew cylinder map, conjugating (Diophantine) quasi-periodic invariant circles of P0 and P"
The map ˚" has no dynamical meaning inside the gaps. The gap dynamics in the case of circle maps can be illustrated by the Arnold family of circle maps [2,7,58], given by P˛;" (x) D x C 2 ˛ C " sin x which exhibits a countable union of open resonance tongues where the dynamics is periodic, see Fig. 3. Note that this map only is a diffeomorphism for j"j < 1. We like to mention that non-perturbative versions of Theorem 3 have been proven in [70,71,139]. For simplicity we formulated Theorem 3 under C 1 -regularity, noting that there exist many ways to generalize this. On the one hand there exist Ck -versions for finite k and on the other hand there exist fine tunings in terms of real-analytic and Gevrey regularity. For details we refer to [24,31] and references therein. This same remark applies to other results in this section and in Sect. “KAM Theory: An Overview” on KAM theory. A possible application of Theorem 3 runs as follows. Consider a system of weakly coupled Van der Pol oscillators y¨1 C c1 y˙1 C a1 y1 C f1 (y1 ; y˙1 ) D "g1 (y1 ; y2 ; y˙1 ; y˙2 ) y¨2 C c2 y˙2 C a2 y2 C f2 (y2 ; y˙2 ) D "g2 (y1 ; y2 ; y˙1 ; y˙2 ) : Writing y˙ j D z j ; j D 1; 2, one obtains a vector field in the four-dimensional phase space R2 R2 D f(y1 ; z1 ); (y2 ; z2 )g. For " D 0 this vector field has an invariant two-torus, which is the product of the periodic motions of the individual Van der Pol oscillations. This two-torus is normally hyperbolic and therefore persistent for j"j 1 [73]. In
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 3 Arnold resonance tongues; for " 1 the maps are endomorphic
Hamiltonian Perturbation Theory (and Transition to Chaos)
fact the torus is an attractor and we can define a Poincaré return map within this torus attractor. If we include some of the coefficients of the equations as parameters, Theorem 3 is directly applicable. The above statements on quasi-periodic circle maps then directly translate to the case of quasi-periodic invariant two-tori. Concerning the resonant cases, generically a tongue structure like in Fig. 3 occurs; for the dynamics corresponding to parameter values inside such a tongue one speaks of phase lock. Remark The celebrated synchronization of Huygens’ clocks [77] is related to a 1 : 1 resonance, meaning that the corresponding Poincaré map would have its parameters in the main tongue with rotation number 0. Compare with Fig. 3. There exist direct generalizations to cases with n-oscillators (n 2 N), leading to families of invariant n-tori carrying quasi-periodic flow, forming a nowhere dense set of positive measure. An alteration with resonance occurs as roughly sketched in Fig. 3. In higher dimension the gap dynamics, apart from periodicity, also can contain strange attractors [112,126]. We shall come back to this subject in a later section. Area-Preserving Maps The above setting historically was preceded by an area preserving analogue [104] that has its origin in the Hamiltonian dynamics of frictionless mechanics. Let R2 n f(0; 0)g be an annulus, with sympectic polar coordinates ('; I) 2 T 1 K, where K is an interval. Moreover, let D d' ^ dI be the area form on . We consider a -preserving smooth map P" : ! of the form P" ('; I) D (' C 2 ˛(I); I) C O(") ; where we assume that the map I 7! ˛(I) is a (local) diffeomorphism. This assumption is known as the twist condition and P" is called a twist map. For the unperturbed case " D 0 we are dealing with a pure twist map and its dynamics are comparable to the unperturbed family of cylinder maps as met in Subsect. “Circle Maps”. Indeed it is again a family of rigid rotations, parametrized by I and where P0 (:; I) has rotation number ˛(I). In this case the question is what will be the fate of this family of invariant circles, as well as with the corresponding rigidly rotational dynamics. Regarding the rotation number we again introduce Diophantine conditions. Indeed, for > 2 and > 0 the subset [0; 1]; is defined as in (5), i. e., it contains all
˛ 2 [0; 1], such that for all rationals p/q ˇ ˇ ˇ ˇ ˇ˛ p ˇ q : ˇ qˇ Pulling back [0; 1]; along the map ˛ we obtain a subset ; . Theorem 4 (Twist Map Theorem [104]) For sufficiently small, and for the perturbation O(") sufficiently small in C 1 -topology, there exists a C 1 transformation ˚" : ! , conjugating the restriction P0 j; to a subsystem of P" . As in the case of Theorem 3 again we chose the formulation of [29,31]. Largely the remarks following Theorem 3 also apply here. Remark Compare the format of the Theorems 3 and 4 and observe that in the latter case the role of the parameter ˛ has been taken by the action variable I. Theorem 4 implies that typically quasi-periodicity occurs with positive measure in phase space. In the gaps typically we have coexistence of periodicity, quasi-periodicity and chaos [6,9,35,107,108,123,137]. The latter follows from transversality of homo- and heteroclinic connections that give rise to positive topological entropy. Open problems are whether the corresponding Lyapunov exponents also are positive, compare with the discussion at the end of the introduction. Similar to the applications of Theorem 3 given at the end of Subsect. “Circle Maps”, here direct applications are possible in the conservative setting. Indeed, consider a system of weakly coupled pendula @U (y1 ; y2 ) @y1 @U y¨2 C ˛22 sin y2 D " (y1 ; y2 ) : @y2 y¨1 C ˛12 sin y1 D "
Writing y˙ j D z j , j D 1; 2 as before, we again get a vector field in the four-dimensional phase space R2 R2 D f(y1 ; y2 ); (z1 ; z2 )g. In this case the energy H" (y1 ; y2 ; z1 ; z2 ) 1 1 D z12 C z22 ˛12 cos y1 ˛22 cos y2 C "U(y1 ; y2 ) 2 2 is a constant of motion. Restricting to a three-dimensional energy surface H"1 D const:, the iso-energetic Poincaré
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Hamiltonian Perturbation Theory (and Transition to Chaos)
map P" is a twist map and application of Theorem 4 yields the conclusion of quasi-periodicity (on invariant two-tori) occurring with positive measure in the energy surfaces of H" .
is the nth convergent in the continued fraction expansion of ˛ then the Bruno-condition reads X log(q nC1 ) n
qn
0 and > 2 we have the Diophantine nonresonance condition p
j e
2i q
j jqj :
The corresponding set of constitutes a set of full measure in T 1 D fg. Yoccoz [141] completely solved the elliptic case using the Bruno-condition. If D e2 i˛
pn and when qn
As an example
F(z) D z C z2 ; where 2 T 1 is not a root of unity. Observe that a point z 2 C is a periodic point of F with period q if and only if F q (z) D z, where obviously q
F q (z) D q z C C z2 : Writing q F q (z) z D z q 1 C C z2 1 ; the period q periodic points exactly are the roots of the right hand side polynomial. Abbreviating N D 2q 1, it directly follows that, if z1 ; z2 ; : : : ; z N are the nontrivial roots, then for their product we have z1 z2 : : : z N D q 1 : It follows that there exists a nontrivial root within radius jq 1j1/N of z D 0. Now consider the set of T 1 defined as follows: 2 whenever lim inf jq 1j1/N D 0 : q!1
It can be directly shown that is residual, again compare with [115]. It also follows that for 2 linearization is impossible. Indeed, since the rotation is irrational, the existence of periodic points in any neighborhood of z D 0 implies zero radius of convergence. Remark Notice that the residual set is in the complement of the full measure set of all Diophantine numbers, again see [115].
Hamiltonian Perturbation Theory (and Transition to Chaos)
Considering 2 T 1 as a parameter, we see a certain analogy of these results on complex linearization with the Theorems 3 and 4. Indeed, in this case for a full measure set of s on a neighborhood of z D 0 the map F D F is conjugated to a rigid irrational rotation. Such a domain in the z-plane often is referred to as a Siegel disc. For a more general discussion of these and of Herman rings, see [101]. KAM Theory: An Overview In Sect. “Invariant Curves of Planar Diffeomorphisms” we described the persistent occurrence of quasi-periodicity in the setting of diffeomorphisms of the circle or the plane. The general perturbation theory of quasi-periodic motions is known under the name Kolmogorov–Arnold–Moser (or KAM) theory and discussed extensively elsewhere in this encyclopedia Kolmogorov–Arnold–Moser (KAM) Theory. Presently we briefly summarize parts of this KAM theory in broad terms, as this fits in our considerations, thereby largely referring to [4,80,81,119,121,143,144], also see [20,24,55]. In general quasi-periodicity is defined by a smooth conjugation. First on the n-torus T n D Rn /(2 Z)n consider the vector field X! D
n X jD1
!j
@ ; @' j
where !1 ; !2 ; : : : ; !n are called frequencies [43,106]. Now, given a smooth (say, of class C 1 ) vector field X on a manifold M, with T M an invariant n-torus, we say that the restriction Xj T is parallel if there exists ! 2 Rn and a smooth diffeomorphism ˚ : T ! T n , such that ˚ (Xj T ) D X! . We say that Xj T is quasi-periodic if the frequencies !1 ; !2 ; : : : ; !n are independent over Q. A quasi-periodic vector field Xj T leads to an integer affine structure on the torus T. In fact, since each orbit is dense, it follows that the self conjugations of X! exactly are the translations of T n , which completely determine the affine structure of T n . Then, given ˚ : T ! T n with ˚ (Xj T ) D X! , it follows that the self conjugations of Xj T determines a natural affine structure on the torus T. Note that the conjugation ˚ is unique modulo translations in T and T n . Note that the composition of ˚ by a translation of T n does not change the frequency vector !. However, the composition by a linear invertible map S 2 GL(n; Z) yields S X! D XS! . We here speak of an integer affine structure [43].
Remark The transition maps of an integer affine structure are translations and elements of GL(n; Z). The current construction is compatible with the integrable affine structure on the Liouville tori of an integrable Hamiltonian system [6]. Note that in that case the structure extends to all parallel tori. Classical KAM Theory The classical KAM theory deals with smooth, nearly integrable Hamiltonian systems of the form '˙ D !(I) C " f (I; '; ") I˙ D "g(I; '; ") ;
(6)
where I varies over an open subset of Rn and ' over the standard torus T n . Note that for " D 0 the phase space as an open subset of Rn T n is foliated by invariant tori, parametrized by I. Each of the tori is parametrized by ' and the corresponding motion is parallel (or multi- periodic or conditionally periodic) with frequency vector !(I). Perturbation theory asks for persistence of the invariant n-tori and the parallelity of their motion for small values of j"j. The answer that KAM theory gives needs two essential ingredients. The first ingredient is that of Kolmogorov non-degeneracy which states that the map I 2 Rn 7! !(I) 2 Rn is a (local) diffeomorphism. Compare with the twist condition of Sect. “Invariant Curves of Planar Diffeomorphisms”. The second ingredient generalizes the Diophantine conditions (5) of that section as follows: for > n 1 and > 0 consider the set n D f! 2 Rn j jh!; kij jkj ; k 2 Z n nf0gg: (7) R; n The following properties are more or less direct. First R; n has a closed half line geometry in the sense that if ! 2 R; n and s 1 then also s! 2 R; . Moreover, the intersection n is a Cantor set of measure S n1 nRn D O( ) S n1 \R; ; as # 0, see Fig. 4. Completely in the spirit of Theorem 4, the classical KAM theorem roughly states that a Kolmogorov non-degenerate nearly integrable system (6)" , for j"j 1 is smoothly conjugated to the unperturbed version (6)0 , provided that the frequency map ! is co-restricted to the Dion . In this formulation smoothness has to phantine set R; be taken in the sense of Whitney [119,136], also compare with [20,24,29,31,55,121]. As a consequence we may say that in Hamiltonian systems of n degrees of freedom typically quasi-periodic invariant (Lagrangian) n-tori occur with positive measure in
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Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 4 The Diophantine set Rn; has the closed half line geometry and the intersection Sn1 \ Rn; is a Cantor set of measure Sn1 n Rn; D O() as # 0
phase space. It should be said that also an iso-energetic version of this classical result exists, implying a similar conclusion restricted to energy hypersurfaces [6,9,21,24]. The Twist Map Theorem 4 is closely related to the iso-energetic KAM Theorem. Remark We chose the quasi-periodic stability format as in Sect. “Invariant Curves of Planar Diffeomorphisms”. For regularity issues compare with a remark following Theorem 3. For applications we largely refer to the introduction and to [24,31] and references therein. Continuing the discussion on affine structures at the beginning of this section, we mention that by means of the symplectic form, the domain of the I-variables in Rn inherits an affine structure [60], also see [91] and references therein. Statistical Mechanics deals with particle systems that are large, often infinitely large. The Ergodic Hypothesis roughly says that in a bounded energy hypersurface, the dynamics are ergodic, meaning that any evolution in the energy level set comes near every point of this set. The taking of limits as the number of particles tends to infinity is a notoriously difficult subject. Here we dis-
cuss a few direct consequences of classical KAM theory for many degrees of freedom. This discussion starts with Kolmogorov’s papers [80,81], which we now present in a slightly rephrased form. First, we recall that for Hamiltonian systems (say, with n degrees of freedom), typically the union of Diophantine quasi-periodic Lagrangian invariant n-tori fills up positive measure in the phase space and also in the energy hypersurfaces. Second, such a collection of KAM tori immediately gives rise to non-ergodicity, since it clearly implies the existence of distinct invariant sets of positive measure. For background on Ergodic Theory, see e. g. [9,27] and [24] for more references. Apparently the KAM tori form an obstruction to ergodicity, and a question is how bad this obstruction is as n ! 1. Results in [5,78] indicate that this KAM theory obstruction is not too bad as the size of the system tends to infinity. In general the role of the Ergodic Hypothesis in Statistical Mechanics has turned out to be much more subtle than was expected, see e. g. [18,64]. Dissipative KAM Theory As already noted by Moser [105,106], KAM theory extends outside the world of Hamiltonian systems, like to volume preserving systems, or to equivariant or reversible systems. This also holds for the class of general smooth systems, often called dissipative. In fact, the KAM theorem allows for a Lie algebra proof, that can be used to cover all these special cases [24,29,31,45]. It turns out that in many cases parameters are needed for persistent occurrence of (Diophantine) quasi-periodic tori. As an example we now consider the dissipative setting, where we discuss a parametrized system with normally hyperbolic invariant n-tori carrying quasi-periodic motion. From [73] it follows that this is a persistent situation and that, up to a smooth (in this case of class Ck for large k) diffeomorphism, we can restrict to the case where T n is the phase space. To fix thoughts we consider the smooth system '˙ D !() C " f ('; ; ") ˙ D0;
(8)
where 2 Rn is a multi-parameter. The results of the classical KAM theorem regarding (6)" largely carry over to (8);" . Now, for " D 0 the product of phase space and parameter space as an open subset of T n Rn is completely foliated by invariant n-tori and since the perturbation does ˙ not concern the -equation, this foliation is persistent. The interest is with the dynamics on the resulting invariant tori that remains parallel after the perturbation; compare
Hamiltonian Perturbation Theory (and Transition to Chaos)
with the setting of Theorem 3. As just stated, KAM theory here gives a solution similar to the Hamiltonian case. The analogue of the Kolmogorov non-degeneracy condition here is that the frequency map 7! !() is a (local) diffeomorphism. Then, in the spirit of Theorem 3, we state that the system (8);" is smoothly conjugated to (8);0 , as before, provided that the map ! is co-restricted to the n . Again the smoothness has to be Diophantine set R; taken in the sense of Whitney [29,119,136,143,144], also see [20,24,31,55]. It follows that the occurrence of normally hyperbolic invariant tori carrying (Diophantine) quasi-periodic flow is typical for families of systems with sufficiently many parameters, where this occurrence has positive measure in parameter space. In fact, if the number of parameters equals the dimension of the tori, the geometry as sketched in Fig. 4 carries over in a diffeomorphic way. Remark Many remarks following Subsect. “Classical KAM Theory” and Theorem 3 also hold here. In cases where the system is degenerate, for instance because there is a lack of parameters, a path formalism can be invoked, where the parameter path is required to n , see be a generic subfamily of the Diophantine set R; Fig. 4. This amounts to the Rüssmann non-degeneracy, that still gives positive measure of quasi-periodicity in the parameter space, compare with [24,31] and references therein. In the dissipative case the KAM theorem gives rise to families of quasi-periodic attractors in a typical way. This is of importance in center manifold reductions of infinite dimensional dynamics as, e. g., in fluid mechanics [125,126]. In Sect. “Transition to Chaos and Turbulence” we shall return to this subject. Lower Dimensional Tori We extend the above approach to the case of lower dimensional tori, i. e., where the dynamics transversal to the tori is also taken into account. We largely follow the setup of [29,45] that follows Moser [106]. Also see [24,31] and references therein. Changing notation a little, we now consider the phase space T n Rm D fx(mod 2); yg, as well a parameter space fg D P Rs . We consider a C 1 -family of vector fields X(x; y; ) as before, having T n f0g T n Rm as an invariant n-torus for D 0 2 P. x˙ D !() C f (y; ) y˙ D ˝() y C g(y; ) ˙ D0;
(9)
with f (y; 0 ) D O(jyj) and g(y; 0 ) D O(jyj2 ), so we assume the invariant torus to be of Floquet type. The system X D X(x; y; ) is integrable in the sense that it is T n -symmetric, i. e., x-independent [29]. The interest is with the fate of the invariant torus T n f0g and its parallel dynamics under small perturbation to a system ˜ X˜ D X(x; y; ) that no longer needs to be integrable. Consider the smooth mappings ! : P ! Rn and ˝ : P ! gl(m; R). To begin with we restrict to the case where all eigenvalues of ˝(0 ) are simple and nonzero. In general for such a matrix ˝ 2 gl(m; R), let the eigenvalues be given by ˛1 ˙ iˇ1 ; : : : ; ˛ N 1 ˙ iˇ N 1 and ı1 ; : : : ; ı N 2 , where all ˛ j ; ˇ j and ı j are real and hence m D 2N1 C N2 . Also consider the map spec : gl(m; R) ! R2N 1 CN 2 , given by ˝ 7! (˛; ˇ; ı). Next to the internal frequency vector ! 2 Rn , we also have the vector ˇ 2 R N 1 of normal frequencies. The present analogue of Kolmogorov non-degeneracy is the Broer–Huitema–Takens (BHT) non-degeneracy condition [29,127], which requires that the product map ! (spec) ı ˝ : P ! Rn gl(m; R) at D 0 has a surjective derivative and hence is a local submersion [72]. Furthermore, we need Diophantine conditions on both the internal and the normal frequencies, generalizing (7). Given > n 1 and > 0, it is required for all k 2 Z n n f0g and all ` 2 Z N 1 with j`j 2 that jhk; !i C h`; ˇij jkj :
(10)
Inside Rn R N 1 D f!; ˇg this yields a Cantor set as before (compare Fig. 4). This set has to be pulled back along the submersion ! (spec) ı ˝, for examples see Subsects. “(n 1)-Tori” and “Quasi-periodic Bifurcations” below. The KAM theorem for this setting is quasi-periodic stability of the n-tori under consideration, as in Subsect. “Dissipative KAM Theory”, yielding typical examples where quasi-periodicity has positive measure in parameter space. In fact, we get a little more here, since the normal linear behavior of the n-tori is preserved by the Whitney smooth conjugations. This is expressed as normal linear stability, which is of importance for quasi-periodic bifurcations, see Subsect. “Quasi-periodic Bifurcations” below. Remark A more general set-up of the normal stability theory [45] adapts the above to the case of non-simple (multiple) eigenvalues. Here the BHT non-degeneracy condition is formulated in terms of versal unfolding of the matrix ˝(0 ) [7]. For possible conditions under which vanishing eigenvalues are admissible see [29,42,69] and references therein.
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This general set-up allows for a structure preserving formulation as mentioned earlier, thereby including the Hamiltonian and volume preserving case, as well as equivariant and reversible cases. This allows us, for example, to deal with quasi-periodic versions of the Hamiltonian and the reversible Hopf bifurcation [38,41,42,44]. The Parameterized KAM Theory discussed here a priori needs many parameters. In many cases the parameters are distinguished in the sense that they are given by action variables, etc. For an example see Subsect. “(n 1)-Tori” on Hamiltonian (n 1)-tori Also see [127] and [24,31] where the case of Rüssmann nondegeneracy is included. This generalizes a remark at the end of Subsect. “Dissipative KAM Theory”.
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 5 Range of the energy-momentum map of the spherical pendulum
Global KAM Theory We stay in the Hamiltonian setting, considering Lagrangian invariant n-tori as these occur in a Liouville integrable system with n degrees of freedom. The union of these tori forms a smooth T n -bundle f : M ! B (where we leave out all singular fibers). It is known that this bundle can be non-trivial [56,60] as can be measured by monodromy and Chern class. In this case global action angle variables are not defined. This non-triviality, among other things, is of importance for semi-classical versions of the classical system at hand, in particular for certain spectrum defects [57,62,134,135], for more references also see [24]. Restricting to the classical case, the problem is what happens to the (non-trivial) T n -bundle f under small, non-integrable perturbation. From the classical KAM theory, see Subsect. “Classical KAM Theory” we already know that on trivializing charts of f Diophantine quasi-periodic n-tori persist. In fact, at this level, a Whitney smooth conjugation exists between the integrable system and its perturbation, which is even Gevrey regular [136]. It turns out that these local KAM conjugations can be glued together so to obtain a global conjugation at the level of quasi-periodic tori, thereby implying global quasi-periodic stability [43]. Here we need unicity of KAM tori, i. e., independence of the action-angle chart used in the classical KAM theorem [26]. The proof uses the integer affine structure on the quasi-periodic tori, which enables taking convex combinations of the local conjugations subjected to a suitable partition of unity [72,129]. In this way the geometry of the integrable bundle can be carried over to the nearlyintegrable one. The classical example of a Liouville integrable system with non-trivial monodromy [56,60] is the spherical pendulum, which we now briefly revisit. The configuration
space is S2 D fq 2 R3 j hq; qi D 1g and the phase space T S2 Š f(q; p) 2 R6 j hq; qi D 1 and hq; pi D 0g. The two integrals I D q1 p2 q2 p1 (angular momentum) and E D 12 hp; pi C q3 (energy) lead to an energy momentum map EM : T S2 ! R2 , given by (q; p) 7! (I; E) D q1 p2 q2 p1 ; 12 hp; pi C q3 . In Fig. 5 we show the image of the map EM. The shaded area B consists of regular values, the fiber above which is a Lagrangian two-torus; the union of these gives rise to a bundle f : M ! B as described before, where f D EMj M . The motion in the twotori is a superposition of Huygens’ rotations and pendulum-like swinging, and the non-existence of global action angle variables reflects that the three interpretations of ‘rotating oscillation’, ‘oscillating rotation’ and ‘rotating rotation’ cannot be reconciled in a consistent way. The singularities of the fibration include the equilibria (q; p) D ((0; 0; ˙1); (0; 0; 0)) 7! (I; E) D (0; ˙1). The boundary of this image also consists of singular points, where the fiber is a circle that corresponds to Huygens’ horizontal rotations of the pendulum. The fiber above the upper equilibrium point (I; E) D (0; 1) is a pinched torus [56], leading to non-trivial monodromy, in a suitable bases of the period lattices, given by
1 0
1 1
2 GL(2; R) :
The question here is what remains of the bundle f when the system is perturbed. Here we observe that locally Kolmogorov non-degeneracy is implied by the non-trivial monodromy [114,122]. From [43,122] it follows that the non-trivial monodromy can be extended in the perturbed case.
Hamiltonian Perturbation Theory (and Transition to Chaos)
Remark The case where this perturbation remains integrable is covered in [95], but presently the interest is with the nearly integrable case, so where the axial symmetry is broken. Also compare [24] and many of its references. The global conjugations of [43] are Whitney smooth (even Gevrey regular [136]) and near the identity map in the C 1 -topology [72]. Geometrically speaking these diffeomorphisms also are T n -bundle isomorphisms between the unperturbed and the perturbed bundle, the basis of which is a Cantor set of positive measure. Splitting of Separatrices KAM theory does not predict the fate of close-to-resonant tori under perturbations. For fully resonant tori the phenomenon of frequency locking leads to the destruction of the torus under (sufficiently rich) perturbations, and other resonant tori disintegrate as well. In the case of a single resonance between otherwise Diophantine frequencies the perturbation leads to quasi-periodic bifurcations, cf. Sect. “Transition to Chaos and Turbulence”. While KAM theory concerns the fate of most trajectories and for all times, a complementary theorem has been obtained in [93,109,110,113]. It concerns all trajectories and states that they stay close to the unperturbed tori for long times that are exponential in the inverse of the perturbation strength. For trajectories starting close to surviving tori the diffusion is even superexponentially slow, cf. [102,103]. Here a form of smoothness exceeding the mere existence of infinitely many derivatives of the Hamiltonian is a necessary ingredient, for finitely differentiable Hamiltonians one only obtains polynomial times. Solenoids, which cannot be present in integrable systems, are constructed for generic Hamiltonian systems in [16,94,98], yielding the simultaneous existence of representatives of all homeomorphy-classes of solenoids. Hyperbolic tori form the core of a construction proposed in [5] of trajectories that venture off to distant points of the phase space. In the unperturbed system the union of a family of hyperbolic tori, parametrized by the actions conjugate to the toral angles, form a normally hyperbolic manifold. The latter is persistent under perturbations, cf. [73,100], and carries a Hamiltonian flow with fewer degrees of freedom. The main difference between integrable and non-integrable systems already occurs for periodic orbits. Periodic Orbits A sharp difference to dissipative systems is that it is generic for hyperbolic periodic orbits on compact energy shells in
Hamiltonian systems to have homoclinic orbits, cf. [1] and references therein. For integrable systems these form together a pinched torus, but under generic perturbations the stable and unstable manifold of a hyperbolic periodic orbit intersect transversely. It is a nontrivial task to actually check this genericity condition for a given non-integrable perturbation, a first-order condition going back to Poincaré requires the computation of the so-called Mel’nikov integral, see [66,137] for more details. In two degrees of freedom normalization leads to approximations that are integrable to all orders, which implies that the Melnikov integral is a flat function. In the real analytic case the Melnikov criterion is still decisive in many examples [65]. Genericity conditions are traditionally formulated in the universe of smooth vector fields, and this makes the whole class of analytic vector fields appear to be nongeneric. This is an overly pessimistic view as the conditions defining a certain class of generic vector fields may certainly be satisfied by a given analytic system. In this respect it is interesting that the generic properties may also be formulated in the universe of analytic vector fields, see [28] for more details. (n 1)-Tori The (n 1)-parameter families of invariant (n 1)-tori organize the dynamics of an integrable Hamiltonian system in n degrees of freedom, and under small perturbations the parameter space of persisting analytic tori is Cantorized. This still allows for a global understanding of a substantial part of the dynamics, but also leads to additional questions. A hyperbolic invariant torus T n1 has its Floquet exponents off the imaginary axis. Note that T n1 is not a normally hyperbolic manifold. Indeed, the normal linear behavior involves the n 1 zero eigenvalues in the direction of the parametrizing actions as well; similar to (9) the format x˙ D !(y) C O(y) C O(z2 ) y˙ D O(y) C O(z3 ) z˙ D ˝(y)z C O(z2 ) in Floquet coordinates yields an x-independent matrix ˝ that describes the symplectic normal linear behavior, cf. [29]. The union fz D 0g over the family of (n1)-tori is a normally hyperbolic manifold and constitutes the center manifold of T n1 . Separatrices splitting yields the dividing surfaces in the sense of Wiggins et al. [138]. The persistence of elliptic tori under perturbation from an integrable system involves not only the internal
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frequencies of T n1 , but also the normal frequencies. Next to the internal resonances the necessary Diophantine conditions (10) exclude the normal-internal resonances hk; !i D ˛ j
(11)
hk; !i D 2˛ j
(12)
hk; !i D ˛ i C ˛ j
(13)
hk; !i D ˛ i ˛ j :
(14)
The first three resonances lead to the quasi-periodic center-saddle bifurcation studied in Sect. “Transition to Chaos and Turbulence”, the frequency-halving (or quasiperiodic period doubling) bifurcation and the quasi-periodic Hamiltonian Hopf bifurcation, respectively. The resonance (14) generalizes an equilibrium in 1 : 1 resonance whence T n1 persists and remains elliptic, cf. [78]. When passing through resonances (12) and (13) the lower-dimensional tori lose ellipticity and acquire hyperbolic Floquet exponents. Elliptic (n 1)-tori have a single normal frequency whence (11) and (12) are the only normal-internal resonances. See [35] for a thorough treatment of the ensuing possibilities. The restriction to a single normal-internal resonance is dictated by our present possibilities. Indeed, already the bifurcation of equilibria with a fourfold zero eigenvalue leads to unfoldings that simultaneously contain all possible normal resonances. Thus, a satisfactory study of such tori which already may form one-parameter families in integrable Hamiltonian systems with five degrees of freedom has to await further progress in local bifurcation theory.
tion [66,75,82], where a periodic solution branches off. In a second transition of similar nature a quasi-periodic two-torus branches off, then a quasi-periodic three-torus, etc. The idea is that the motion picks up more and more frequencies and thus obtains an increasingly complicated power spectrum. In the early 1970s this idea was modified in the Ruelle–Takens route to turbulence, based on the observation that, for flows, a three-torus can carry chaotic (or strange) attractors [112,126], giving rise to a broad band power spectrum. By the quasi-periodic bifurcation theory [24,29,31] as sketched below these two approaches are unified in a generic way, keeping track of measure theoretic aspects. For general background in dynamical systems theory we refer to [27,79]. Another transition to chaos was detected in the quadratic family of interval maps f (x) D x(1 x) ; see [58,99,101], also for a holomorphic version. This transition consists of an infinite sequence of period doubling bifurcations ending up in chaos; it has several universal aspects and occurs persistently in families of dynamical systems. In many of these cases also homoclinic bifurcations show up, where sometimes the transition to chaos is immediate when parameters cross a certain boundary, for general theory see [13,14,30,117]. There exist quite a number of case studies where all three of the above scenarios play a role, e. g., see [32,33,46] and many of their references. Quasi-periodic Bifurcations
Transition to Chaos and Turbulence One of the main interests over the second half of the twentieth century has been the transition between orderly and complicated forms of dynamics upon variation of either initial states or of system parameters. By ‘orderly’ we here mean equilibrium and periodic dynamics and by complicated quasi-periodic and chaotic dynamics, although we note that only chaotic dynamics is associated to unpredictability, e. g. see [27]. As already discussed in the introduction systems like a forced nonlinear oscillator or the planar three-body problem exhibit coexistence of periodic, quasi-periodic and chaotic dynamics, also compare with Fig. 1. Similar remarks go for the onset of turbulence in fluid dynamics. Around 1950 this led to the scenario of Hopf– Landau–Lifschitz [75,76,83,84], which roughly amounts to the following. Stationary fluid motion corresponds to an equilibrium point in an 1-dimensional state space of velocity fields. The first transition is a Hopf bifurca-
For the classical bifurcations of equilibria and periodic orbits, the bifurcation sets and diagrams are generally determined by a classical geometry in the product of phase space and parameter space as already established by, e. g., [8,133], often using singularity theory. Quasi-periodic bifurcation theory concerns the extension of these bifurcations to invariant tori in nearly-integrable systems, e. g., when the tori lose their normal hyperbolicity or when certain (strong) resonances occur. In that case the dense set of resonances, also responsible for the small divisors, leads to a Cantorization of the classical geometries obtained from Singularity Theory [29,35,37,38,39,41,44,45, 48,49,67,68,69], also see [24,31,52,55]. Broadly speaking, one could say that in these cases the Preparation Theorem [133] is partly replaced by KAM theory. Since the KAM theory has been developed in several settings with or without preservation of structure, see Sect. “KAM Theory: An Overview”, for the ensuing quasi-periodic bifurcation theory the same holds.
Hamiltonian Perturbation Theory (and Transition to Chaos)
Hamiltonian Cases To fix thoughts we start with an example in the Hamiltonian setting, where a robust model for the quasi-periodic center-saddle bifurcation is given by H!1 ;!2 ;;" (I; '; p; q) D !1 I1 C !2 I2 C 12 p2 C V (q) C " f (I; '; p; q)
(15)
with V (q) D 13 q3 q, compare with [67,69]. The unperturbed (or integrable) case " D 0, by factoring out the T 2 -symmetry, boils down to a standard center-saddle bifurcation, involving the fold catastrophe [133] in the potential function V D V (q). This results in the existence of two invariant two-tori, one elliptic and the other hyperbolic. For 0 ¤ j"j 1 the dense set of resonances complicates this scenario, as sketched in Fig. 6, determined by the Diophantine conditions jhk; !ij jkj ; jhk; !i C `ˇ(q)j jkj
;
for q < 0 ;
(16)
for q > 0
for all k p 2 Z n n f0g and for all ` 2 Z with j`j 2. Here ˇ(q) D 2q is the normal frequency of the elliptic torus p given by q D for > 0. As before, (cf. Sects. “Invariant Curves of Planar Diffeomorphisms”, “KAM Theory: An Overview”), this gives a Cantor set of positive measure [24,29,31,45,69,105,106]. For 0 < j"j 1 Fig. 6 will be distorted by a nearidentity diffeomorphism; compare with the formulations of the Theorems 3 and 4. On the Diophantine Cantor set the dynamics is quasi-periodic, while in the gaps generically there is coexistence of periodicity and chaos, roughly
comparable with Fig. 1, at left. The gaps at the border furthermore lead to the phenomenon of parabolic resonance, cf. [86]. Similar programs exist for all cuspoid and umbilic catastrophes [37,39,68] as well as for the Hamiltonian Hopf bifurcation [38,44]. For applications of this approach see [35]. For a reversible analogue see [41]. As so often within the gaps generically there is an infinite regress of smaller gaps [11,35]. For theoretical background we refer to [29,45,106], for more references also see [24]. Dissipative Cases In the general dissipative case we basically follow the same strategy. Given the standard bifurcations of equilibria and periodic orbits, we get more complex situations when invariant tori are involved as well. The simplest examples are the quasi-periodic saddle-node and quasi-periodic period doubling [29] also see [24,31]. To illustrate the whole approach let us start from the Hopf bifurcation of an equilibrium point of a vector field [66,75,82,116] where a hyperbolic point attractor loses stability and branches off a periodic solution, cf. Subsect. “Dissipative Perturbations”. A topological normal form is given by
y˙1 y˙2
D
˛ ˇ
ˇ ˛
y1 y2
y12 C y22
y1 y2
(17) where y D (y1 ; y2 ) 2 R2 , ranging near (0; 0). In this representation usually one fixes ˇ D 1 and lets ˛ D (near 0) serve as a (bifurcation) parameter, classifying modulo topological equivalence. In polar coordinates (17) so gets the form '˙ D 1 ; r˙ D r r3 : Figure 7 shows an amplitude response diagram (often called the bifurcation diagram). Observe the occurrence of the attracting periodic solution for > 0 of amplitude p . Let us briefly consider the Hopf bifurcation for fixed points of diffeomorphisms. A simple example has the form
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 6 Sketch of the Cantorized Fold, as the bifurcation set of the quasiperiodic center-saddle bifurcation for n D 2 [67], where the horizontal axis indicates the frequency ratio !2 : !1 , cf. (15). The lower part of the figure corresponds to hyperbolic tori and the upper part to elliptic ones. See the text for further interpretations
P(y) D e2(˛Ciˇ ) y C O(jyj2 ) ;
(18)
y 2 C Š R2 , near 0. To start with ˇ is considered a constant, such that ˇ is not rational with denominator less than five, see [7,132], and where O(jyj2 ) should contain generic third order terms. As before, we let ˛ D serve as a bifurcation parameter, varying near 0. On one side of the
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We now discuss the quasi-periodic Hopf bifurcation [17,29], largely following [55]. The unperturbed, integrable family X D X (x; y) on T n R2 has the form X (x; y) D [!() C f (y; )]@x C [˝()y C g(y; )]@ y ; (19)
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 7 Bifurcation diagram of the Hopf bifurcation
bifurcation value D 0, this system has by normal hyperbolicity and [73], an invariant circle. Here, due to the invariance of the rotation numbers of the invariant circles, no topological stability can be obtained [111]. Still this bifurcation can be characterized by many persistent properties. Indeed, in a generic two-parameter family (18), say with both ˛ and ˇ as parameters, the periodicity in the parameter plane is organized in resonance tongues [7,34,82]. (The tongue structure is hardly visible when only one parameter, like ˛, is used.) If the diffeomorphism is the return map of a periodic orbit for flows, this bifurcation produces an invariant two-torus. Usually this counterpart for flows is called Ne˘ımark–Sacker bifurcation. The periodicity as it occurs in the resonance tongues, for the vector field is related to phase lock. The tongues are contained in gaps of a Cantor set of quasi-periodic tori with Diophantine frequencies. Compare the discussion in Subsect. “Circle Maps”, in particular also regarding the Arnold family and Fig. 3. Also see Sect. “KAM Theory: An Overview” and again compare with [115]. Quasi-periodic versions exist for the saddle-node, the period doubling and the Hopf bifurcation. Returning to the setting with T n Rm as the phase space, we remark that the quasi-periodic saddle-node and period doubling already occur for m D 1, or in an analogous center manifold. The quasi-periodic Hopf bifurcation needs m 2. We shall illustrate our results on the latter of these cases, compare with [19,31]. For earlier results in this direction see [52]. Our phase space is T n R2 D fx(mod 2); yg, where we are dealing with the parallel invariant torus T n f0g. In the integrable case, by T n -symmetry we can reduce to R2 D fyg and consider the bifurcations of relative equilibria. The present interest is with small non-integrable perturbations of such integrable models.
were f D O(jyj) and g D O(jyj2 ) as before. Moreover 2 P is a multi-parameter and ! : P ! Rn and ˝ : P ! gl(2; R) are smooth maps. Here we take ˛() ˇ() ˝() D ; ˇ() ˛() which makes the @ y component of (19) compatible with the planar Hopf family (17). The present form of Kolmogorov non-degeneracy is Broer–Huitema–Takens stability [29,42,45], requiring that there is a subset P on which the map 2 P 7! (!(); ˝()) 2 Rn gl(2; R) is a submersion. For simplicity we even assume that is replaced by (!; (˛; ˇ)) 2 Rn R2 : Observe that if the non-linearity g satisfies the well-known Hopf non-degeneracy conditions, e. g., compare [66,82], then the relative equilibrium y D 0 undergoes a standard planar Hopf bifurcation as described before. Here ˛ again plays the role of bifurcation parameter and a closed orbit branches off at ˛ D 0. To fix thoughts we assume that y D 0 is attracting for ˛ < 0. and that the closed orbit occurs for ˛ > 0, and is attracting as well. For the integrable family X, qualitatively we have to multiply this planar scenario with T n , by which all equilibria turn into invariant attracting or repelling n-tori and the periodic attractor into an attracting invariant (n C 1)-torus. Presently the question is what happens to both the n- and the (n C 1)-tori, when we apply a small near-integrable perturbation. The story runs much like before. Apart from the BHT non-degeneracy condition we require Diophantine conditions (10), defining the Cantor set (2) ; D f(!; (˛; ˇ)) 2 j jhk; !i C `ˇj jkj ;
8k 2 Zn n f0g ; 8` 2 Z with j`j 2g ;
(20)
(2) In Fig. 8 we sketch the intersection of ;
Rn R2 with 2 a plane f!g R for a Diophantine (internal) frequency vector !, cf. (7).
Hamiltonian Perturbation Theory (and Transition to Chaos)
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 8 (2) Planar section of the Cantor set ;
From [17,29] it now follows that for any family X˜ on R2 P, sufficiently near X in the C 1 -topology a near-identity C 1 -diffeomorphism ˚ : T n R2 ! T n R2 exists, defined near T n f0g , that con(2) . jugates X to X˜ when further restricting to T n f0g ; So this means that the Diophantine quasi-periodic invariant n-tori are persistent on a diffeomorphic image of the (2) Cantor set ; , compare with the formulations of the Theorems 3 and 4. Similarly we can find invariant (n C 1)-tori. We first have to develop a T nC1 symmetric normal form approximation [17,29] and Normal Forms in Perturbation Theory. For this purpose we extend the Diophantine conditions (20) by requiring that the inequality holds for all j`j N for N D 7. We thus find another large Cantor set, again see Fig. 8, where Diophantine quasi-periodic invariant (n C 1)-tori are persistent. Here we have to restrict to ˛ > 0 for our choice of the sign of the normal form coefficient, compare with Fig. 7. In both the cases of n-tori and of (n C 1)-tori, the nowhere dense subset of the parameter space containing the tori can be fattened by normal hyperbolicity to open subsets. Indeed, the quasi-periodic n- and (n C 1)-tori are infinitely normally hyperbolic [73]. Exploiting the normal form theory [17,29] and Normal Forms in Perturbation Theory to the utmost and using a more or less standard contraction argument [17,53], a fattening of the parameter domain with invariant tori can be obtained that leaves out only small ‘bubbles’ around the resonances, as sketched and explained in Fig. 9 for the n-tori. For earlier results in the same spirit in a case study of the quasi-periodic saddlenode bifurcation see [49,50,51], also compare with [11].
Tn
Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 9 Fattening by normal hyperbolicity of a nowhere dense parameter set with invariant n-tori in the perturbed system. The curve H is the Whitney smooth (even Gevrey regular [136]) image of the ˇ-axis in Fig. 8. H interpolates the Cantor set Hc that contains the non-hyperbolic Diophantine quasi-periodic invari(2) , see (20). To the points 1;2 2 ant n-tori, corresponding to ; Hc discs A1;2 are attached where we find attracting normally hyperbolic n-tori and similarly in the discs R1;2 repelling ones. The contact between the disc boundaries and H is infinitely flat [17,29]
A Scenario for the Onset of Turbulence Generally speaking, in many settings quasi-periodicity constitutes the order in between chaos [31]. In the Hopf– Landau–Lifschitz–Ruelle–Takens scenario [76,83,84,126] we may consider a sequence of typical transitions as given by quasi-periodic Hopf bifurcations, starting with the standard Hopf or Hopf–Ne˘ımark–Sacker bifurcation as described before. In the gaps of the Diophantine Cantor sets generically there will be coexistence of periodicity, quasi-periodicity and chaos in infinite regress. As said earlier, period doubling sequences and homoclinic bifurcations may accompany this. As an example consider a family of maps that undergoes a generic quasi-periodic Hopf bifurcation from circle to two-torus. It turns out that here the Cantorized fold of Fig. 6 is relevant, where now the vertical coordinate is
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a bifurcation parameter. Moreover compare with Fig. 3, where also variation of " is taken into account. The Cantor set contains the quasi-periodic dynamics, while in the gaps we can have chaos, e. g., in the form of Hénon like strange attractors [46,112]. A fattening process as explained above, also can be carried out here. Future Directions One important general issue is the mathematical characterization of chaos and ergodicity in dynamical systems, in conservative, dissipative and in other settings. This is a tough problem as can already be seen when considering two-dimensional diffeomorphisms. In particular we refer to the still unproven ergodicity conjecture of [9] and to the conjectures around Hénon like attractors and the principle ‘Hénon everywhere’, compare with [22,32]. For a discussion see Subsect. “A Scenario for the Onset of Turbulence”. In higher dimension this problem is even harder to handle, e. g., compare with [46,47] and references therein. In the conservative case a related problem concerns a better understanding of Arnold diffusion. Somewhat related to this is the analysis of dynamical systems without an explicit perturbation setting. Here numerical and symbolic tools are expected to become useful to develop computer assisted proofs in extended perturbation settings, diagrams of Lyapunov exponents, symbolic dynamics, etc. Compare with [128]. Also see [46,47] for applications and further reference. This part of the theory is important for understanding concrete models, that often are not given in perturbation format. Regarding nearly-integrable Hamiltonian systems, several problems have to be considered. Continuing the above line of thought, one interest is the development of Hamiltonian bifurcation theory without integrable normal form and, likewise, of KAM theory without action angle coordinates [87]. One big related issue also is to develop KAM theory outside the perturbation format. The previous section addressed persistence of Diophantine tori involved in a bifurcation. Similar to Cremer’s example in Subsect. “Cremer’s Example in Herman’s Version” the dynamics in the gaps between persistent tori displays new phenomena. A first step has been made in [86] where internally resonant parabolic tori involved in a quasi-periodic Hamiltonian pitchfork bifurcation are considered. The resulting large dynamical instabilities may be further amplified for tangent (or flat) parabolic resonances, which fail to satisfy the iso-energetic non-degeneracy condition. The construction of solenoids in [16,94] uses elliptic periodic orbits as starting points, the simplest exam-
ple being the result of a period-doubling sequence. This construction should carry over to elliptic tori, where normal-internal resonances lead to encircling tori of the same dimension, while internal resonances lead to elliptic tori of smaller dimension and excitation of normal modes increases the torus dimension. In this way one might be able to construct solenoid-type invariant sets that are limits of tori with varying dimension. Concerning the global theory of nearly-integrable torus bundles [43], it is of interest to understand the effects of quasi-periodic bifurcations on the geometry and its invariants. Also it is of interest to extend the results of [134] when passing to semi-classical approximations. In that case two small parameters play a role, namely Planck’s constant as well as the distance away from integrability.
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Hamilton–Jacobi Equations and Weak KAM Theory
Hamilton–Jacobi Equations and Weak KAM Theory ANTONIO SICONOLFI Dip. di Matematica, “La Sapienza” Università di Roma, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Subsolutions Solutions First Regularity Results for Subsolutions Critical Equation and Aubry Set An Intrinsic Metric Dynamical Properties of the Aubry Set Long-Time Behavior of Solutions to the Time-Dependent Equation Main Regularity Result Future Directions Bibliography Glossary Hamilton–Jacobi equations This class of first-order partial differential equations has a central relevance in several branches of mathematics, both from a theoretical and an application point of view. It is of primary importance in classical mechanics, Hamiltonian dynamics, Riemannian and Finsler geometry, and optimal control theory, as well. It furthermore appears in the classical limit of the Schrödinger equation. A connection with Hamilton’s equations, in the case where the Hamiltonian has sufficient regularity, is provided by the classical Hamilton–Jacobi method which shows that the graph of the differential of any regular, say C1 , global solution to the equation is an invariant subset for the corresponding Hamiltonian flow. The drawback of this approach is that such regular solutions do not exist in general, even for very regular Hamiltonians. See the next paragraph for more comments on this issue. Viscosity solutions As already pointed out, Hamilton– Jacobi equations do not have in general global classical solutions, i. e. everywhere differentiable functions satisfying the equation pointwise. The method of characteristics just yields local classical solutions. This explains the need of introducing weak solutions. The idea for defining those of viscosity type is to con-
sider C1 functions whose graph, up to an additive constant, touches that of the candidate solution at a point and then stay locally above (resp. below) it. These are the viscosity test functions, and it is required that the Hamiltonian satisfies suitable inequalities when its first-order argument is set equal to the differential of them at the first coordinate of the point of contact. Similarly it is defined the notion of viscosity sub, supersolution. Clearly a viscosity solution satisfies pointwise the equation at any differentiability points. A peculiarity of the definition is that a viscosity solution can admit no test function at some point, while the nonemptiness of both classes of test functions is equivalent to the solution being differentiable at the point. Nevertheless powerful existence, uniqueness and stability results hold in the framework of viscosity solution theory. The notion of viscosity solutions was introduced by Crandall and Lions at the beginning of the 1980s. We refer to Bardi and Capuzzo Dolcetta [2], Barles [3], Koike [24] for a comprehensive treatment of this topic. Semiconcave and semiconvex functions These are the appropriate regularity notions when working with viscosity solution techniques. The definition is given by requiring some inequalities, involving convex combinations of points, to hold. These functions possess viscosity test functions of one of the two types at any point. When the Hamiltonian enjoys coercivity properties ensuring that any viscosity solution is locally Lipschitz-continuous then a semiconcave or semiconcave function is the solution if and only if it is classical solution almost everywhere, i. e. up to a set of zero Lebesgue measure. Metric approach This method applies to stationary Hamilton–Jacobi equations with the Hamiltonian only depending on the state and momentum variable. This consists of defining a length functional, on the set of Lipschitz-continuous curves, related to the corresponding sublevels of the Hamiltonian. The associated length distance, obtained by performing the infimum of the intrinsic length of curves joining two given points, plays a crucial role in the analysis of the equation and, in particular, enters in representation formulae for any viscosity solution. One important consequence is that only the sublevels of the Hamiltonian matter for determining such solutions. Accordingly the convexity condition on the Hamiltonian can be relaxed, just requiring quasiconvexity, i. e. convexity of sublevels. Note that in this case the metric is of Finsler type and the sublevels are the unit cotangent balls of it.
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Critical equations To any Hamiltonian is associated a one-parameter family of Hamilton–Jacobi equations obtained by fixing a constant level of Hamiltonian. When studying such a family, one comes across a threshold value under which no subsolutions may exist. This is called the critical value and the same name is conferred to the corresponding equation. If the ground space is compact then the critical equation is unique among those of the family for which viscosity solutions do exist. When, in particular, the underlying space is a torus or, in other terms, the Hamiltonian is Z N periodic then such functions play the role of correctors in related homogenization problems. Aubry set The analysis of the critical equation shows that the obstruction for getting subsolutions at subcritical levels is concentrated on a special set of the ground space, in the sense that no critical subsolution can be strict around it. This is precisely the Aubry set. This is somehow compensated by the fact that critical subsolutions enjoy extra regularity properties on the Aubry set. Definition of the Subject The article aims to illustrate some applications of weak KAM theory to the analysis of Hamilton–Jacobi equations. The presentation focuses on two specific problems, namely the existence of C1 classical subsolutions for a class of stationary (i. e. independent of the time) Hamilton–Jacobi equations, and the long-time behavior of viscosity solutions of an evolutive version of it. The Hamiltonian is assumed to satisfy mild regularity conditions, under which the corresponding Hamilton equations cannot be written. Consequently PDE techniques will be solely employed in the analysis, since the powerful tools of the Hamiltonian dynamics are not available. Introduction Given a continuous or more regular Hamiltonian H(x; p) defined on the cotangent bundle of a boundaryless manifold M, where x and p are the state and the momentum variable, respectively, and satisfying suitable convexity and coercivity assumptions, is considered the family of Hamilton–Jacobi equations H(x; Du) D a
x 2 M;
(1)
with a a real parameter, as well as the time-dependent version w t C H(x; Dw) D 0
x 2 M ; t 2 (0; C1) ;
(2)
where Du and ut stand for the derivative with respect to state and time variable, respectively. As a matter of fact, it will be set, for the sake of simplicity, to either M D R N (noncompact case) or M D T N (compact case), where T N indicates the flat torus endowed with the Euclidean metric and with the cotangent bundle identified to T N RN . The main scope of this article is to study the existence of the C1 classical subsolution to (1), and the long-time behavior of viscosity solutions to (2) by essentially employing tools issued from weak KAM theory. Some of the results that will be outlined are valid with the additional assumption of compactness for the underlying manifold, in particular those concerning the asymptotics of solutions to (2). For both issues it is crucial to perform a qualitative analysis of (1) for a known value of the parameter a, qualified as critical; accordingly Eq. (1) is called critical when a is equal to the critical value. This analysis leads to detection of a special closed subset of the ground space, named after Aubry, where any locally Lipschitz-continuous subsolution to (1) enjoys some additional regularity properties, and behaves in a peculiar way. This will have a central role in the presentation. The requirements on H will be strengthened to obtain some theorems, but we remain in a setting where the corresponding Hamilton equations cannot be written. Consequently PDE techniques will be solely employed in the analysis, since the powerful tools of Hamiltonian dynamics are not available. Actually the notion of critical value was independently introduced by Ricardo Mañé at the beginning of the 1980s, in connection with the analysis of integral curves of the Euler–Lagrange flow enjoying some global minimizing properties, and by P.L. Lions, S.R.S. Varadhan and G. Papanicolaou [25] in 1987 in the framework of viscosity solutions theory, for studying the periodic homogenization of Hamilton–Jacobi equations. The Aubry set was determined and analyzed by Serge Aubry, in a pure dynamical way, as the union of the supports of integral curves of Euler–Lagrange flow possessing suitable minimality properties. John Mather defined (1986), in a more general framework, a set, contained in the Aubry set, starting from special probability measures invariant with respect to the flow. See Contreras and Iturriaga, [9], for an account on this theory. The first author to point out the link between Aubry– Mather theory and weak solutions to the critical Hamilton–Jacobi equation was Albert Fathi, see [16,17], with the so-called weak KAM theory (1996); he thoroughly investigated the PDE counterpart of the dynamical phenom-
Hamilton–Jacobi Equations and Weak KAM Theory
ena occurring at the critical level. However, his investigation is still within the framework of the dynamical systems theory, requires the Hamiltonian to be at least C2 , and requires the existence of associated Hamiltonian flow as well. The work of Fathi and Siconolfi (2005) [19,20], completely bypassed such assumptions and provided a geometrical analysis of the critical equation independent of the flow, which made it possible to deal with nonregular Hamiltonians. The new idea being the introduction of a length functional, intrinsically related to H, for any curve, and of the related distance, as well. The notion of the Aubry set was suitably generalized to this broader setting. Other important contributions in bridging the gap between PDE and the dynamical viewpoint have been made by Evans and Gomes [14,15]. The material herein is organized as follows: the Sects. “Subsolutions”, “Solutions” are of introductory nature and illustrate the notions of viscosity (sub)solution with their basic properties. Some fundamental techniques used in this framework are introduced as well. Section “First Regularity Results for Subsolutions” deals with issues concerning regularity of subsolutions to (1). The key notion of the Aubry set is introduced in Sect. “Critical Equation and Aubry Set” in connection with the investigation of the critical equation, and a qualitative analysis of it, specially devoted to looking into metric and dynamical properties, is performed in Sects. “An Intrinsic Metric”, “Dynamical Properties of the Aubry Set”. Sections “Long-Time Behavior of Solutions to the Time-Dependent Equation”, “Main Regularity Result” present the main results relative to the long-time behavior of solutions to (2) and the existence of C1 subsolutions to (1). Finally, Sect. “Future Directions” gives some ideas of possible developments in the topic.
@fp : H(x; p) ag D fp : H(x; p) D ag
(6)
where @, in the above formula, indicates the boundary. The a-sublevel of the Hamiltonian appearing in (5) will be denoted by Z a (x). It is a consequence of the coerciveness and convexity assumptions on H that the set-valued map x 7! Z a (x) possesses convex compact values and, in force of (3), is upper semicontinuous; it is in addition continuous at any point x where int Z a (x) ¤ ;. Here (semi)continuous must be understood with respect to the Hausdorff metric. Next will be given four different definitions of weak subsolutions to Eq. (1) and their equivalence will be proved. From this it can be seen that the family of functions so detected is intrinsically related to the equation. As a matter of fact, it will be proved, under more stringent assumptions, that this family is the closure of the classical (i. e. C1 ) subsolutions in the locally uniform topology. Some notations and definitions must be preliminarily introduced. Given two continuous functions u and v, it is said that v is a (strict) supertangent to u at some point x0 if such point is a (strict) local maximizer of u v. The notion of subtangent is obtained by replacing a maximizer with a minimizer. Since (sub, super)tangents are involved in the definition of viscosity solution, they will be called in the sequel viscosity test functions. It is necessary to check: Proposition 1 Let u be a continuous function possessing both C1 supertangents and subtangents at a point x0 , then u is differentiable at x0 . Recall that by Rademacher Theorem a locally Lipschitz function is differentiable almost everywhere (for short a.e.), with respect to the Lebesgue measure. For such a function w the (Clarke) generalized gradient at any point x is defined by @w(x) D cofp D lim D w(x i ) : x i i
Subsolutions First I will detail the basic conditions postulated throughout the paper for H. Additional properties required for obtaining some particular result, will be introduced when needed. The Hamiltonian is assumed to be continuous in both variables ;
(3)
to satisfy the coercivity assumption f(x; p) : H(y; p) ag is compact for any a
(4)
and the following quasiconvexity conditions for any x 2 M, a 2 R fp : H(x; p) ag is convex
(5)
differentiability point of u ; lim x i D xg ; i
where co denotes the convex hull. Remark 2 Record for later use that this set of weak derivatives can be retrieved even if the differentiability points are taken not in the whole ground space, but just outside a set of vanishing Lebesgue measure. The generalized gradient is nonempty at any point; if it reduces to a singleton at some x, then the function w is strictly differentiable at x, i. e. it is differentiable and Du is continuous at x. The set-valued function x 7! @w(x) possesses convex compact values and is upper semicontinu-
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ous. The following variational property holds: 0 2 @w(x) at any local minimizer or maximizer of w ; (7) furthermore, if @(w
is C1 then
)(x) D @w(x) D (x) :
(8)
First definition of weak subsolution A function u is said to be an a.e. subsolution to (1) if it is locally Lipschitz-continuous and satisfies
˛ :D inffjx yj ; x 2 B1 ; y 2 @B2 g > 0 ; and choose an l0 such that
H(x; Du(x)) a for x in a subset of M with full measure : Second definition of weak subsolution A function u is said to be a viscosity subsolution of first type to (1) if it is continuous and H(x0 ; D (x0 )) a ;
jZ a j1;B 2 < l0
(10)
sup u inf u l0 ˛ < 0
(11)
B2
B1
Since u is not Lipschitz-continuous on B1 , a pair of points x0 , x1 in B1 can be found satisfying u(x1 ) u(x0 ) > l0 jx1 x0 j ;
or equivalently D (x0 ) 2 Z a (x0 ) ; for any x0 2 M, any C1 supertangent
to u at x0 .
The previous definition can be equivalently rephrased by taking test functions of class Ck , 1 < k C1, or simply differentiable, instead of C1 . Third definition of weak subsolution A function u is a viscosity subsolution of second type if it satisfies the previous definition with subtangent in place of supertangent. Fourth definition of weak subsolution A function u is a subsolution in the sense of Clarke if @u(x) Z a (x) for all x 2 M : Note that this last definition is unique based on a condition holding at any point of M, and not just at the differentiability points of u or at points where some test function exists. This fact will be exploited in the forthcoming definition of strict subsolution. Proposition 3 The previous four definitions are equivalent. We first show that the viscosity subsolutions of both types are locally Lipschitz-continuous. It is exploited that Za , being upper semicontinuous, is also locally bounded, namely for any bounded subset B of M there is a positive r with Z a (x) B(0; r)
where B(0; r) is the Euclidean ball centered at 0 with radius r. The minimum r for which (9) holds true will be indicated by jZ a j1;B . The argument is given for a viscosity subsolution of first type, say u; the proof for the others is similar. Assume by contradiction that there is an open bounded domain B1 where u is not Lipschitz-continuous, then consider an open bounded domain B2 containing B1 such that
for x 2 B ;
(9)
(12)
which shows that the function x 7! u(x) u(x0 ) l0 jx x0 j), has a positive supremum in B2 . On the other hand such a function is negative on @B2 by (11), and so it attains its maximum in B2 at a point x 2 B2 , or, in other terms, x 7! u(x0 ) C l0 jx x0 j) is supertangent to u at x ¤ x0 . Consequently l0
x1 x0 2 Z a (x1 ) ; jx1 x0 j
in contradiction with (10). Since at every differentiability point u can be taken as a test function of itself, then any viscosity subsolution of both types is also an a.e. subsolution. The a.e. subsolutions, in turn, satisfy the fourth definition above, thanks to the definition of generalized gradient, Remark 2, and the fact that H is convex in p and continuous in both arguments. Finally, exploit (7) and (8) to see that the differential of any viscosity test function to u at some point x0 is contained in @u(x0 ). This shows that any subsolution in the Clarke sense is also a viscosity subsolution of the first and second type. In view of Proposition 3, from now on any element of this class of functions will be called a subsolution of (1) without any further specification, similarly the notion of a subsolution in an open subset of M can be given. Note that for any subsolution u, any bounded open domain B, the quantity jZ a j1;B is a Lipschitz constant in B for every subsolution, consequently the family of all subsolutions to (1) is locally equiLipschitz-continuous.
Hamilton–Jacobi Equations and Weak KAM Theory
A conjugate Hamiltonian Hˇ can be associated to H, it is defined by ˇ H(x; p) D H(x; p) for any x, p :
(13)
Note that Hˇ satisfies, as H does, assumptions (3)–(5). The two corresponding conjugate Hamilton–Jacobi equations have the same family of subsolutions, up to a change of sign, as is apparent looking at the first definition of subsolution. Next we will have a closer look at the family of subsolutions to (1), denoted from now on by S a ; the properties deduced will be exploited in the next sections. Advantage is taken of this to illustrate a couple of basic arguments coming from viscosity solutions theory. Proposition 4 The family S a is stable with respect to the local uniform convergence. The key point in the proof of this result is to use the same C1 function, at different points, for testing the limit as well as the approximating functions. This is indeed the primary trick to obtaining stability properties in the framework of viscosity solutions theory. Let un be a sequence in S a and u n ! u locally uniformly in M. Let be a supertangent to u at some point x0 , it can be assumed, without loss of generality, that is a strict subtangent, by adding a suitable quadratic term. Therefore, there is a compact neighborhood U of x0 where x0 itself is the unique maximizer of u . Any sequence xn of maximizers of u n in U converges to a maximizer of u , and so x n ! x0 , and consequently lies in the interior of U for n large enough. In other terms is supertangent to un at xn , when n is sufficiently large. Consequently H(x n ; D (x n ) a, which implies, exploiting the continuity of the Hamiltonian and passing at the limit, H(x0 ; D (x0 )) a, as desired. Take into account the equiLipschitz character of subsolutions to (1), and the fact that the subsolution property is not affected by addition of a constant, to obtain by slightly adjusting the previous argument, and using the Ascoli Theorem: Proposition 5 Let u n 2 S a n , with an converging to some a. Then the sequence un converges to u 2 S a , up to addition of constants and extraction of a subsequence. Before ending the section, a notion which will have some relevance in what follows will be introduced. A subsolution u is said to be strict in some open subset ˝ M if @u(x) int Z a (x) for any x 2 ˝ ;
where int stands for interior. Since the multivalued map x 7! @u(x) is upper semicontinuous, this is equivalent to ess sup˝ 0 H(x; Du(x)) < a for any ˝ 0 compactly contained in ˝ ; where the expression compactly contained means that the closure of ˝ 0 is compact and is contained in ˝. Accordingly, the maximal (possibly empty) open subset W u where u is strict is given by the formula Wu :D fx : @u(x) int Z a (x)g :
(14)
Solutions Unfortunately the relevant stability properties pointed out in the previous section for the family of subsolutions, do not hold for the a.e. solutions, namely the locally Lipschitzcontinuous functions satisfying the equation up to a subset of M with vanishing measure. Take, for instance, the sequence un in T 1 , obtained by linear interpolation of un un
k 2n
k 2n
D 0 for k even, 0 k 2n
D
1 2n
for k odd, 0 k 2n ;
then it comprises a.e. solutions of (1), with H(x; p) D jpj and a D 1. But its uniform limit is the null function, which is an a.e. subsolution of the same equation, according to Proposition 4, but fails to be an a.e. solution. This lack of stability motivates the search for a stronger notion of weak solution. The idea is to look at the properties of S a with respect to the operations of sup and inf. Proposition 6 Let S˜ S a be a family of locally equibounded functions, then the function defined as the pointwise supremum, or infimum, of the elements of S˜ is a subsolution to (1). Set u(x) D inffv(x) : v 2 S˜g. Let , un be a C1 subtangent to u at a point x0 , and a sequence of functions in S˜ with u n (x0 ) ! u(x0 ), respectively. Since the sequence un is made up of locally equibounded and locally equiLipschitz-continuous functions, it locally uniformly converges, up to a subsequence, by Ascoli Theorem, to a function w which belongs, in force of Proposition 4, to S a . In addition w(x0 ) D u(x0 ), and w is supertangent to u at x0 by the very definition of u. Therefore, is also subtangent to v at x0 and so H(x0 ; D (x0 )) a, which shows the assertion. The same
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Hamilton–Jacobi Equations and Weak KAM Theory
proof, with obvious adaptations, allows us to handle the case of the pointwise supremum. Encouraged by the previous result, consider the subsolutions of (1) enjoying some extremality properties. A definition is preliminary. A family S˜ of locally equibounded subsolutions to (1) is said to be complete at some point x0 if there exists "x 0 such that if two subsolutions u1 , u2 agree outside some neighborhood of x0 with radius less than "x 0 and u1 2 S˜ then u2 2 S˜. The interesting point in the next proposition is that the subsolutions which are extremal with respect to a complete family, possess an additional property involving the viscosity test functions. Proposition 7 Let u be the pointwise supremum (infimum) of a locally equibounded family S˜ S a complete at a point x0 , and let be a C1 subtangent (supertangent) to u at x0 . Then H(x0 ; D (x0 )) D a. Only the case where u is a pointwise supremum will be discussed. The proof is based on a push up method that will be again used in the sequel. If, in fact, the assertion were not true, there should be a C1 strict subtangent at x0 , with (x0 ) D u(x0 ), such that H(x0 ; D (x0 )) < a. The function , being C1 , is a (classical) subsolution of (1) in a neighborhood U of x0 . Push up a bit the test function to define ( maxf C " ; ug in B(x0 ; ") vD (15) u otherwise with the positive constant " chosen so that B(x0 ; ") U and " < "x 0 , where "x 0 is the quantity appearing in the definition of a complete family of subsolutions at x0 . By the Proposition 6, the function v belongs to S a , and is equal to u 2 S˜ outside B(x0 ; "). Therefore, v 2 S˜, which is in contrast with the maximality of u because v(x0 ) > u(x0 ). Proposition 7 suggests the following two definitions of weak solution in the viscosity sense, or viscosity solutions for Eq. (1). The function u is a viscosity solution of the first type if it is a subsolution and for any x0 , any C1 subtangent to u at x0 one has H(x0 ; D (x0 )) D a. The viscosity solutions of the second type are definite by replacing the subtangent with supertangent. Such functions are clearly a.e. solutions. The same proof of Proposition 4, applied to C1 subtangents as well as supertangents, gives: Proposition 8 The family of viscosity solutions (of both types) to (1) is stable with respect the local uniform convergence. Moreover, the argument of Proposition 6, with obvious adaptations, shows:
Proposition 9 The pointwise infimum (supremum) of a family of locally equibounded viscosity solutions of the first (second) type is a viscosity solution of the first (second) type to (1). Morally, it can be said that the solutions of the first type enjoy some maximality properties, and some minimality properties hold for the others. Using the notion of the strict subsolution, introduced in the previous section, the following partial converse of Proposition 7 can be obtained: Proposition 10 Let ˝, u, ' be a bounded open subset of M, a viscosity solution to (1) of the first (second) type, and a strict subsolution in ˝ coincident with u on @˝, respectively. Then u ' (u ') in ˝. The proof rests on a regularization procedure of ' by mollification. Assume that u is a viscosity solution of the first type, the other case can be treated similarly. The argument is by contradiction, then admit that the minimizers of u' in ˝ (the closure of ˝) are in an open subset ˝ 0 compactly contained in ˝. Define for x 2 ˝ 0 , ı > 0, Z 'ı (x) D ı (y x)'(y) dy ; where ı is a standard C 1 mollifier supported in B(0; ı). By using the convex character of the Hamiltonian and Jensen Lemma, it can be found Z H(x; D'ı (x)) ı (y x)H(x; D'(y)) dy: Therefore, taking into account the stability of the set of minimizers under the uniform convergence, and that ' is a strict subsolution, ı can be chosen so small that 'ı is a C 1 strict subsolution of (1) in ˝ 0 and, in addition, is subtangent to u at some point of ˝ 0 . This is in contrast with the very definition of viscosity solution of the first type. The above argument will be used again, and explained with some more detail, in the next section. The family of viscosity solutions of first and second type coincide for conjugate Hamilton–Jacobi equations, ˇ up to a change of sign. More with Hamiltonian H and H, precisely: Proposition 11 A function u is a viscosity solution of the first (second) type to ˇ H(x; Du) D a
in M
(16)
if and only if u is a viscosity solution of the second (first) type to (1).
Hamilton–Jacobi Equations and Weak KAM Theory
In fact, if u, are a viscosity solution of the first type to (16) and a C1 supertangent to u at a point x0 , respectively, then u is a subsolution to (1), and is supertangent to u at x0 so that ˇ 0 ; D (x0 )) D H(x0 ; D (x0 )) ; a D H(x which shows that u is indeed a viscosity solution of the second type to (1). The other implications can be derived analogously. The choice between the two types of viscosity solutions is just a matter of taste, since they give rise to two completely equivalent theories. In this article those of the first type are selected, and they are referred to from now on as (viscosity) solutions, without any further specification. Next a notion of regularity is introduced, called semiconcavity (semiconvexity), which fits the viscosity solutions framework, and that will be used in a crucial way in Sect. “Main Regularity Result”. The starting remark is that even if the notion of viscosity solution of the first (second) type is more stringent than that of the a.e. solution, as proved above, the two notions are nevertheless equivalent for concave (convex) functions. In fact a function of this type, say u, which is locally Lipschitz-continuous, satisfies the inequality u(y) () u(x0 ) C p (y x) ; for any x0 , y, p 2 @u(x0 ). It is therefore apparent that it admits (global) linear supertangents (subtangents) at any point x0 . If there were also a C1 subtangent (supertangent), say , at x0 , then u should be differentiable at x0 by Proposition 1, and Du(x0 ) D D (x0 ), so that if u were an a.e. solution then H(x0 ; D (x0 )) D a, as announced. In the above argument the concave (convex) character of u was not exploited to its full extent, but it was just used to show the existence of C1 super(sub)tangents at any point x and that the differentials of such test functions make up @u. Clearly such a property is still valid for a larger class of functions. This is the case for the family of so-called strongly semiconcave (semiconvex) functions, which are concave (convex) up to the subtraction (addition) of a quadratic term. This point will now be outlined for strongly semiconcave functions, a parallel analysis could be performed in the strongly semiconvex case. A function u is said to be strongly semiconcave if u(x) kjx x0 j2 is concave for some positive constant k, some x0 2 M. Starting from the inequality u(x1 C (1 )x2 ) kjx1 C (1 )x2 x0 j2 (u(x1 ) kjx1 x0 j2 ) C (1 )(u(x2 ) kjx2 x0 j2 )
which holds for any x1 , x2 , 2 [0; 1], it is derived through straightforward calculations u(x1 C (1 )x2 ) u(x1 ) (1 )u(x2 ) k(1 )jx1 x2 j2 ;
(17)
which is actually a property equivalent to strong semiconcavity. This shows that for such functions the subtraction of kjx x0 j2 , for any x0 2 M, yields the concavity property. Therefore, given any x0 , and taking into account (8), one has u(x) kjx x0 j2 u(x0 ) C p(x x0 ) ; for any x, any p 2 @u(x0 ) which proves that the generalized gradient of u at x0 is made up by the differentials of the C1 supertangents to u at x0 . The outcome of the previous discussion is summarized in the following statement. Proposition 12 Let u be a strongly semiconcave (semiconvex) function. For any x, p 2 @u(x) if and only if it is the differential of a C1 supertangent (subtangent) to u at x. Consequently, the notions of a.e. solution to (1) and viscosity solution of the first (second) type coincide for this class of functions. In Sect. “Main Regularity Result” a weaker notion of semiconcavity will be introduced, obtained by requiring a milder version of (17), which will be crucial to proving the existence of C1 subsolutions to (1). Even if the previous analysis, and in particular the part about the equivalent notions of subsolutions, is only valid for (1), one can define a notion of viscosity solution for a wider class of Hamilton–Jacobi equations than (1), and even for some second-order equations. To be more precise, given a Hamilton–Jacobi equation G(x; u; ; Du) D 0, with G nondecreasing with respect to the second argument, a continuous function u is called the viscosity solution of it, if for any x0 , any C1 supertangent (subtangent) to u at x0 the inequality G(x0 ; u(x0 ); D (x0 )) () 0 holds true. Loosely speaking the existence of comparison principles in this context is related to the strict monotonicity properties of the Hamiltonian with respect to u or the presence in the equation of the time derivative of the unknown. For instance such principles hold for (2), see Sect. “Long-Time Behavior of Solutions to the Time-Dependent Equation”. Obtaining uniqueness properties for viscosity solutions to (1) is a more delicate matter. Such properties are
689
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Hamilton–Jacobi Equations and Weak KAM Theory
actually related to the existence of strict subsolutions, since this, in turn, allows one to slightly perturb any solution obtaining a strict subsolution. To exemplify this issue, Proposition 10 is exploited to show: Proposition 13 Let ˝, g be an open-bounded subset of M and a continuous function defined on @˝, respectively. Assume that there is a strict subsolution ' to (1) in ˝. Then there is at most one viscosity solution to (1) in ˝ taking the datum g on the boundary. Assume by contradiction the existence of two viscosity solutions u and v with v > u C " at some point of ˝, where " is a positive constant. The function v :D ' C (1 )v is a strict subsolution to (1) for any 2]0; 1], by the convexity assumption on H. Further, can be taken so small that the points of ˝ for which v > u C "2 make up a nonempty set, say ˝ 0 , are compactly contained in ˝. This goes against Proposition 10, because u and v C "2 agree on @˝ 0 and the strict subsolution v C "2 exceeds u in ˝ 0 . First Regularity Results for Subsolutions A natural question is when does a classical subsolution to (1) exist? The surprising answer is that it happens whenever there is a (locally Lipschitz) subsolution, provided the assumptions on H introduced in the previous section are strengthened a little. Furthermore, any subsolution can be approximated by regular subsolutions in the topology of locally uniform convergence. This theorem is postponed to Sect. “Main Regularity Result”. Some preliminary results of regularity for subsolution, holding under the assumptions (3)–(6), are presented below. Firstly the discussion concerns the existence of subsolutions to (1) that are regular, say C1 , at least on some distinguished subset of M. More precisely an attempt is made to determine when such functions can be obtained by mollification of subsolutions. The essential output is that this smoothing technique works if the subsolution one starts from is strict, so that, loosely speaking, some room is left to perturb it locally, still obtaining a subsolution. A similar argument has already been used in the proof of Proposition 10. In this analysis it is relevant the critical level of the Hamiltonian which is defined as the one for which the corresponding Hamilton–Jacobi equation possesses a subsolution, but none of them are strict on the whole ground space. It is also important subset of M, named after Aubry and indicated by A, made up of points around which no critical subsolution (i. e. subsolution to (1) with a D c) is strict.
According to what was previously outlined, to smooth up a critical subsolution around the Aubry set, seems particulary hard, if not hopeless. This difficulty will be overcome by performing a detailed qualitative study of the behavior of critical subsolutions on A. The simple setting to be first examined is when there is a strict subsolution, say u, to (1) satisfying H(x; Du(x)) a "
for a.e. x 2 M, and some " > 0 ; (18)
and, in addition, H is uniformly continuous on M B
T M, whenever B is a bounded subset of R N . In this case the mollification procedure plainly works to supply a regular subsolution. The argument of Proposition 10 can be adapted to show this. Define, for any x, any ı > 0 Z uı (x) D ı (y x)u(y) dy ; where ı is a standard C 1 mollifier supported in B(0; ı), and by using the convex character of the Hamiltonian and Jensen Lemma, get Z H(x; Duı (x)) ı (y x)H(x; Du(y)) dy ; so that if o() is a continuity modulus of H in M B(0; r), with r denoting a Lipschitz constant for u, a ı can be selected in such a way that o(ı) "2 , and consequently uı is the desired smooth subsolution, and is, in addition, still strict on M. Even if condition (18) does not hold on the whole underlying space, the previous argument can be applied locally, to provide a smoothing of any subsolution u, at least in the open subset W u where it is strict (see (14) for the definition of this set), by introducing countable open coverings and associated C 1 partition of the unity. The uniform continuity assumption on H as well as the global Lipschitz-continuity of u can be bypassed as well. It can be proved: Proposition 14 Given u 2 S a with W u nonempty, there exists v 2 S a , which is strict and of class C 1 on W u . Note that the function v appearing in the statement is required to be a subsolution on the whole M. In the proof of Proposition 14 an extension principle for subsolutions will be used that will be explained later. Extension principle Let v and C be a subsolution to (1) and a closed subset of M, respectively. Any continuous extension of vjC which is a subsolution on M n C is also a subsolution in the whole M.
Hamilton–Jacobi Equations and Weak KAM Theory
The argument for showing Proposition 14 will also provide the proof, with some adjustments, of the main regularity result, i. e. Theorem 35 in Sect. “Main Regularity Result”. By the very definition of W u an open neighborhood U x0 , compactly contained in Wu , can be found, for all x 2 Wu , in such a way that H(y; Du(y)) < a"x
for a.e. y 2 U x0 and some "x > 0: (19)
Through regularization of u by means of a C 1 mollifier ı supported in B(0; ı), for ı > 0 suitably small, a smooth function can be then constructed still satisfying (19) in a neighborhood of x slightly smaller than U x0 , say U x . The next step is to extract from fU x g, x 2 Wu , a countable locally finite cover of W u , say fU x i g, i 2 N. In the sequel the notations U i , "i are adopted in place of U x i , "x i , respectively. The regularized function is denoted by ui . Note that such functions are not, in general, subsolutions to (1) on M, since their behavior outside U i cannot be controlled. To overcome this difficulty a C 1 partition of the unity ˇ i subordinated to U i is introduced. The crucial point here is that the mollification parameters, denoted by ı i , can be adjusted in such a way that the uniform distance ju u i j1;U i is as small as desired. This quantity, more precisely, is required to be small with respect to jDˇ1i j1 , 21i and the "j corresponding to indices j such that U j \ U i ¤ ;. In place of 21i one could take the terms of any positive convergent series with sum 1. Define v via the formula (P ˇ i u i in Wu vD (20) u otherwise : Note that a finite number of terms are involved in the sum defining v in W u since the cover fU i g is locally finite. It can P be surprising at first sight that the quantity ˇ i u i , with ui subsolution in U i and ˇ i supported in U i , represents a subsolution to (1), since, by differentiating, one gets X X X D ˇi ui D ˇ i Du i C Dˇ i u i ; and the latter term does not seem easy to handle. The trick is to express it through the formula X X Dˇ i (u i u) ; (21) Dˇ i u i D P which holds true because ˇ i 1, by the very definition P of partition of unity, and so Dˇ i 0. From (21) deduce ˇX ˇ X ˇ ˇ Dˇ i u i ˇ jDˇ i j1 ju u i j1;U i ; ˇ
and consequently, recalling that ju u i j1;U i is small with respect to jDˇ1i j1 D
X
X ˇ i Du i : ˇi ui
Since the Hamiltonian is convex in p, ˇ i is supported in U i and ui is a strict subsolution to (1) in U i , we finally discover that v, defined by (20), is a strict subsolution in W u . Taking into account the extension principle for subsolutions, it is left to show, for proving Proposition 14, that v is continuous. For this, first observe that for any n 2 N the set [ in U i is compact and disjoint from @Wu , and consequently minfi : x 2 U i g ! C1 when x 2 Wu approaches @Wu : This, in turn, implies, since ju u i j1;U i is small compared to 21i , ˇX ˇ X ˇ ˇ ˇ i (x)u i (x) u(x)ˇ ˇ i (x)ju u i j1;U i ˇ fi : x2U i g
X fi : x2U i g
ˇ i (x)
1 ! 0; 2i
whenever x 2 Wu approaches @Wu . This shows the assertion. The next step is to look for subsolutions that are strict in a subset of M as large as possible. In particular a strict subsolution to (1) on the whole M does apparently exist at any level a of the Hamiltonian with a > inffb : H(x; Du) D b has a subsolutiong :
(22)
The infimum on the right-hand side of the previous formula is the critical value of H; it will be denoted from now on by c. Accordingly the values a > c (resp. a < c) will be qualified as supercritical (resp. subcritical). The inf in (22) is actually a minimum in view of Proposition 5. By the coercivity properties of H, the quantity min p H(x; p) is finite for any x, and clearly c sup min H(x; p) ; x
p
which shows that c > 1, but it can be equal to C1 if M is noncompact. In this case no subsolutions to (1) should exist for any a. In what follows it is assumed that c is finite. Note that the critical value for the conjugate Hamiltonian Hˇ does not change, since, as already noticed in
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Hamilton–Jacobi Equations and Weak KAM Theory
Sect. “Subsolutions”, the family of subsolutions of the two corresponding Hamilton–Jacobi equations are equal up to a change of sign. From Proposition 14 can be derived: Theorem 15 There exists a smooth strict subsolution to (1) for any supercritical value a.
If x belongs to the full measure set where v and all the vi are differentiable, one finds, by exploiting the convex character of H 1 0 X X i Du i (x)A i H(x; Du i (x)) H @x; in
in
0
C @1
Critical Equation and Aubry Set
(23)
A significant progress in the analysis is achieved by showing that there is a critical subsolution v with W v , see (14) for the definition, enjoying a maximality property. More precisely the following statement holds: Proposition 16 There exists v 2 Sc with Wv D W0 :D
[
fWu : u is a critical subsolutiong :
This result, combined with Proposition 14, gives the Proposition 17 There exists a subsolution to (23) that is strict and of class C 1 on W 0 . To construct v appearing in the statement of Proposition 16 a covering technique to W 0 , as in Proposition 14, is applied and then the convex character of H is exploited. Since no regularity issues are involved, there is no need to introduce smoothing procedures and partitions of unity, so the argument is altogether quite simple. Any point y 2 W0 possesses a neighborhood U y where some critical subsolution vy satisfies H(x; Dv y (x) c" y
1 i A H(x; 0) ;
in
Here the attention is focused on the critical equation H(x; Du) D c :
X
for a.e. x 2 U y , some positive " y :
A locally finite countable subcover fU y i g, i 2 N, can be extracted, the notations U i , vi , "i are used in place of U y i , v y i , " y i . The function v is defined as an infinite convex P 1 v. combination of ui , more precisely v D 2i i To show that v has the properties asserted in the statement, note that the functions vi are locally equiLipschitz-continuous, being critical subsolutions, and can be taken, in addition, locally equibounded, up to addition of P P a constant. The series i u i , i Du i are therefore locally uniformly convergent by the Weierstrass M-test. This shows that the function v is well defined and is Lipschitzcontinuous, in addition X Dv(x) D i Du i (x) for a.e. x :
for any fixed n. This implies, passing to the limit for n ! C1 ! 1 X i Du i (x) H(x; Dv(x)) D H x; iD1
1 X
i H(x; Du i (x)) :
iD1
The function v is thus a critical subsolution, and, in addition, one has X H(x; Dv(x)) H(x; Dv i (x)) C j H(x; Dv j (x)) i¤ j (24) c j"j ; for any j and a.e. x 2 U j . This yields Proposition 16 since fU j g, j 2 N, is a locally finite open cover of W 0 , and so it comes from (24) that the essential sup of v, on any open set compactly contained in W 0 , is strictly less that c. The Aubry set A is defined as M n W0 . According to Propositions 14, 16 it is made up by the bad points around which no function of Sc can be regularized through mollification still remaining a critical subsolution. The points of A are actually characterized by the fact that no critical subsolution is strict around them. Note that a local as well as a global aspect is involved in such a property, for the subsolutions under investigation must be subsolutions on the whole space. Note further that A is also the Aubry set ˇ for the conjugate critical equation with Hamiltonian H. A qualitative analysis of A is the main subject of what follows. Notice that the Aubry set must be nonempty if M is compact, since, otherwise, one could repeat the argument used for the proof of Proposition 16 to get a finite open cover fU i g of M and a finite family ui of critical subsolutions satisfying H(x; Du i (x) c " i
for a.e. x 2 U i and some " i > 0; P and to have for a finite convex combination u D i i u i H(x; Du(x)) c minf i " i g ; i
Hamilton–Jacobi Equations and Weak KAM Theory
in contrast with the very definition of critical value. If, on the contrary, M is noncompact Hamiltonian such that the corresponding Aubry set is empty, then it can be easily exhibited. One example is given by H(x; p) D jpj f (x), in the case where the potential f has no minimizers. It is easily seen that the critical level is given by inf M f , since, for a less than this value, the sublevels Z a () are empty at some x 2 M and consequently the corresponding Hamilton–Jacobi equation does not have any subsolution; on the other side M
which shows that any constant function is a strict critical subsolution on M. This, in turn, implies the emptiness of A. In view of Proposition 14, one has: Proposition 18 Assume that M is noncompact and the Aubry set is empty, then there exists a smooth strict critical subsolution. The points y of A are divided into two categories according to whether the sublevel Z c (y) has an empty or nonempty interior. It is clear that a point y with int Z c (y) D ; must belong to A because for such a point for all p 2 Z c (y) ;
y
S˜a D fu 2 S a : u(y) D 0g ;
(26)
and to define y
w a (x) D sup u(x) ;
(27)
y
S˜a
y
H(x; 0) D f (x) < inf f ;
H(y; p) D c
the push-up argument introduced in Sect. “Subsolutions” for proving Proposition 7. By using the previous proposition, the issue of the existence of (viscosity) solutions to (23) or, more generally, to (1) can be tackled. The starting idea is to fix a point y in M, to consider the family
(25)
and, since any critical subsolution u must satisfy @u(y)
Z c (y), it cannot be strict around y. These points are called equilibria, and E indicates the set of all equilibria. The reason for this terminology is that if the regularity assumptions on H are enough to write the Hamilton’s equations on T M, then (y; p0 ) is an equilibrium of the related flow with H(y; p0 ) D c if and only if y 2 E and Z c (y) D fp0 g. This point of view will not be developed further herein. From now on the subscript c will be omitted to ease notations. Next the behavior of viscosity test functions of any critical subsolution at points belonging to the Aubry set is investigated. The following assertion holds true: Proposition 19 Let u, y, be a critical subsolution, a point of the Aubry set and a viscosity test function to u at y, respectively. Then H(y; D (y)) D c. Note that the content of the proposition is an immediate consequence of (25) if, in addition, y is an equilibrium. In the general case it is not restrictive to prove the statement when is a strict subtangent. Actually, if the inequality H(y; D (y)) < c takes place, a contradiction is reached by constructing a subsolution v strict around y by means of
Since S˜a is complete (this terminology was introduced in y Sect. “Solutions”) at any x ¤ y, the function w a is a subsolution to (1) on M, and a viscosity solution to M n fyg, by Propositions 6, 7. If a D c, and the point y belongs to A then, in view y of Proposition 19, w c is a critical solution on the whole M. On the contrary, the fact that y 62 A i. e. y 2 W0 , prevents this function from being a global solution. In fact in this case, according to Propositions 14, 16, there is a critical subsolution ', which is smooth and strict around y, and it can be also assumed, without any loss of generality, y to vanish at y. Therefore, ' is subtangent to w c at y for the y maximality property of w c and H(y; D'(y)) < c. A characterization of the Aubry set then follows: First characterization of A A point y belongs to the y Aubry set if and only if the function w c , defined in (27) with a D c, is a critical solution on the whole M. If A ¤ ;, which is true when M is compact, then the existence of a critical solution can be derived. Actually in the compact case the critical level is the unique one for which a viscosity solution to (1) does exist. If, in fact a > c, then, by Theorem 15, the equation possesses a smooth strict critical subsolution, say ', which is subtangent to any other function f defined on M at the minimizers of f u, which do exist since M is assumed to be compact. This rules out the possibility of having a solution of (1) since H(x; D'(x)) < a for any x. Next is discussed the issue that in the noncompact case a solution does exist at the critical as well as at any supercritical level. Let a be supercritical. The idea is to exploit the noncompact setting, and to throw away the points where the property of being a solution fails, by letting them go to infinity. y Let w n :D w a n be a sequence of subsolutions given by (27), with jy n j ! C1. The wn are equiLipschitzcontinuous, being subsolutions to (1), and locally equibounded, up to the addition of a constant. One then gets,
693
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Hamilton–Jacobi Equations and Weak KAM Theory
using Ascoli Theorem and arguing along subsequences, a limit function w. Since the wn are solutions around any fixed point, for n suitably large, then, in view of the stability properties of viscosity solutions, see Proposition 8, w is a solution to (1) around any point of M, which means that w is a viscosity solution on the whole M, as announced. The above outlined properties are summarized in the next statement. Proposition 20
at any point possessing a sublevel with a nonempty interior. The positive homogeneity property implies that the line integral in (28) is invariant under change of parameter preserving the orientation. The intrinsic length ` a is moreover lower semicontinuous for the uniform convergence of a equiLipschitz-continuous sequence of curves, by standard variational argument, see [7]. Let S a denote the length distance associated to ` a , namely S a (y; x) D inff` a () : connects y to xg :
(i) If M is compact then a solution to (1) does exist if and only if a D c. (ii) If M is noncompact then (1) can be solved in the viscosity sense if and only if a c.
An Intrinsic Metric Formula (27) gives rise to a nonsymmetric semidistance S a (; ) by simply putting y
S a (y; x) D w a (x) : This metric viewpoint will allow us to attain a deeper insight into the structure of the subsolutions to (1) as well as of the geometric properties of the Aubry set. It is clear that Sa satisfies the triangle inequality and S a (y; y) D 0 for any y. It fails, in general to be symmetric and non-negative. It will be nevertheless called, from now on, distance to ease terminology. An important point to be discussed is that Sa is a length distance, in the sense that a suitable length functional ` a can be introduced in the class of Lipschitz-continuous curves of M in such a way that, for any pairs of points x and y, S a (y; x) is the infimum of the lengths of curves joining them. Such a length, will be qualified from now on as intrinsic to distinguish it from the natural length on the ground space, denoted by `. It only depends on the corresponding sublevels of the Hamiltonian. More precisely one defines for a (Lipschitz-continuous) curve parametrized in an interval I Z ˙ dt ; (28) ` a () D a (; ) I
where a stands for the support function of the a-sublevel of H. More precisely, the function a is defined, for any (x; q) 2 TM as a (x; q) D maxfp q : p 2 Z a (x)g ; it is accordingly convex and positively homogeneous in p, upper semicontinuous in x, and, in addition, continuous
The following result holds true: Proposition 21 S a and Sa coincide. Note that by the coercivity of the Hamiltonian ` a () r`() for some positive r. Taking into account that the Euclidean segment is an admissible junction between any pair of points, deduce the inequality jS a (y; x)j rjy xj for any y, x ; which, combined with the triangle inequality, implies that the function x 7! S a (y; x) is locally Lipschitz-continuous, for any fixed y. Let y0 , x0 be a pair of points in M. Since u :D S a (y0 ; ) is locally Lipschitz-continuous, one has Z d S a (y0 ; x0 ) D u(x0 ) u(y0 ) D u((t) dt dt I for any curve connecting y0 to x0 , defined in some interval I. It is well known from [8] that d ˙ u((t)) D p(t) (t) dt for a.e. t 2 I, some p(t) 2 @u((t)) ; and, since @u(x) Z a (x) for any x, derive S a (y0 ; x0 ) ` a () for all every curve joining y0 to x0 ; which, in turn, yields the inequality S a (y0 ; x0 ) S a (x0 ; y0 ). The converse inequality is obtained by showing that the function w :D S a (y0 ; ) is a subsolution to (1), see [20,29]. From now on the subscript from Za , Sa and a will be omitted in the case where a D c. It is clear that, in general, the intrinsic length of curves can have any sign. However, if the curve is a cycle such a length must be non-negative, according to Proposition 21, otherwise going several times through the same loop the identity S a 1 would be obtained. This remark will have some relevance in what follows. Proposition 21 allows us to determine the intrinsic metric related to the a-sublevel of the conjugate Hamilˇ denoted by Zˇ a (). Since Zˇ a (x) D Z a (x), for tonian H,
Hamilton–Jacobi Equations and Weak KAM Theory
ˇ the correspondany x, because of the very definition of H, ing support function ˇ satisfies ˇ a (x; q) D a (x; q)
for any x, q :
Therefore, the intrinsic lengths ` a and `ˇa coincide up to a change of orientation. In fact, given , defined in [0; 1], and denoted by (s) D (1 s) the curve with opposite orientation, one has Z 1 ˙ ds ; `ˇa () D a (; ) 0
and using r D 1 t as a new integration variable one obtains Z 1 Z 1 ˙ ds D a (; ) a ( ; ˙ ) dr D ` a ( ): 0
(i) if a curve connects two points belonging to C then the corresponding variation of u is estimated from above by the its intrinsic length because of Proposition 23, and since u coincides with a subsolution to (1) on C, (ii) the same estimate holds true for any pair of points if the curve joining them lies outside C in force of the property that u is a subsolution in M n C. Let " be a positive constant, x, y a pair of points and a curve joining them, whose intrinsic length approximates S a (y; x) up to ". The interval of definition of can be partitioned in such a way that the portion of the curve corresponding to each subinterval satisfies the setting of one of the previous items (i) and (ii). By exploiting the additivity of the intrinsic length one finds
0
u(x) u(y) ` a () S a (y; x) C " ;
This yields Sˇa (x; y) D S a (y; x) for any x, y ; where Sˇa stands for the conjugate distance. The function S a (; y) is thus the pointwise supremum of the family fv : v is a subsolution to (16) and v(y) D 0g ; and accordingly S a (; y) the pointwise infimum of fu 2 S a : u(y) D 0g. Summing up:
Proposition 22 Given u 2 S a , y 2 M, the functions S a (y; ), S a (; y) are supertangent and subtangent, respectively, to u at y. The Extension Principle for subsolutions can now be proved. Preliminarily the fifth characterization of the family of subsolutions is given. Proposition 23 A continuous function u is a subsolution to (1) if and only if u(x) u(y) S a (y; x) for any x, y :
(29)
It is an immediate consequence that any subsolution satisfies the inequality in the statement. Conversely, let be a C1 subtangent to u at some point y. By the inequality (29) the subsolution x 7! S a (y; x) is supertangent to u at y, for any y. Therefore, is subtangent to x 7! S a (y; x) at the same point, and so one has H(y; D (y)) a, which shows the assertion taking into account the third definition of subsolution given in Sect. “Subsolutions”. To prove the Extension Lemma, one has to show that a function w coincident with some subsolution to (1) on a closed set C, and being a subsolution on M n C, is a subsolution on the whole M. The intrinsic length will play a main role here. Two facts are exploited:
and the conclusion is reached taking into account the characterization of a subsolution given by Proposition 23, and the fact that " is arbitrary. To carry on the analysis, it is in order to discuss an apparent contradiction regarding the Aubry set. Let y0 2 A n E and p0 2 int Z(y), then p0 is also in the interior of the c-sublevels at points suitably close to y, say belonging to a neighborhood U of y, since Z is continuous at y. This implies p (x y0 ) `c () ;
(30)
for any x 2 U, any curve joining y to X and lying in U. However, this inequality does not imply by any means that the function x 7! p(x y) is subtangent to x 7! S(y; x) at y. This, in fact, should go against Proposition 19, since H(y; p0 ) < c. The unique way to overcome the contradiction is to admit that, even for points very close to y, the critical distance from y is realized by the intrinsic lengths of curves going out of U. In this way one could not deduce from the inequality (30) the previously indicated subtangency property. This means that S is not localizable with respect to the natural distance, and the behavior of the Hamiltonian in points far from y in the Euclidean sense can affect it. There thus exist a sequence of points xn converging to y and a sequence of curves joining y to xn with intrinsic length approximating S(y; x n ) up to n1 and going out U. By juxtaposition of n and the Euclidean segment from xn to y, a sequence of cycles n can be constructed based on y (i. e. passing through y) satisfying `c ( n ) ! 0 ;
inf `( n ) > 0 : n
(31)
695
696
Hamilton–Jacobi Equations and Weak KAM Theory
This is a threshold situation, since the critical length of any cycle must be non-negative. Next it is shown that (31) is indeed a metric characterization of the Aubry set. Metric characterization of the Aubry set A point y belongs to A if and only if there is a sequence n of cycles based on y and satisfying (31). What remains is to prove that the condition (31) holds at any equilibrium point and, conversely, that if it is true at some y, then such a point belongs to A. If y 2 E then this can be directly proved exploiting that int Z(y) is empty and, consequently the sublevel, being convex, is contained in the orthogonal of some element, see [20]. Conversely, let y 2 M n A, according to Proposition 17, there is a critical subsolution u which is of class C 1 and strict in a neighborhood U of y. One can therefore find a positive constant ı such that Du(x) q (x; q)ı
for any x 2 U, any unit vector q: (32)
Let now be a cycle based on y and parametrized by the Euclidean arc-length in [0; `()], then (t) 2 U, for t belonging to an interval that can be assumed without loss of generality of the form [0; t1 ] for some t1 dist(y; @U) (where dist indicates the Euclidean distance of a point from a set). This implies, taking into account (32) and that is a cycle ˇ ˇ `c () D `c ˇ[0;t1 ] C `c ˇ[t1 ;T] (u((t1 ) u((0)) C ıt1 C (u((T) u((t1 )) ı dist(y; @U))) : This shows that the condition (31) cannot hold for sequences of cycles passing through y. By slightly adapting the previous argument, a further property of the intrinsic critical length, to be used later in Sect. “Long-Time Behavior of Solutions to the Time-Dependent Equation”, can be deduced. Proposition 24 Let M be compact. Given ı > 0, there are two positive constants ˛, ˇ such that any curve lying at a distance greater than ı from A satisfies `c () ˛ C ˇ`() : An important property of the Aubry set is that it is a uniqueness set for the critical equation, at least when the ground space is compact. This means that two critical solutions coinciding on A must coincide on M. More precisely it holds:
Proposition 25 Let M be compact. Given an admissible trace g on A, i. e. satisfying the compatibility condition g(y2 ) g(y1 ) S(y1 ; y2 ) ; the unique viscosity solution taking the value g on A is given by minfg(y) C S(y; ) : y 2 Ag : The representation formula yields indeed a critical solution thanks to the first characterization of the Aubry set and Proposition 9. The uniqueness can be obtained taking into account that there is a critical subsolution which is strict and C 1 in the complement of A (see Proposition 17), and arguing as in Proposition 13. Some information on the Aubry set in the one-dimensional case can be deduced from both the characterizations of A Proposition 26 Assume M to have dimension 1, then (i) if M is compact then either A D E or A D M, (ii) if M is noncompact then A D E , and, in particular, A D ; if E D ;. In the one-dimensional case the c-sublevels are compact intervals. Set Z(x) D [˛(x); ˇ(x)] with ˛, ˇ continuous, and consider the Hamiltonian H(x; p) D H(x; p ˛(x)) : It is apparent that u is a critical (sub)solution for H if and only if u C F, where F is any antiderivative of ˛, is a (sub)solution to H D c, and in addition, u is strict as a subsolution in some ˝ M if and only if u C F is strict in the same subset. This proves that c is also the critical value for H. y y Further, u 2 S˜c for some y, with S˜c defined as in (26), if and only if u C F0 , where F 0 is the antiderivative of ˛ vanishing at y, is in the corresponding family of subsolutions to H D c. Bearing in mind the first characterization of A, it comes that the Aubry sets of the two Hamiltonians H and H coincide. The advantage of using H is that the corresponding critical sublevels Z(x) equal [0; ˇ(x) ˛(x)], for any x, and accordingly the support function, denoted by , satisfies
q (ˇ(x) ˛(x)) if q > 0 (x; q) D 0 if q 0 for any x, q. This implies that the intrinsic critical length related to H, say `c , is non-negative for all curves. Now,
Hamilton–Jacobi Equations and Weak KAM Theory
assume M to be noncompact and take y 62 E , the claim is that y 62 A. In fact, let " > 0 be such that m :D inffˇ(x) ˛(x) : x 2 I" :D]y "; y C "[ g > 0 ; given a cycle based on y, there are two possibilities: either intersects @I" or is entirely contained in I" . In the first case `c () m " ;
(33)
in the second case can be assumed, without loosing generality, to be parametrized by the Euclidean arc-length; since it is a cycle one has Z
`()
˙ ds D 0 ;
0
˙ D 1 for t belonging to a set of one-dimensional so that (t) measure `() 2 . One therefore has `c () m
`() : 2
(34)
Inequalities (33), (34) show that `c () cannot be infinitesimal unless `() is infinitesimal. Hence item (ii) of Proposition 26 is proved. The rest of the assertion is obtained by suitably adapting the previous argument.
curve is contained in A. In fact if such a curve is supported on a point, say x0 , then L(x0 ; 0) D c and, consequently, the critical value is the minimum of p 7! H(x; p). This, in turn, implies, in view of (35), that the sublevel Z(x0 ) has an empty interior so that x0 2 E A. If, on the contrary, a critical curve is nonconstant and x1 , x2 are a pair of different points lying in its support, then S(x1 ; x2 ) C S(x2 ; x1 ) D 0 and there exist two sequences of curves, n and n whose intrinsic length approximates the S(x1 ; x2 ) and S(x2 ; x1 ), respectively. Hence the trajectories obtained through juxtaposition of n and n are cycles with critical length infinitesimal and natural length estimated from below by a positive constant, since they contain x1 and x2 , with x1 ¤ x2 . This at last implies that such points, and so the whole support of the critical curve, are contained in the Aubry set. Next the feature of the parametrization of a critical curve is investigated, since it apparently matters for a curve to be critical. For this purpose let x0 , q0 ¤ 0, and p0 2 Z(x0 ) with (x0 ; q0 ) D p0 q0 , it comes from the definition of the Lagrangian (x0 ; q0 ) c D p0 q0 H(x0 ; p0 ) L(x0 ; q0 ) ; by combining this formula with (37), and recalling the relationship between intrinsic length and distance, one gets L( ; ˙ ) C c D ( ; ˙ )
Dynamical Properties of the Aubry Set In this section the convexity and the coercivity assumptions on H are strengthened and it is required, in addition to (3), H is convex in p lim
jpj!C1
(35)
H(x; p) D C1 uniformly in x: jpj
(36)
The Lagrangian L can be therefore defined through the formula L(x; q) D maxfp q H(x; p) : p 2 R N g : A curve , defined in some interval I, is said to be critical provided that Z
t2
L( ; ˙ )C c ds D S( (t2 ); (t1 )) ;
S( (t1 ); (t2 )) D t1
(37) for any t1 , t2 in I. It comes from the metric characterization of A, given in the previous section, that any critical
for a.e. t :
(38)
A parametrization is called Lagrangian if it satisfies the above equality. As a matter of fact it is possible to prove that any curve , which stays far from E , can be endowed with such a parametrization, see [10]. A relevant result to be discussed next is that the Aubry set is fully covered by critical curves. This property allows us to obtain precious information on the behavior of critical subsolution on A, and will be exploited in the next sections in the study of long-time behavior of solutions to (2). More precisely the following result can be shown: Theorem 27 Given y0 2 A, there is a critical curve, defined in R, taking the value y0 at t D 0. If y0 2 E then the constant curve (t) y0 is critical, as pointed out above. It can therefore be assumed y0 62 E . It is first shown that a critical curve taking the value y0 at 0, and defined in a bounded interval can be constructed. For this purpose start from a sequence of cycles n , based on y0 , satisfying the properties involved in the metric characterization of A, and parametrized by (Euclidean) arc length in [0; Tn ], with Tn D `( n ). By exploiting Ascoli Theorem, and arguing along subsequences, one obtains a uniform limit curve of the n , with (0) D y0 ,
697
698
Hamilton–Jacobi Equations and Weak KAM Theory
in an interval [0; T], where T is strictly less than infn Tn . It is moreover possible to show that is nonconstant. A new sequence of cycles n can be defined through juxtapositionˇ of , the Euclidean segment joining (T) to n (T) and n ˇ[T;Tn ] . By the lower semicontinuity of the intrinsic length, the fact that n (T) converges to (T), and consequently the segment between them has infinitesimal critical length, one gets lim `c ( n ) D 0 :
(39)
n
The important thing is that all the n coincide with in [0; T], so that if t1 < t2 < T, S((t1 ); (t2 )) is estimated from above by ˇ `c ˇ[t ;t ] D `c n j[t1 ;t2 ] ; 1
2
and S((t2 ); (t1 )) by the intrinsic length of the portion of
n joining (t2 ) to (t1 ). Taking into account (39) one gets 0 D lim `c ( n ) S((t1 ); (t2 )) C S((t2 ); (t1 )) ; n
which yields the crucial identity S((t2 ); (t1 )) D S((t1 ); (t2 )) :
(40)
ˇ In addition the previous two formulae imply that ˇ
[t 1 ;t 2 ]
is a minimal geodesic whose intrinsic length realizes S((t1 ; (t2 )), so that (40) can be completed as follows: Z t2 ˙ ds D S((t1 ); (t2 )) : (41) (; ) S((t2 ); (t1 )) D
Next is presented a further result on the behavior of critical subsolutions on A. From this it appears that, even in the broad setting presently under investigation (the Hamiltonian is supposed to be just continuous), such subsolutions enjoy some extra regularity properties on A. Proposition 29 Let be a critical curve, there is a negligible set E in R such that, for any critical subsolution u the function u ı is differentiable whenever in R n E and d ˙ u((t)) D ((t); (t)) : dt More precisely E is the complement in R of the set of ˙ where, in addition, is differLebesgue points of (; ) entiable. E has a vanishing measure thanks to Rademacher and Lebesgue differentiability theorem. See [10] for a complete proof of the proposition. The section ends with the statement of a result, that will be used for proving the forthcoming Theorem 34. The proof can be obtained by performing Lagrangian reparametrizations. Proposition 30 Let be a curve defined in [0; 1]. Denote by ) the set of curves obtained through reparametrization of in intervals with right endpoint 0, and for 2 ) indicate by [0; T( )] its interval of definition. One has (Z ) T( )
(L( ; ˙ ) C c) ds : 2 )
` c () D inf
:
0
t1
Finally, has a Lagrangian parametrization, up to a change of parameter, so that it is obtained in the end Z t2 Z t2 ˙ ds D ˙ C c) ds : (; ) (L(; ) t1
t1
This shows that is a critical curve. By applying Zorn lemma can be extended to a critical curve defined in R, which concludes the proof of Theorem 27. As a consequence a first, perhaps surprising, result on the behavior of a critical subsolution on the Aubry set is obtained. Proposition 28 All critical subsolutions coincide on any critical curve, up to an additive constant. If u is such a subsolution and any curve, one has S((t1 ); (t2 )) u((t2 )) u((t1 )) S((t2 ); (t1 )) ; by Proposition 22. If in addition is critical then the previous formula holds with equality, which proves Proposition 28.
Long-Time Behavior of Solutions to the Time-Dependent Equation In this section it is assumed, in addition to (3), (36), M is compact
(42)
H is strictly convex in p :
(43)
A solution of the time-dependent Eq. (2) is said to be stationary if it has the variable-separated form u0 (x) at ;
(44)
for some constant a. Note that if is a supertangent (subtangent) to u0 at some point x0 , then at is supertangent (subtangent) to u0 at at (x0 ; t) for any t, so that the inequality a C H(x0 ; D (x0 )) () 0 holds true. Therefore, u0 is a solution to (1) in M. Since such a solution does exist only when a D c, see Proposition 20, it is the case that in (44) u0 is a critical solution
Hamilton–Jacobi Equations and Weak KAM Theory
and a is equal to c. The scope of this section is to show that any solution to the time-dependent equation uniformly converge to a stationary solution, as t goes to C1. In our setting there is a comparison principle for (2), stating that two solutions v, w, issued from initial data v0 , w0 , with v0 w0 , satisfies v w, from any x 2 M, t > 0. In addition there exists a viscosity solution v, for any continuous initial datum v0 , which is, accordingly, unique, and is given by the Lax–Oleinik representation formula:
Z
t
v(x; t) D inf v0 ((0)) C
for any ". Hence, we exploit the stability properties that have been illustrated in Sects “Subsolutions”, “Solutions”, to prove the claim. The following inequality will be used L(x; q) C c (x; q)
for any x, q;
which yields, by performing a line integration Z
t
(L( ; ˙ ) C c) ds `c ( ) ;
0
is a curve with (t) D x :
(45)
This shows that Eq. (2) enjoys the semigroup property, namely if w and v are two solutions with w(; 0) D v(; t0 ), for some t0 > 0, then w(x; t) D v(x; t0 C t) : It is clear that the solution of (2), taking a critical solution u0 as initial datum, is stationary and is given by (44). For any continuous initial datum v0 it can be found, since the underlying manifold is compact, a critical solution u0 and a pair of constants ˛ > ˇ such that u0 C ˛ > v0 > u0 C ˇ ; and consequently by the comparison principle for (2) u0 C ˛ > v(; t) C ct > u0 C ˇ
(46)
0
˙ ds : L(; )
for any t :
This shows that the family of functions x 7! v(x; t) C ct, for t 0, is equibounded. It can also be proved, see [10], that it is also equicontinuous, so that, by Ascoli Theorem, every sequence v(; t n ) C ct n , for t n ! C1, is uniformly convergent in M, up to extraction of a subsequence. The limits obtained in this way will be called !-limit of v C ct. The first step of the analysis is to show:
for any t > 0, any curve defined in [0; t], moreover, taking into account Lax–Oleinik formula and that M is assumed in this section to be compact v(x; t) C c t v0 (y) C S(y; x) for some y depending on x ;
(47)
for a solution v of (2) taking a function v0 as initial datum. If, in addition, v0 is a critical subsolution, invoke Proposition 23 to derive from (47) v(x; t) v0 (x) c t
for any x, t :
(48)
A crucial point to be exploited, is that, for such a v0 , the evolution induced by (2) on the Aubry set takes place on the critical curves. Given t > 0 and x 2 A, pick a critical curve with (t) D x0 , whose existence is guaranteed by Theorem 27, and then employ Proposition 29, about the behavior of the subsolution to (23) on critical curves, to obtain Z t ˙ ds v(x; t) : (49) L(; ) v0 (x) c t D v0 ((0)) C 0
By combining (49) and (48), one finally has Z
t
v(x; t) D v0 ((0)) C
˙ ds D v0 (x) c t ; L(; )
0
Proposition 31 Let v be a solution to (2). The pointwise supremum and infimum of the !-limit of v C ct are critical subsolutions. The trick is to introduce a parameter " small and consider the functions v " (x; t) D v(x; t/")
which actually shows the announced optimal character of critical curves with respect to the Lax–Oleinik formula, and, at the same time, the following Proposition 32 Let v be a solution to (2) taking a critical subsolution v0 as initial datum at t D 0. Then
for " > 0 :
Arguing as above, it can be seen that the family v " C ct is equibounded, moreover v " C ct is apparently the solution to "w t C H(x; Dw) D c ;
v(x; t) D v0 (x) c t
for any x 2 A :
Summing up: stationary solutions are derived by taking as initial datum solutions to (23); more generally, solutions issued from a critical subsolution are stationary at least on
699
700
Hamilton–Jacobi Equations and Weak KAM Theory
the Aubry set. The next step is to examine the long-time behavior of such solutions on the whole M. Proposition 33 Let v be a solution to (2) taking a critical subsolution v0 as initial datum at t D 0. One has lim v(x; t) C c t D u0 (x) uniformly in x;
t!C1
Theorem 34 Let v be a viscosity solution to (2) taking a continuous function v0 as initial datum for t D 0, then lim v(x; t) C c t D u0 (x) uniformly in x;
t!C1
where u0 is the critical solution given by the formula u0 (x) D inf inf v0 (z) C S(z; y) C S(y; x) : y2A z2M
where u0 is the critical solution with trace v0 on A. The starting remark for getting the assertion is that, for any given x0 , an "-optimal curve for v(x0 ; t0 ), say , must be close to A for some t 2 [0; t0 ], provided " is sufficiently small and t0 large enough. If in fact stayed far from A for any t 2 [0; t0 ] then L((s); 0) C c could be estimated from below by a positive constant, since E A, and the same should hold true, by continuity, for L((s); q) C c if jqj is small. One should then deduce that `(), and consequently (in view of Proposition 24) `c () were large. On the other side
(51)
The claim is that u0 , as defined in (51), is the critical solution with trace w0 :D inf v0 (z) C S(z; )
(52)
z2M
on the Aubry set. This can indeed be deduced from the representation formula given in Proposition 25, once it is proved that w0 is a critical subsolution. This property, in turn, comes from the characterization of critical subsolutions in terms of critical distance, presented in Proposition 23, the triangle inequality for S, and the inequalities w0 (x1 ) w0 (x2 )
u0 (x0 ) v(x0 ; t0 ) C c t0 v0 ((0)) C `c () " ; (50) by (46) and the comparison principle for (2), which shows that the critical length of is bounded from above, yielding a contradiction. It can be therefore assumed that, up to a slight modification, the curve intersects A at a time s0 2 [0; t0 ] and satisfies Z t0 ˙ ds " v(x0 ; t0 ) v0 ((0)) C L(; ) 0 Z t0 ˙ ds " : v((s0 ); s0 )) C L(; ) s0
It is known from Proposition 32 that v((s0 ); s0 )) D v0 ((s0 )) c s0 , so we have from the previous inequality, in view of (46) ˇ v(x0 ; t0 ) v0 ((s0 )) c t0 C `c ˇ[s 0 ;t0 ] " v0 ((s0 )) c t0 C S((s0 ); x0 ) " : Bearing in mind the representation formula for u0 given in Proposition 25, we obtain in the end v(x0 ; t0 ) u0 (x0 ) c t0 " ; and conclude exploiting u0 v0 and the comparison principle for (2). The previous statement can be suitably generalized by removing the requirement of v0 being a critical subsolution. One more precisely has:
v0 (z2 )CS(z2 ; x1 )v0 (z2 )S(z2 ; x2 ) S(x2 ; x1 ) ; which hold true if z2 is a point realizing the infimum for w0 (x2 ). If, in particular, v0 itself is a critical subsolution, then it coincides with w0 , so that, as announced, Theorem 34 includes Proposition 33. In the general case, it is apparent that w0 v0 , moreover if z 2 M and w 0 is a critical subsolution with w 0 v0 , one deduces from Proposition 23 w 0 (x) v0 (z) C S(z; x)
for any z;
which tells that w 0 w0 , therefore w0 is the maximal critical subsolution not exceeding v0 . A complete proof of Theorem 34 is beyond the scope of this presentation. To give an idea, we consider the simplified case where the equilibria set E is a uniqueness set for the critical equation. Given x0 2 E , " > 0, take a z0 realizing the infimum for w0 (x0 ), and a curve , connecting z0 to x0 , whose intrinsic length approximates S(z0 ; x0 ) up to ". By invoking Proposition 30 one deduces that, up to a change of the parameter, such a curve, defined in [0; T], for some T > 0, satisfies Z T ˙ C c) ds < `c () C " : (L(; ) 0
Therefore, taking into account the Lax–Oleinik formula, one discovers Z T ˙ C c) ds 2" w0 (x0 ) v0 (z0 ) C (L(; ) (53) 0 v(x0 ; T) C cT 2" :
Hamilton–Jacobi Equations and Weak KAM Theory
Since L(x0 ; 0) C c D 0, by the very definition of equilibrium, it can be further derived from Lax–Oleinik formula that t 7! v(x0 ; t) is nonincreasing, so that the inequality (53) still holds if T is replaced by every t > T. This, together with the fact that " in (53) is taken arbitrarily, shows in the end, that any !-limit of v C ct satisfies w0 (x0 )
(x0 ) for any x0 2 E :
(54)
On the other side, the initial datum v0 is greater than or equal to w0 , and the solution to (2) with initial datum w0 , say w, has as unique !-limit the critical solution u0 with trace w0 on E [recall that E is assumed to be a uniqueness set for (23)]. Consequently, by the comparison principle for (2), one obtains u0
in M ;
(55)
which, combined with (54), implies that and w0 coincide on E . Further, by Proposition 31, the maximal and minimal are critical subsolutions, and u0 is the maximal critical subsolution taking the value w0 on E . This finally yields D u0 for any , and proves the assertion of Theorem 34. In the general case the property of the set of !-limits of critical curves (i. e. limit points for t ! C1) of being a uniqueness set for the critical equation must be exploited. In this setting the strict convexity of H is essential, in [4,10] there are examples showing that Theorem 34 does not hold for H just convex in p. Main Regularity Result This section is devoted to the discussion of Theorem 35 If the Eq. (1) has a subsolution then it also admits a C1 subsolution. Moreover, the C1 subsolutions are dense in S a with respect to the local uniform convergence. Just to sum up: it is known that a C1 subsolution does exist when a is supercritical, see Theorem 15, and if the Aubry set is empty, see Proposition 18. So the case where a D c and A ¤ ; is left. The starting point is the investigation of the regularity properties of any critical subsolution on A. For this, and for proving Theorem 35 conditions (43), (35) on H are assumed, and (3) is strengthened by requiring H is locally Lipschitz-continuous in both arguments. (56) This regularity condition seems unavoidable to show that the functions S a (y; ) and Sˇa (y; ) D S a (; y) enjoy a weak form of semiconcavity in M n fyg, for all y, namely,
if v is any function of this family, x1 , x2 are different from y and 2 [0; 1], then v(x1 C (1 )x2 ) v(x1 ) (1 )v(x2 ) can be estimated from below by a quantity of the same type as that appearing on the right-hand side of (17), with jx1 x2 j2 replaced by a more general term which is still infinitesimal for jx1 x2 j ! 0. Of course the fundamental property of possessing C1 supertangents at any point different from y and that the set made up by their differentials coincides with the generalized gradient is maintained in this setting. To show the validity of this semiconcavity property, say for S a (y; ), at some point x it is crucial that for any neighborhood U of x suitably small there are curves joining y to x and approximating S a (y; x) which stays in U for a (natural) length greater than a fixed constant depending on U. This is clearly true if x ¤ y and explains the reason why the initial point y has been excluded. However, if a D c and y 2 A, exploiting the metric characterization of the Aubry set, it appears that this restriction on y can be removed, so that the following holds: Proposition 36 Let y 2 A, then the functions S(y; ) and S(; y) are semiconcave (in the sense roughly explained above) on the whole M. This, in particular, implies that both functions possess C1 supertangents at y and their differentials comprise the generalized gradient. From this the main regularity property of critical subsolutions on A are deduced. This reinforces the results given in Propositions 28, 29 under less stringent assumptions. Theorem 37 Every critical subsolution is differentiable on A. All have the same differential, denoted by p(y), at any point y 2 A, and H(y; p(y)) D c : Furthermore, the function y 7! p(y) is continuous on A. It is known from Proposition 22 that S(y; ), S(; y) are supertangent and subtangent, respectively, to every critical subsolution u at any y. If, in particular, y 2 A then S(y; ) and S(; y) admit C1 supertangents and subtangents, respectively, thanks to Proposition 36. This, in turn, implies that u is differentiable at y by Proposition 1 and, in addition, that all the differentials of supertangents to S(y; ) coincide with Du(y), which shows that its generalized gradient reduces to a singleton and so S(y; ) is strictly differentiable at y by Proposition 36. If p(y) denotes the differential of S(y; ) at y, then Du(y) D p(y) for any critical subsolution and, in addition, H(y; p(y)) D c by Proposition 19.
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Finally, the strict differentiability of S(y; ) at y gives that p() is continuous on A. The first application of Theorem 37 is relative to critical curves, and confirm their nature of generalized characteristics. If is such a curve (contained in A by the results of Sect. “Dynamical Properties of the Aubry Set”) and u a critical subsolution, it is known from Proposition 29 that d ˙ D ((t); (t)) ˙ u((t) D p((t)) (t) dt ˙ D L((t); (t)) Cc; for a.e. t 2 R. Bearing in mind the definition of L, it ˙ is deduced that p((t)) is a maximizer of p 7! p (t) H((t); p), then by invoking (7) one obtains: Proposition 38 Any critical curve satisfies the differential inclusion ˙ 2 @ p H(; p()) for a.e. t 2 R : In the statement @ p denotes the generalized gradient with respect to the variable p. The proof of Theorem 35 is now attacked. Combining Proposition 17 and Theorem 37, it is shown that there exists a critical subsolution, say w, differentiable at any point 1 of M, ˇ strict and of class C outside the Aubry set, and with ˇ Dw A D p() continuous. The problem is therefore to adjust the proof of Proposition 14 in order to have continuity of the differential on the whole M. The first step is to show a stronger version of the Proposition 16 asserting that it is possible to find a critical subsolution u, which is not only strict on W0 D M n A, but also strictly differentiable on A. Recall that this means that if y 2 A and xn are differentiability points of u with x n ! y, then Du(x n ) ! Du(y) D p(y). This implies that if p() is a continuous extension of p() in M, then p(x) and Du(x) are close at every differentiability point of u close to A. Starting from a subsolution u enjoying the previous property, the idea is then to use for defining the sought C1 subsolution, say v, the same formula (20) given in Proposition 14, i. e. (P ˇ i u i in W0 vD u in A where ˇ i is a C 1 partition of unity subordinated to a countable locally finite open covering U i of W 0 , and the ui are obtained from u through suitable regularization in U i by mollification. Look at the sketch of the proof of
Proposition 16 for the precise properties of these objects. It must be shown X D ˇ i u i (x n ) ! Du(y) D p(y) ; for any sequence xn of elements of W 0 converging to y 2 A, or equivalently
ˇ X ˇ ˇ ˇ ˇ i u i (x n ) ˇ ! 0 : ˇp(x n ) D One has ˇ X ˇ X ˇ ˇ ˇ i u i (x n ) ˇ ˇ i (x n )jDu i (x n ) ˇp(x n ) D i
p(x n )jDˇ i (x n )ju i (x n ) u(x n )j : The estimates given in the proof of Proposition 14 show that the second term of the right-hand side of the formula is small. To estimate the first term calculate Z jDu i (x n ) p(x n )j ı i (z x n ) jDu(z) p(z)j C jp(z) p(x n )j dz ; and observe first that jDu(z) p(z)j is small, as previously explained, since x n ! y 2 A and z is close to xn , second, that the mollification parameter ı i can be chosen in such a way that jp(z) p(x n )j is also small. What is left is to discuss the density issue of the C1 subsolutions. This is done still assuming a D c and A nonempty. The proof in the other cases is simpler and goes along the same lines. It is clear from what was previously outlined that the initial subsolution u and v obtained as a result of the regularization procedure are close in the local uniform topology. It is then enough to show that any critical subsolution w can be approximated in the same topology by a subsolution enjoying the same property of u, namely being strict in W 0 and strictly differentiable on the Aubry set. The first property is easy to obtain by simply performing a convex combination of w with a C1 subsolution, strict in W 0 , whose existence has been proved above. It can be, in turn, suitably modified in a neighborhood of A in order to obtain the strict differentiability property, see [20]. Future Directions This line of research seems still capable of relevant developments. In particular to provide exact and approximate correctors for the homogenization of Hamilton–Jacobi equations in a stationary ergodic environment, see Lions and Souganidis [26] and Davini and Siconolfi [11,12], or
Hamilton–Jacobi Equations and Weak KAM Theory
in the direction of extending the results about long-time behavior of solutions of time-dependent problems to the noncompact setting, see Ishii [22,23]. Another promising field of utilization is in mass transportation theory, see Bernard [5], Bernard and Buffoni [6], and Villani [30]. The generalization of the model in the case where the Hamiltonian presents singularities should also make it possible to tackle through these techniques the N-body problem. With regard to applications, the theory outlined in the paper could be useful for dealing with topics such as the analysis of dielectric breakdown as well as other models in fracture mechanics. Bibliography 1. Arnold WI, Kozlov WW, Neishtadt AI (1988) Mathematical aspects of classical and celestial mechanics. In: Encyclopedia of Mathematical Sciences: Dynamical Systems III. Springer, New York 2. Bardi M, Capuzzo Dolcetta I (1997) Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston 3. Barles G (1994) Solutions de viscosité des équations de Hamilton–Jacobi. Springer, Paris 4. Barles G, Souganidis PE (2000) On the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J Math Anal 31:925–939 5. Bernard P (2007) Smooth critical subsolutions of the Hamilton– Jacobi equation. Math Res Lett 14:503–511 6. Bernard P, Buffoni B (2007) Optimal mass transportation and Mather theory. J Eur Math Soc 9:85–121 7. Buttazzo G, Giaquinta M, Hildebrandt S (1998) One-dimensional Variational Problems. In: Oxford Lecture Series in Mathematics and its Applications, 15. Clarendon Press, Oxford 8. Clarke F (1983) Optimization and nonsmooth analysis. Wiley, New York 9. Contreras G, Iturriaga R (1999) Global Minimizers of Autonomous Lagrangians. In: 22nd Brazilian Mathematics Colloquium, IMPA, Rio de Janeiro 10. Davini A, Siconolfi A (2006) A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J Math Anal 38:478–502 11. Davini A, Siconolfi A (2007) Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case. to appear in Mathematische Annalen 12. Davini A, Siconolfi A (2007) Hamilton–Jacobi equations in the stationary ergodic setting: existence of correctors and Aubry set. (preprint)
13. Evans LC (2004) A survey of partial differential methods in weak KAM theory. Commun Pure Appl Math 57 14. Evans LC, Gomes D (2001) Effective Hamiltonians and averaging for Hamilton dynamics I. Arch Ration Mech Anal 157: 1–33 15. Evans LC, Gomes D (2002) Effective Hamiltonians and averaging for Hamilton dynamics II. Arch Rattion Mech Anal 161: 271–305 16. Fathi A (1997) Solutions KAM faibles et barrières de Peierls. C R Acad Sci Paris 325:649–652 17. Fathi A (1998) Sur la convergence du semi–groupe de Lax– Oleinik. C R Acad Sci Paris 327:267–270 18. Fathi A () Weak Kam Theorem in Lagrangian Dynamics. Cambridge University Press (to appear) 19. Fathi A, Siconolfi A (2004) Existence of C 1 critical subsolutions of the Hamilton–Jacobi equations. Invent Math 155:363–388 20. Fathi A, Siconolfi A (2005) PDE aspects of Aubry–Mather theory for quasiconvex Hamiltonians. Calc Var 22:185–228 21. Forni G, Mather J (1994) Action minimizing orbits in Hamiltonian systems. In: Graffi S (ed) Transition to Chaos in Classical and Quantum Mechanics. Lecture Notes in Mathematics, vol 1589. Springer, Berlin 22. Ishii H (2006) Asymptotic solutions for large time Hamilton– Jacobi equations. In: International Congress of Mathematicians, vol III. Eur Math Soc, Zürich, pp 213–227 23. Ishii H () Asymptotic solutions of Hamilton–Jacobi equations in Euclidean n space. Anal Non Linéaire, Ann Inst H Poincaré (to appear) 24. Koike S (2004) A beginner’s guide to the theory of viscosity solutions. In: MSJ Memoirs, vol 13. Tokyo 25. Lions PL (1987) Papanicolaou G, Varadhan SRS, Homogenization of Hamilton–Jacobi equations. Unpublished preprint 26. Lions PL, Souganidis T (2003) Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Commun Pure Appl Math 56:1501–1524 27. Roquejoffre JM (2001) Convergence to Steady States of Periodic Solutions in a Class of Hamilton–Jacobi Equations. J Math Pures Appl 80:85–104 28. Roquejoffre JM (2006) Propriétés qualitatives des solutions des équations de Hamilton–Jacobi et applications. Séminaire Bourbaki 975, 59ème anné. Société Mathématique de France, Paris 29. Siconolfi A (2006) Variational aspects of Hamilton–Jacobi equations and dynamical systems. In: Encyclopedia of Mathematical Physics. Academic Press, New York 30. Villani C () Optimal transport, old and new. http://www.umpa. ens-lyon.fr/cvillani/. Accessed 28 Aug 2008 31. Weinan E (1999) Aubry–Mather theory and periodic solutions of the forced Burgers equation. Commun Pure and Appl Math 52:811–828
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Hybrid Control Systems ANDREW R. TEEL1 , RICARDO G. SANFELICE2 , RAFAL GOEBEL3 1 Electrical and Computer Engineering Department, University of California, Santa Barbara, USA 2 Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, USA 3 Department of Mathematics and Statistics, Loyola University, Chicago, USA Article Outline Glossary Notation Definition of the Subject Introduction Well-posed Hybrid Dynamical Systems Modeling Hybrid Control Systems Stability Theory Design Tools Applications Discussion and Final Remarks Future Directions Bibliography Glossary Global asymptotic stability The typical closed-loop objective of a hybrid controller. Often, the hybrid controller achieves global asymptotic stability of a compact set rather than of a point. This is the property that solutions starting near the set remain near the set for all time and all solutions tend toward the set asymptotically. This property is robust, in a practical sense, for well-posed hybrid dynamical systems. (Well-posed) Hybrid dynamical system System that combines behaviors typical of continuous-time and discrete-time dynamical systems, that is, combines both flows and jumps. The system is said to be well-posed if the data used to describe the evolution (consisting of a flow map, flow set, jump map, and jump set) satisfy mild regularity conditions; see conditions (C1)–(C3) in Subsect. “Conditions for Wellposedness”. Hybrid controller Algorithm that takes, as inputs, measurements from a system to be controlled (called the plant) and combines behaviors of continuous-time and discrete-time controllers (i. e. flows and jumps) to produce, as outputs, signals that are to control the plant.
Hybrid closed-loop system The hybrid system resulting from the interconnection of a plant and a controller, at least one of which is a hybrid dynamical system. Invariance principle A tool for studying asymptotic properties of bounded solutions to (hybrid) dynamical systems, applicable when asymptotic stability is absent. It characterizes the sets to which such solutions must converge, by relying in part on invariance properties of such sets. Lyapunov stability theory A tool for establishing global asymptotic stability of a compact set without solving for the solutions to the hybrid dynamical system. A Lyapunov function is one that takes its minimum, which is zero, on the compact set, that grows unbounded as its argument grows unbounded, and that decreases in the direction of the flow map on the flow set and via the jump map on the jump set. Supervisor of hybrid controllers A hybrid controller that coordinates the actions of a family of hybrid controllers in order to achieve a certain stabilization objective. Patchy control Lyapunov functions provide a means of constructing supervisors. Temporal regularization A modification to a hybrid controller to enforce a positive lower bound on the amount of time between jumps triggered by the hybrid control algorithm. Zeno (and discrete) solutions A solution (to a hybrid dynamical system) that has an infinite number of jumps in a finite amount of time. It is discrete if, moreover, the solution never flows, i. e., never changes continuously. Notation Rn denotes n-dimensional Euclidean space. R denotes the real numbers. R denotes the nonnegative real numbers, i. e., R D [0; 1). Z denotes the integers. N denotes the natural numbers including 0, i. e., N D f0; 1; : : :g. Given a set S, S denotes its closure. Given a vector x 2 Rn , jxj denotes its Euclidean vector norm. B is the closed unit ball in the norm j j. Given a set S Rn and a point x 2 Rn , jxj S :D inf y2S jx yj. Given sets S1 ; S2 subsets of Rn , S1 C S2 :D fx1 C x2 j x1 2 S1 ; x2 2 S2 g. A function is said to be positive definite with respect to a given compact set in its domain if it is zero on that compact set and positive elsewhere. When the compact set is the origin, the function will be called positive definite.
Hybrid Control Systems
Given a function h : Rn ! R, h1 (c) denotes its c-level set, i. e. h1 (c) :D fz 2 Rn jh(z) D c g. The double-arrow notation e. g., g : D Rn , indicates a set-valued mapping, in contrast to a single arrow used for functions. Definition of the Subject Control systems are ubiquitous in nature and engineering. They regulate physical systems to desirable conditions. The mathematical theory behind engineering control systems developed over the last century. It started with the elegant stability theory of linear dynamical systems and continued with the more formidable theory of nonlinear dynamical systems, rooted in knowledge of stability theory for attractors in nonlinear differential or difference equations. Most recently, researchers have recognized the limited capabilities of control systems modeled only by differential or difference equations. Thus, they have started to explore the capabilities of hybrid control systems. Hybrid control systems contain dynamical states that sometimes change continuously and other times change discontinuously. These states, which can flow and jump, together with the output of the system being regulated, are used to produce a (hybrid) feedback control signal. Hybrid control systems can be applied to classical systems, where their added flexibility permits solving certain challenging control problems that are not solvable with other methods. Moreover, a firm understanding of hybrid dynamical systems allows applying hybrid control theory to systems that are, themselves, hybrid in nature, that is, having states that can change continuously and also change discontinuously. The development of hybrid control theory is in its infancy, with progress being marked by a transition from ad-hoc methods to systematic design tools. Introduction This article will present a general framework for modeling hybrid control systems and analyzing their dynamical properties. It will put forth basic tools for studying asymptotic stability properties of hybrid systems. Then, particular aspects of hybrid control will be described, as well as approaches to successfully achieving control objectives via hybrid control even if they are not solvable with classical methods. First, some examples of hybrid control systems are given. Hybrid dynamical systems combine behaviors typical of continuous-time dynamical systems (i. e., flows) and behaviors typical of discrete-time dynamical systems (i. e., jumps). Hybrid control systems exploit state variables that may flow as well as jump to achieve control objectives that
are difficult or impossible to achieve with controllers that are not hybrid. Perhaps the simplest example of a hybrid control system is one that uses a relay-type hysteresis element to avoid cycling the system’s actuators between “on” and “off ” too frequently. Consider controlling the temperature of a room by turning a heater on and off. As a good approximation, the room’s temperature T is governed by the differential equation T˙ D T C T0 C T u ;
(1)
where T 0 represents the natural temperature of the room, T represents the capacity of the heater to raise the temperature in the room by being always on, and the variable u represents the state of the heater, which can be either 1 (“on”) or 0 (“off ”). A typical temperature control task is to keep the temperature between two specified values Tmin and Tmax where T0 < Tmin < Tmax < T0 C T : For purposes of illustration, consider the case when Tmin D 70ı F, Tmax D 80ı F. An algorithm that accomplishes this control task is input T, u if u=1 and T >= 80 then u=0 elseif u = 0 and T >
> > > x > 0 (; e) 12 > > x22 = x11 x21 D 0 and > > > x12 x22 0 :
> > > > x ; 0 (; e) 12 > x22
A controller designed to accomplish stabilization of the ball to a periodic pattern will only be able to measure the ball’s state at impacts. During flows, it will be able to control the robot’s velocity through u. Regardless of the nature of the controller, the closed-loop system will be a hybrid system by virtue of the dynamics of the one degree-of-freedom system. Figure 3 shows a trajectory to the closed-loop system with a controller that stabilizes the ball state to the periodic pattern in Fig. 2b (note the discontinuity in the velocity of the ball at impacts); see the control strategy in [55]. Following the modeling techniques illustrated by the examples above, the next section introduces a general modeling framework for hybrid dynamical systems. The framework makes possible the development of a robust stability theory for hybrid dynamical systems and prepares the way for insights into the design of robust hybrid control systems.
For numerous mathematical problems, well-posedness refers to the uniqueness of a solution and its continuous dependence on parameters, for example on initial conditions. Here, well-posedness will refer to some mild regularity properties of the data of a hybrid system that enable the development of a robust stability theory. Hybrid Behavior and Model Hybrid dynamical systems combine continuous and discrete dynamics. Such a combination may emerge when controlling a continuous-time system with a control algorithm that incorporates discrete dynamics, like in the temperature control problem in Sect. “Introduction”, when controlling a system that features hybrid phenomena, like in the juggling problem in Sect. “Introduction”, or as a modeling abstraction of complex dynamical systems. Solutions to hybrid systems (sometimes referred to as trajectories, executions, runs, or motions) can evolve both continuously, i. e. flow, and discontinuously, i. e. jump. Figure 4 depicts a representative behavior of a solution to a hybrid system.
Hybrid Control Systems, Figure 4 Evolution of a hybrid system: continuous motion during flows (solid), discontinuous motion at jumps (dashed)
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For a purely continuous-time system, flows are usually modeled by differential equations, and sometimes by differential inclusions. For a purely discrete-time system, jumps are usually modeled by difference equations, and sometimes by difference inclusions. Set-valued dynamics naturally arise as regularizations of discontinuous difference and differential equations and represent the effect of state perturbations on such equations, in particular, the effect of state measurement errors when the equations represent a system in closed loop with a (discontinuous) feedback controller. For the continuous-time case, see the work by Filippov [22] and Krasovskii [34], as well as [26,27]; for the discrete-time case, see [33]. When working with hybrid control systems, it is appropriate to allow for set-valued discrete dynamics in order to capture decision making capabilities, which are typical in hybrid feedback. Difference inclusions, rather than equations, also arise naturally in modeling of hybrid automata, for example when the discrete dynamics are generated by multiple so-called “guards” and “reset maps”; see, e. g. [8,11,40] or [56] for details on modeling guards and resets in the current framework. Naturally, differential equations and difference inclusions will be featured in the model of a hybrid system. In most hybrid systems, or even in some purely continuoustime systems, the flow modeled by a differential equation is allowed to occur only on a certain subset of the state space Rn . Similarly, the jumps modeled by a difference equation may only be allowed from a certain subset of the state space. Hence, the model of the hybrid system stated above will also feature a flow set, restricting the flows, and a jump set, restricting the jumps. More formally, a hybrid system will be modeled with the following data:
The flow set C Rn ; The flow map f : C ! Rn ; The jump set D Rn ; The (set-valued) jump map G : D Rn .
A shorthand notation for a hybrid system with this data will be H D ( f ; C; G; D). Such systems can be written in the suggestive form ( x˙ D f (x) ; x2C n H : x2R (8) C x 2 G(x) ; x 2 D ; where x 2 Rn denotes the state of the system, x˙ denotes its derivative with respect to time, and x C denotes its value after jumps. In several control applications, the state x of the hybrid system can contain logic states that take value in discrete sets (representing, for example, “on” or “off ”
states, like in the temperature control problem in Sect. “Introduction”). Two parameters will be used to specify “time” in solutions to hybrid systems: t, taking values in R0 , and representing the elapsed “real” time; and j, taking values in N, and representing the number of jumps that have occurred. For each solution, the combined parameters (t; j) will be restricted to belong to a hybrid time domain, a particular subset of R0 N. Hybrid time domains corresponding to different solutions may differ. Note that with such a parametrization, both purely continuous-time and purely discrete-time dynamical systems can be captured. Furthermore, for truly hybrid solutions, both flows and jumps are parametrized “symmetrically” (cf. [8,40]). A subset E of R0 N is a hybrid time domain if it is the union of infinitely many intervals of the form [t j ; t jC1 ] f jg, where 0 D t0 t1 t2 : : : , or of finitely many such intervals, with the last one possibly of the form [t j ; t jC1 ] f jg, [t j ; t jC1 ) f jg, or [t j ; 1) f jg. On each hybrid time domain there is a natural ordering of points: we write (t; j) (t 0 j0 ) for (t; j); (t 0 ; j0 ) 2 E if t t 0 and j j0 . Solutions to hybrid systems are given by functions, which are called hybrid arcs, defined on hybrid time domains and satisfying the dynamics and the constraints given by the data of the hybrid system. A hybrid arc is a function x : dom x ! Rn , where dom x is a hybrid time domain and t 7! x(t; j) is a locally absolutely continuous function for each fixed j. A hybrid arc x is a solution to H D ( f ; C; G; D) if x(0; 0) 2 C [ D and it satisfies Flow condition: x˙ (t; j) D f (x(t; j)) and
(9)
x(t; j) 2 C
for all j 2 N and almost all t such that (t; j) 2 dom x; Jump condition: x(t; j C 1) 2 G(x(t; j)) x(t; j) 2 D
and
(10)
for all (t; j) 2 dom x such that (t; j C 1) 2 dom x. Figure 5 shows a solution to a hybrid system H D ( f ; C; G; D) flowing (as solutions to continuous-time systems do) while in the flow set C and jumping (as solutions to discrete-time systems do) from points in the jump set D. A hybrid arc x is said to be nontrivial if dom x contains at least one point different from (0; 0) and complete
Hybrid Control Systems
Hybrid Control Systems, Figure 5 Evolution of a solution to a hybrid system. Flows and jumps of the solution x are allowed only on the flow set C and on the jump set D, respectively
if dom x is unbounded (in either the t or j direction, or both). It is said to be Zeno if it has an infinite number of jumps in a finite amount of time, and discrete if it has an infinite number of jumps and never flows. A solution x to a hybrid system is maximal if it cannot be extended, i. e., there is no solution x 0 such that dom x is a proper subset of dom x 0 and x agrees with x 0 on dom x. Obviously, complete solutions are maximal.
between solutions. In contrast to purely continuous-time systems or purely discrete-time systems, the uniform metric is not a suitable indicator of distance: two solutions experiencing jumps at close but not the same times will not be close in the uniform metric, even if (intuitively) they represent very similar behaviors. For example, Fig. 6 shows two solutions to the juggling problem in Sect. “Introduction” starting at nearby initial conditions for which the velocities are not close in the uniform metric. A more appropriate distance notion should take possibly different jump times into account. We use the following notion: given T; J; " > 0, two hybrid arcs x : dom x ! Rn and y : dom y ! Rn are said to be (T; J; ")-close if: (a) for all (t; j) 2 dom x with t T, j J there exists s such that (s; j) 2 dom y, jt sj < ", and jx(t; j) y(s; j)j < " ; (b) for all (t; j) 2 dom y with t T, j J there exists s such that (s; j) 2 dom x, jt sj < ", and jy(t; j) x(s; j)j < " :
Conditions for Well-Posedness Many desired results in stability theory for dynamical systems, like invariance principles, converse Lyapunov theorems, or statements about generic robustness of stability, hinge upon some fundamental properties of the space of solutions to the system. These properties may involve continuous dependence of solutions on initial conditions, completeness and sequential compactness of the space of solutions, etc. To begin addressing these or similar properties for hybrid systems, one should establish a concept of distance
An appealing geometric interpretation of (T; J; ")closeness of x and y can be given. The graph of a hybrid arc x : dom x ! Rn is the subset of RnC2 given by ˚ gph x :D (t; j; z) j (t; j) 2 dom x; z D x(t; j) : Hybrid arcs x and y are (T; J; ")-close if the restriction of the graph of x to t T, j J, i. e., the set f(t; j; z)j(t; j) 2 dom x; t T; j J; z D x(t; j)g, is in the "neighborhood of gph y, and vice versa: the restriction of the graph of y is in the "-neighborhood of gph x. (The
Hybrid Control Systems, Figure 6 Two solutions to the juggling system in Sect. “Introduction” starting at nearby initial conditions. Velocities are not close in the uniform metric near the jump times. a Ball’s heights. b Ball’s velocities
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neighborhoods of gph x and gphy should be understood in the norm for which the unit ball is [1; 1][1; 1]B.) The (T; J; ")-closeness can be used to quantify the concept of graphical convergence of a sequence of hybrid arcs; for details, see [23]. Here, it is only noted that graphical convergence of a sequence of mappings is understood as convergence of the sequence of graphs of these mappings. Such a convergence concept does have solid intuitive motivation when the mappings considered are associated with solutions to hybrid systems. It turns out that when (T; J; ")-closeness and graphical convergence are used to study the properties of the space of solutions to a hybrid system H D ( f ; C; G; D), only mild and easy to verify conditions on the data of H are needed to ensure that H is “well-posed”. These conditions are: (C1) the flow set C and jump set D are closed; (C2) the flow map f : C ! Rn is continuous; (C3) the jump map G : D Rn is outer semicontinuous and locally bounded. Only (C3) requires further comment: G : D Rn is outer semicontinuous if for every convergent sequence x i 2 D with x i ! x and every convergent sequence y i 2 G(x i ) with y i ! y, one has y 2 G(x); G is locally bounded if for each compact set K Rn there exists a compact set K 0 Rn such that G(x) K 0 for all x 2 K. Any system H D ( f ; C; G; D) meeting (C1), (C2), and (C3) will be referred to as well-posed. One important consequence of a hybrid system H being well-posed is the following: (?) Every sequence of solutions to H has a subsequence that graphically converges to a solution to H , which holds under very mild boundedness assumptions about the sequence of solutions in question. The assumptions hold, for example, if the sequence is uniformly bounded, i. e., there exists a compact set K Rn such that, for each i, x i (t; j) 2 K for all (t; j) 2 dom x i , where fx i g1 iD1 is a sequence of solutions. Another important consequence of well-posedness is the following outer-semicontinuity (or upper-semicontinuity) property: (??) For every x 0 2 Rn , every desired level of closeness of solutions " > 0, and every (T; J), there exists a level of closeness for initial conditions ı > 0 so that for every solution xı to H with jxı (0; 0) x 0 j < ı, there exist a solution x to H with x(0; 0) D x 0 such that xı and x are (T; J; ")-close.
This property holds at each x 0 2 Rn from which all maximal solutions to H are either complete or bounded. Properties (?) and (??), while being far weaker than any kind of continuous dependence of solutions on initial conditions, are sufficient to develop basic stability characterizations. Continuous dependence of solutions on initial conditions is rare in hybrid systems, as in its more classical meaning it entails uniqueness of solutions from each initial point. If understood in a set-valued sense, in order for inner-semicontinuity (or lower-semicontinuity) to be present, it still requires many further assumptions on the data. Property (?) is essentially all that is needed to establish invariance principles, which will be presented in Subsect. “Invariance Principles”. Property (??) is useful in describing, for example, uniformity of convergence and of overshoots in an asymptotically stable hybrid system. For the analysis of robustness properties of well-posed hybrid systems, strengthened versions of (?) and (??) are available. They take into account the effect of small perturbations; for example a stronger version of (?) makes the same conclusion not about a sequence of solutions to H , but about a sequence of solutions to H generated with vanishing perturbations. (More information can be found in [23].) It is the stronger versions of the two properties that make converse Lyapunov results possible; see Subsect. “Converse Lyapunov Theorems and Robustness”, where more precise meaning to perturbations is also given. In the rest of this article, the analysis results will assume that (C1)–(C3) hold, i. e., the hybrid system under analysis is well-posed. The control algorithms will be constructed so that the corresponding closed-loop systems are well-posed. Such algorithms will be called well-posed controllers. Modeling Hybrid Control Systems Hybrid Controllers for Classical Systems Given a nonlinear control system of the form ( P:
x˙ D f (x; u) ;
x 2 CP
y D h(x) ;
(11)
where CP is a subset of the state space where the system is allowed to evolve, a general output-feedback hybrid controller K D (; ; C K ; ; D K ) takes the form
K:
8 ˆ < u D (y; ) ˙ D (y; ) ; ˆ : C 2 (y; ) ;
(y; ) 2 C K (y; ) 2 D K ;
Hybrid Control Systems
During the rest of the sampling period, it will hold the values of u and z constant.
Hybrid Control Systems, Figure 7 Sample-and-hold control of a nonlinear system
where the output of the plant y 2 R p is the input to the controller, the input to the plant u 2 Rm is the output of the controller, and 2 R k is the controller state. When system (11) is controlled by K , their interconnection results in a hybrid closed-loop system given by ) x˙ D f (x; (h(x); )) (x; ) 2 C ˙ D (h(x); ) (12) ) xC D x (x; ) 2 D ; C 2 (h(x); ) where C :D f(x; ) jx 2 C P ; (h(x); ) 2 C K g D :D f(x; ) j(h(x); ) 2 D K g : We now cast some specific situations into this framework. Sample-and-hold Control Perhaps the simplest example of a hybrid system that arises in control system design is when a continuous-time plant is controlled via a digital computer connected to the plant through a sample-and-hold device. This situation is ubiquitous in feedback control applications. A hybrid system emerges by considering the nonlinear control system (11), with C P D Rn , where the measurements y are sampled every T > 0 seconds, producing a sampled signal ys that is processed through a discrete-time algorithm
C z D (z; y s ) us to generate a sequence of input values us each of which is held for T seconds to generate the input signal u. When combined with the continuous-time dynamics, this algorithm will do the following: At the beginning of the sample period, it will update the value of u and the value of the controller’s internal state z, based on the values of z and y (denoted ys ) at the beginning of the sampling period.
A complete model of this behavior is captured by defining the controller state to be 2 3 z 6 7 :D 4 5 ; where keeps track of the input value to hold during a sampling period and is a timer state that determines when the state variables z and should be updated. The hybrid controller is specified in the form of the previous section as 9 u D > > > > z˙ D 0 = (y; z; ; ) 2 C K ˙ D 0 > > > > ; ˙ D 1 9
C > z D (z; y) = (y; z; ; ) 2 D K ; > ; C D0 where C K :D f(y; z; ; ) j 2 [0; T] g ; D K :D f(y; z; ; ) j D T g : The overall closed-loop hybrid system is given by 9 x˙ D f (x; ) > > > > = z˙ D 0 (x; z; ; ) 2 C > ˙ D 0 > > > ; ˙ D 1 9 xC D x > > > >
C = z (x; z; ; ) 2 D ; D (z; h(x)) > > > > ; C D 0 where C :D f(x; z; ; ) j 2 [0; T] g ; D :D f(x; z; ; ) j D T g : Notice that if the function is discontinuous then this may fail to be a well-posed hybrid system. In such a case, it becomes well-posed by replacing the function by its setvalued regularization \ ¯ y) :D (z; ((z; y) C ıB) : ı>0
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This corresponds to allowing, at points of discontinuity, values that can be obtained with arbitrarily small perturbations of z and y. Allowing these values is reasonable in light of inevitable control systems perturbations, like measurement noise and computer round-off error. Networked Control Systems Certain classes of networked control systems can be viewed as generalizations of systems with a sample-andhold device. The networked control systems generalization allows for multiple sample-and-hold devices operating simultaneously and asynchronously, and with a variable sampling period. Compared to the sample-and-hold closed-loop model in Subsect. “Sample-and-hold Control”, one can think of u as a large vector of inputs to a collection of i plants, collectively modeled by x˙ D f (x; u). The update rule for u may only update a certain part of u at a given jump time. This update rule may depend not only on z and y but perhaps also on u and a logic variable, which we denote by `, that may be cycling through the list of i indices corresponding to connections to different plants. Several common update protocols use algorithms that are discontinuous functions, so this will be modeled explicitly by allowing a set-valued update rule. Finally, due to time variability in the behavior of the network connecting the plants, the updates may occur at any time in an interval [Tmin ; Tmax ] where Tmin > 0 represents the minimum amount of time between transmissions in the network and Tmax > Tmin represents the maximum amount of time between transmissions. The overall control system for the network of plants is given by 9 u D z˙ D 0 > = ˙ D 0 `˙ D 0 (y; z; ; `; ) 2 C K > ; ˙ D 1 9
C > z > 2 (z; y; ; `) > > = (y; z; ; `; ) 2 D K ; `C D (` mod i) C 1 > > > > ; C D 0 where C K :D f(y; z; ; `; ) j 2 [0; Tmax ] g ; D K :D f(y; z; ; `; ) j 2 [Tmin ; Tmax ] g : The closed-loop networked control system has the form 9 > x˙ D f (x; ) > = z˙ D 0 ˙ D 0 (x; z; ; `; ) 2 C > > ; `˙ D 0 ˙ D 1
9 xC D x > > > >
C > > > z 2 (z; h(x); ; `) = > > > `C D (` mod i) C 1 > > > > ; C D0
(x; z; ; `; ) 2 D ;
where C :D f(x; z; ; `; ) j 2 [0; Tmax ] g ; D :D f(x; z; ; `; ) j 2 [Tmin ; Tmax ] g : Reset Control Systems The first documented reset controller was created by Clegg [19]. Consisting of an operational amplifier, resistors, and diodes, Clegg’s controller produced an output that was the integral of its input subject to the constraint that the sign of the output and input agreed. This was achieved by forcing the state of the circuit to jump to zero, a good approximation of the behavior induced by the diodes, when the circuit’s input changed sign with respect to its output. Consider such a circuit in a negative feedback loop with a linear control system x˙ D Ax C Bu ; y D Cx ;
x 2 Rn ; u 2 R
y 2R:
Use to denote the state of the integrator. Then, the hybrid model of the Clegg controller is given by uD ˙ D y ; C
D0;
(y; ) 2 C K (y; ) 2 D K ;
where C K :D f(y; ) jy 0 g ; D K :D f(y; ) jy 0 g : One problem with this model is that it exhibits discrete solutions, as defined in Subsect. “Hybrid Behavior and Model”. Indeed, notice that the jump map takes points with D 0, which are in the jump set DK , back to points with D 0. Thus, there are complete solutions that start with D 0 and never flow, which corresponds to the definition of a discrete solution. There are several ways to address this issue. When a reset controller like the Clegg integrator is implemented through software, a temporal regularization, as discussed next in Subsect. “Zeno Solutions and Temporal Regularization”, can be used to force a small amount of flow time
Hybrid Control Systems
between jumps. Alternatively, and also for the case of an analog implementation, one may consider a more detailed model of the reset mechanism, as the model proposed above is not very accurate for the case where and y are small. This modeling issue is analogous to hybrid modeling issues for a ball bouncing on a floor where the simplest model is not very accurate for small velocities. Zeno Solutions and Temporal Regularization Like for reset control systems, the closed-loop system (12) may exhibit Zeno solutions, i. e., solutions with an infinite number of jumps in a finite amount of time. These are relatively easy to detect in systems with bounded solutions, as they exist if and only if there exist discrete solutions, i. e., solutions with an infinite number of jumps and no flowing time. Discrete solutions in a hybrid control system are problematic, especially from an implementation point of view, but they can be removed by means of temporal regularization. The temporal regularization of a hybrid controller K D (; ; C K ; ; D K ) is generated by introducing a timer variable that resets to zero at jumps and that must pass a threshold defined by a parameter ı 2 (0; 1) before another jump is allowed. The regularization produces a well-posed hybrid controller Kı : D ˜ C K;ı ; ˜ ; D K;ı ) with state ˜ :D (; ) 2 R kC1 , where (˜ ; ; ˜ (˜ ) :D () ˜ (˜ ) :D () f1 g ;
in a practical sense. This aspect is discussed in more detail in Subsect. “Zeno Solutions, Temporal Regularization, and Robustness”. Hybrid Controllers for Hybrid Systems Another interesting scenario in hybrid control is when a hybrid controller 8 ˆ < u D (y; ) (y; ) 2 C K K ˙ D (y; ) ; ˆ : C 2 (y; ) ; (y; ) 2 D K is used to control a plant that is also hybrid, perhaps modeled as x˙ D f (x; u) ; x
C
D g(x) ;
x 2 CP x 2 DP
y D h(x) : This is the situation for the juggling example presented in Sect. “Introduction”. In this scenario, the hybrid closedloop system is modeled as ) x˙ D f (x; (h(x); )) (x; ) 2 C ˙ D (h(x); )
C x 2 G(x; ) ; (x; ) 2 D where
˜ (˜ ) :D
C :D f(x; ) jx 2 C P ; (h(x); ) 2 C K g
C K;ı
D :D f(x; ) jx 2 D P or (h(x); ) 2 D K g
() f0g ; :D C K R0 [ (R k [0; ı]) ;
D K;ı :D D K [ı; 1] : This regularization is related to one type of temporal regularization introduced in [32]. The variable is initialized in the interval [0; 1] and remains there for all time. When ı D 0, the controller accepts flowing only if (y; ) 2 C K , since ˙ D 1 and the flow condition for when (y; ) … C K is D 0. Moreover, jumping is possible for ı D 0 if and only if (y; ) 2 D K . Thus, the controller with ı D 0 has the same effect on the closed loop as the original controller K . When ı > 0, the controller forces at least ı seconds between jumps since ˙ 1 for all 2 [0; ı]. In particular, Zeno solutions, if there were any, are eliminated. Based on the remarks at the end of Subsect. “Conditions for Well-Posedness”, we expect the effect of the controller for small ı > 0 to be close to the effect of the controller with ı D 0. In particular, we expect that this temporal regularization for small ı > 0 will not destroy the stability properties of the closed-loop hybrid system, at least
and
G P (x; ) :D
G K (x; ) :D
g(x)
fxg (h(x); )
8 ˆ ˆ G P (x; ) ; x 2 D P ; (h(x); ) … D K ˆ ˆ < G (x; ) ; x … D ; (h(x); ) 2 D K P K G(x; ) :D ˆ G (x; ) [ G (x; ) ; ˆ P K ˆ ˆ : x 2 D P ; (h(x); ) 2 D K : As long as the data of the controller and plant are wellposed, the closed-loop system is a well-posed hybrid system. Also, it can be verified that the only way this model can exhibit discrete solutions is if either the plant exhibits discrete solutions or the controller, with constant y, exhibits discrete solutions. Indeed, if the plant does not exhibit discrete solutions then a discrete solution for the
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Hybrid Control Systems
closed-loop system would eventually have to have x, and thus y, constant. Then, if there are no discrete solutions to the controller with constant y, there can be no discrete solutions to the closed-loop system. Stability Theory Lyapunov stability theory for dynamical systems typically states that asymptotically stable behaviors can be characterized by the existence of energy-like functions, which are called Lyapunov functions. This theory has served as a powerful tool for stability analysis of nonlinear dynamical systems and has enabled systematic design of robust control systems. In this section, we review some recent advances on Lyapunov-based stability analysis tools for hybrid dynamical systems. Global (Pre-)Asymptotic Stability In a classical setting, say of differential equations with Lipschitz continuous right-hand sides, existence of solutions and completeness of maximal ones can be taken for granted. These properties, together with a Lyapunov inequality ensuring that the Lyapunov function decreases along each solution, lead to a classical concept of asymptotic stability. On its own, the Lyapunov inequality does not say anything about the existence of solutions. It is hence natural to talk about a concept of asymptotic stability that is related only to the Lyapunov inequality. This appears particularly natural for the case of hybrid systems, where existence and completeness of solutions can be problematic. The compact set A Rn is stable for H if for each " > 0 there exists ı > 0 such that any solution x to H with jx(0; 0)jA ı satisfies jx(t; j)jA " for all (t; j) 2 dom x; it is globally pre-attractive for H if any solution x to H is bounded and if it is complete then x(t; j) ! A as t C j ! 1; it is globally pre-asymptotically stable if it is both stable and globally pre-attractive. When every maximal solution to H is complete, the prefix “pre” can be dropped and the “classical” notions of stability and asymptotic stability are recovered. Globally Pre-Asymptotically Stable ˝-Limit Sets Suppose all solutions to the hybrid system are bounded, there exists a compact set S such that all solutions eventually reach and remain in S, and there exists a neighborhood of S from which this convergence to S is uniform. (We are not assuming that the set S is forward invariant and thus it may not be stable.) Moreover, assume there is at least one complete solution starting in S. In this case, the hybrid system admits a nonempty compact set A S that is
globally pre-asymptotically stable. Indeed, one such set is the so-called ˝-limit set of S, defined as ˇ 9 8 ˇ y D lim x i (t i ; j i ) ; > ˆ ˇ > ˆ > ˆ i!1 > ˆ ˇ > ˆ = < ˇ n ˇ t i C j i ! 1 ; (t i ; j i ) 2 dom x i : ˝H (S) :D y 2 R ˇ > ˆ ˇ x is a solution to H > ˆ i > ˆ ˇ > ˆ > ˆ ˇ ; : ˇ with x (0; 0) 2 S i
In fact, this ˝-limit set is the smallest compact set in S that is globally pre-asymptotically stable. To illustrate this concept, consider the temperature control system in Sect. “Introduction” with additional heater dynamics given by h˙ D 3h C (2h C h)u ; where h is the heater temperature and h is a constant that determines how hot the heater can get due to being on. That is, when the heater is “on” (u D 1), its temperature rises asymptotically towards h . There is a maximum temperature h ¯ < h for which the heater can operate safely, and another temperature h < h ¯ corresponding to a temperature far enough below h ¯ that it is considered safe to turn the heater back on. For the desired range of temperatures for T given by Tmin D 70ı F, Tmax D 80ı F and T D 30ı F, let the overheating constant be h D 200ı F, the maximum safe temperature be h ¯ D 150ı F, and the lower temperature h D 50ı F. Then, to keep the temperature T in the desired range and prevent overheating, the following algorithm is used: When the heater is “on” (u D 1) and either T 80 or h 150, then turn the heater off (uC D 0). When the heater is “off ” (u D 0) and T 70 and h 50, then turn the heater on (uC D 1). These rules define the jump map for u, which is given by uC D 1u, and the jump set of the hybrid control system. The resulting hybrid closed-loop system, denoted by H T , is given by
T˙ D T C T0 C T u h˙ D 3h C (2h C h)u u˙ D 0 TC D T hC D h
9 > > =
> > uC D 1 u ;
9 > = > ;
8 u D 1; ˆ ˆ ˆ ˆ ˆ ˆ < (T 80 and h 150) or ˆ ˆ ˆ ˆ u D 0; ˆ ˆ : (T 70 or h 50)
8 ˆ < u D 1; (T 80 or h 150) or ˆ : u D 0; (T 70 and h 50) :
Hybrid Control Systems
For this system, the set S :D [70; 80] [0; 150] f0; 1g
(13)
is not forward invariant. Indeed, consider the initial condition (T; h; u) D (70; 150; 0) which is not in the jump set, so there will be some time where the heater remains off, cooling the room to a value below 70ı F. Nevertheless, all trajectories converge to the set S and a neighborhood of initial conditions around S produce solutions that reach the set S in a uniform amount of time. Thus, the set ˝H T (S) S is a compact globally pre-asymptotically stable set for the system H T . Converse Lyapunov Theorems and Robustness For purely continuous-time and discrete-time systems satisfying some regularity conditions, global asymptotic stability of a compact set implies the existence of a smooth Lyapunov function. Such results, known as converse Lyapunov theorems, establish a necessary condition for global asymptotic stability. For hybrid systems H D ( f ; C; G; D), the conditions for well-posedness also lead to a converse Lyapunov theorem. If a compact set A Rn is globally pre-asymptotically stable for the hybrid system H D ( f ; C; G; D), then there exists a smooth Lyapunov function; that is, there exists a smooth function V : Rn ! R0 that is positive definite with respect to A, radially unbounded, and satisfies hrV(x); f (x)i V(x) 8x 2 C ; V(x) max V(g) 2 g2G(x)
8x 2 D :
Converse Lyapunov theorems are not of mere theoretical interest as they can be used to characterize robustness of asymptotic stability. Suppose that H D ( f ; C; G; D) is well-posed and that V : Rn ! R0 is a smooth Lyapunov function for some compact set A. The smoothness of V and the regularity of the data of H imply that V decreases along solutions when the data is perturbed. More precisely: Global pre-asymptotic stability of the compact set A Rn for H is equivalent to semiglobal practical pre-asymptotic stability of A in the size of perturbations to H , i. e., to the following: there exists a con-
tinuous, nondecreasing in the first argument, nonincreasing in second argument function ˇ : R0 R0 ! R0 with the property that lims&0 ˇ(s; t) D
lim t!1 ˇ(s; t) D 0 and, for each " > 0 and each compact set K Rn , there exists > 0, such that for each perturbation level 2 (0; ] each solution x, x(0; 0) 2 K, to the -perturbed hybrid system ( H :
x2R
n
x˙ 2 F (x) ; x
C
2 G (x) ;
x 2 C
x 2 D ;
where, for each x 2 Rn , F (x) :D co f ((x C B) \ C) C B ; G (x) :D fv 2 Rn jv 2 z C B; z 2 G((x C B) \ D) g ; and C :D fz 2 Rn j(z C B) \ C ¤ ; g ; D :D fz 2 Rn j(z C B) \ D ¤ ; g ; satisfies jx(t; j)jA maxfˇ(jx(0; 0)jA ; tC j); "g 8(t; j) 2 dom x: The above result can be readily used to derive robustness of (pre-)asymptotic stability to various types of perturbations, such as slowly-varying and weakly-jumping parameters, “average dwell-time” perturbations (see [16] for details), and temporal regularizations, as introduced in Subsect. “Zeno Solutions and Temporal Regularization”. We clarify the latter robustness now. Zeno Solutions, Temporal Regularization, and Robustness Some of the control systems we will design later will have discrete solutions that evolve in the set we are trying to asymptotically stabilize. So, these solutions do not affect asymptotic stability adversely, but they are somewhat problematic from an implementation point of view. We indicated in Subsect. “Zeno Solutions and Temporal Regularization” how these solutions arising in hybrid control systems can be removed via temporal regularization. Here we indicate how doing so does not destroy the asymptotic stability achieved, at least in a semi-global practical sense. The assumption is that stabilization via hybrid control is achieved as a preliminary step. In particular, assume that there is a well-posed closed-loop hybrid system H :D ( f ; G; C; D) with state 2 Rn , and suppose that the compact set A Rn is globally pre-asymptotically stable. Following the prescription for a temporal regularization of a hybrid controller in Subsect. “Zeno Solutions and Temporal Regularization”, we consider the hybrid system ˜ Dı ) with the state x : D (; ) 2 RnC1 , Hı :D ( f˜; Cı ; G;
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Hybrid Control Systems
where ı 2 (0; 1) and f˜(x) :D f () f1 g ; ˜ G(x) :D G() f0g ; Cı :D C R0 [ (Rn [0; ı]) ; Dı :D D [ı; 1] : ˜ D0 ) has As observed before, the system H0 D ( f˜; C0 ; G; ˜ the compact set A :D A [0; 1] globally pre-asymptotically stable. When ı > 0, in each hybrid time domain of each solution, each time interval is at least ı seconds long, since ˙ 1 for all 2 [0; ı]. In particular, Zeno solutions, if there were any, have been eliminated. Regarding pre-asymptotic stability, note that ˇ ˚ Cı z 2 RnC1 ˇ (z C ıB) \ C0 6D ; (with B RnC1 ), while Dı D0 . Hence, following the ˜ discussion above, one can conclude that for Hı , the set A is semi-globally practically asymptotically stable in the size of the temporal regularization parameter ı. Broadly speaking, temporal regularization does not destroy (practical) pre-asymptotic stability of A. Lyapunov Stability Theorem A Lyapunov function is not only necessary for asymptotic stability but also sufficient. It is a convenient tool for establishing asymptotic stability because it eliminates the need to solve explicitly for solutions to the system. In its sufficiency form, the requirements of a Lyapunov function can be relaxed somewhat compared to the conditions of the previous subsection. For a hybrid system H D ( f ; C; G; D) the compact set A Rn is globally pre-asymptotically stable if there exists a continuously differentiable function V : Rn ! R that is positive definite with respect to A, radially unbounded, and, with the definitions, ( u c (x) :D ud (x) :D
hrV(x); f (x)i
x2C
otherwise ( 1 max g2G(x) V(g) V(x) x 2 D 1
otherwise ;
(14) (15)
satisfies ud (x) 0 8x 2 Rn u c (x) < 0 ;
ud (x) < 0
8x 2 Rn n A :
(16) (17)
In light of the converse theorem of the previous subsection, this sufficient condition for global asymptotic stability is reasonable. Nevertheless, finding a Lyapunov function is often difficult to do. Thus, there is motivation for stability analysis tools that relax the Lyapunov conditions. There are several directions in which to go. One is in the direction of invariance principles, which are presented next. Invariance Principles An important tool to study the convergence of solutions to dynamical systems is LaSalle’s invariance principle. LaSalle’s invariance principle [35,36] states that bounded and complete solutions converge to the largest invariant subset of the set where the derivative or the difference (depending whether the system is continuous-time or discrete-time, respectively) of a suitable energy function is zero. In situations where the condition (17) holds with nonstrict inequalities, the invariance principle provides a tool to extract information about convergence of solutions. By relying on the sequential compactness property of solutions in Subsect. “Conditions for Well-Posedness”, several versions of LaSalle-like invariance principles can be stated for hybrid systems. Like for continuous-time and discrete-time systems, to make statements about the convergence of a solution one typically assumes that the solution is bounded, that its hybrid time domain is unbounded (i. e., the solution is complete), and that a Lyapunov function does not increase along it. To obtain information about the set to which the solution converges, an invariant set is to be computed. For hybrid systems, since solutions may not be unique, the standard concept of invariance needs to be adjusted appropriately. Following [35], but in the setting of hybrid systems, we will insist that a (weakly) invariant set be both weakly forward invariant and weakly backward invariant. The word “weakly” indicates that only one solution, rather than all, needs to meet some invariance conditions. By requiring both forward and backward invariance we refine the sets to which solutions converge. For a given set M and a hybrid system H , these notions were defined, in [54], as follows: Forward invariance: if for each point x 0 2 M there exists at least one complete solution x to H that starts at x0 and stays in the set M for all (t; j) 2 dom x. Backward invariance: if for each point q 2 M and every positive number N there exists a point x0 from which there exists at least one solution x to H and (t ; j ) 2 dom x such that x(t ; j ) D q and x(t; j) 2 M for all (t; j) 2 dom x ; (t; j) (t ; j ).
Hybrid Control Systems
Then, the following invariance principle can be stated: Let V : Rn ! R be continuously differentiable and suppose that U Rn is nonempty. Let x be a bounded and complete solution to a hybrid system H :D ( f ; C; G; D). If x satisfies x(t; j) 2 U for each (t; j) 2 dom x and u c (z) 0;
ud (z) 0
for all z 2 U ;
then, for some constant r 2 V(U), the solution x approaches the largest weakly invariant set contained in 1 1 u c (0) [ u1 \ V 1 (r) \ U : d (0) \ G u d (0) (18) Note that the statement and the conclusion of this invariance principle resemble the ones by LaSalle for differential/difference equations. In particular, the definition of the set (18) involves both the zero-level set of uc and ud as the continuous and discrete-time counterparts of the principle. For more details and other invariance principles for hybrid systems, see [54]. The invariance principle above leads to the following corollary on global pre-asymptotic stability: For a hybrid system H D ( f ; C; G; D) the compact set A Rn is globally pre-asymptotically stable if there exists a continuously differentiable function V : Rn ! R that is positive definite with respect to A, radially unbounded, such that, with the definitions (14)–(15), u c (x) 0;
ud (x) 0 8x 2 Rn ;
and, for every r > 0, the largest weakly invariant subset in (18) is empty. This corollary will be used to establish global asymptotic stability in the control application in Subsect. “Source Localization”.
and suppose that we have constructed a finite family of well-posed hybrid controllers Kq , that work well individually, on a particular region of the state space. We will make this more precise below. For simplicity, the controllers will share the same state. This can be accomplished by embedding the states of the individual controllers into a common state space. Each controller Kq , with q 2 Q and Q being a finite index set not containing 0, is given by 8 ˆ < u D q (x; ) ˙ D q (x; ) ; Kq ˆ : C 2 q (x; ) ;
(20)
(x; ) 2 D q :
The sets Cq and Dq are such that (x; ) 2 C q [ D q implies x 2 C0 . The union of the regions over which these controllers operate is the region over which we want to obtain robust, asymptotic stability. We define this set as :D [q2Q C q [ D q . To achieve our goal, we will construct a hybrid supervisor that makes decisions about which of the hybrid controllers should be used based on the state’s location relative to a collection of closed sets q C q [D q that cover . The hybrid supervisor will have its own state q 2 Q. The composite controller, denoted K with flow set C and jump set D, should be such that 1) C [ D D Q, 2) all maximal solutions of the interconnection of K with (19), denoted H , starting in Q are complete, 3) and the compact set A Q, a subset of Q, is globally asymptotically stable for the system H . We now clarify what we mean by the family of hybrid controllers working well individually. Let H q denote the closed-loop interconnection of the system (19) with the hybrid controller (20). For each q 2 Q, the solutions to the system H q satisfy: I. The set A is globally pre-asymptotically stable. II. Each maximal solution is either complete or ends in 0 @
Design Tools
(x; ) 2 C q
[
1
i A [ n C q [ D q :
i2Q;i>q
Supervisors of Hybrid Controllers In this section, we discuss how to construct a single, globally asymptotically stabilizing, well-posed hybrid controller from several individual well-posed hybrid controllers that behave well on particular regions of the statespace but that are not defined globally. Suppose we are trying to control a nonlinear system x˙ D f (x; u) ;
x 2 C0
(19)
III. No maximal solution starting in q reaches 2
0
n 4C q [ D q [ @
[
13 i A5 n A :
i2Q;i>q
Item III holds for free for the minimum index qmin since q min C q min [ D q min and [ i2Q i D . The combina-
717
718
Hybrid Control Systems
tion of the three items for the maximum index qmax implies that the solutions to H q max that start in q max converge to A. Intuitively, the hybrid supervisor will attempt to reach its goal by guaranteeing completeness of solutions and making the evolution of q eventually monotonic while (x; ) does not belong to the set A. In this way, the (x; ) component of the solutions eventually converges to A since q is eventually constant and because of the first assumption above. The hybrid supervisor can be content with sticking with controller q as long as (x; ) 2 C q [ D q , it can increment q if (x; ) 2 [ i2Q;i>q i , and it can do anything it wants if (x; ) 2 A. These are the only three situations that should come up when starting from q . Otherwise, the hybrid controller would be forced to decrease the value of q, taking away any guarantee of convergence. This provides the motivation for item III above. Due to a disturbance or unfortunate initialization of q, it may be that the state reaches a point that would not otherwise be reached from q . From such conditions, the important thing is that the solution is either complete (and thus converges to A) or else reaches a point where either q can be incremented or where q is allowed to be decremented. This is the motivation for item II above. The individual hybrid controllers are combined into a single, well-posed hybrid controller K as follows: Define ˚q :D [ i2Q;i>q i and then 8 u D q (x; ) ˆ ˆ ˆ ˆ < ˙ D (x; ) ; (x; ) 2 C˜ q q K
C ˆ ˆ ˆ ˜q ; ˆ 2 G q (x; ) ; (x; ) 2 D : q
(21)
where ˜ q : D D q [ ˚q [ n(C q [ D q ) ; D C˜ q is closed and satisfies C q n˚q C˜ q C q ; and the set-valued mapping Gq is constructed via the following definitions: D q;a :D ˚q D q n˚q D q;b D q D q;c :D n C q [ D q [ ˚q
and
f g fi 2 Q ji > q ; (x; ) 2 i g
q (x; ) G q;b (x; ) :D fqg
f g G q;c (x; ) :D fi 2 Q j(x; ) 2 i g [ G q; j (x; ) : G q (x; ) :D f j2fa;b;cg ; (x;)2D q; j g
G q;a (x; ) :D
(22)
This hybrid controller is well-posed and induces complete solutions from Q and global asymptotic stability of the compact set A Q. Uniting Local and Global Controllers As a simple illustration, consider a nonlinear control system x˙ D f (x; u), x 2 Rn , and the task of globally asymptotically stabilizing the origin using state feedback while insisting on using a particular state feedback 2 in a neighborhood of the origin. In order to solve this problem, one can find a state feedback 1 that globally asymptotically stabilizes the origin and then combine it with 2 using a hybrid supervisor. Suppose that the feedback 2 is defined on a closed neighborhood of the origin, denoted C2 , and that if the state x starts in the closed neighborhood 2 C2 of the origin then the closed-loop solutions when using 2 do not reach the boundary of C2 . Then, using the notation of this section, we can take 1 D C1 D Rn and D1 D D2 D ;. With these definitions, the assumptions above are satisfied and a hybrid controller can be constructed to solve the posed problem. The controller need not use the additional variable . Its data is defined as G q (x) :D 3 q, ˜ 1 :D 2 D ˚1 , D ˜ 2 :D Rn nC2 , C˜1 :D C1 n2 and D ˜ C2 :D C2 . Additional examples of supervisors will appear in the applications section later. Patchy Control Lyapunov Functions A key feature of (smooth) control Lyapunov functions (CLFs) is that their decrease along solutions to a given control system can be guaranteed by an appropriate choice of the control value, for each state value. It is known that, under mild assumptions on the control system, the existence of a CLF yields the existence of a robust (non hybrid) stabilizing feedback. It is also known that many nonlinear control systems do not admit a CLF. This can be illustrated by considering the question of robust stabilization of a single point on a circle, which faces a similar obstacle as the question of robust stabilization of the set A D f0; 1g for
Hybrid Control Systems
the control system on R given by x˙ D f (x; u) :D u. Any differentiable function on R that is positive definite with respect to A must have a maximum in the interval (0; 1). At such a maximum, say x¯ , one has rV(x¯ ) D 0 and no choice of u can lead to hrV(x¯ ); f (x¯ ; u)i < 0. (Smooth) patchy control Lyapunov functions (PCLFs) are, broadly speaking, objects consisting of several local CLFs the domains of which cover Rn and have certain weak invariance properties. PCLFs turn out to exist for far broader classes of nonlinear systems than CLFs, especially if an infinite number of patches (i. e., of local CLFs) is allowed. They also lead to robust hybrid stabilizing feedbacks. This will be outlined below. A brief illustration of the concept, for the control system on R mentioned above, would be to consider functions V1 (x) D x 2 on (1; 2/3) and V2 (x) D (x 1)2 on (1/3; 1). These functions are local CLFs for the points, respectively, 0 and 1; their domains cover R; and for each function, an appropriate choice of control will not only lead to the function’s decrease, but will also ensure that solutions starting in the function’s domain will remain there. While the example just mentioned outlines a general idea of a PCLF, the definition is slightly more technical. For the purposes of this article, a smooth patchy control Lyapunov function for a nonlinear system x˙ D f (x; u)
x 2 Rn ; u 2 U Rm
(23)
with respect to the compact set A consists of a finite set Q Z and a collection of functions V q and sets ˝ q , ˝q0 for each q 2 Q, such that: (i)
f˝q gq2Q and f˝q0 gq2Q are families of nonempty open subsets of Rn such that [ [ ˝q D ˝q0 ; Rn D q2Q
q2Q
and for all q 2 Q, the unit (outward) normal vector S to ˝ q is continuous on @˝q n i>q ˝ i0 , and ˝q0 ˝q ; (ii) for each q, V q is a smooth function defined on S a neighborhood of ˝q n i>q ˝ i0 ; and the following conditions are met: There exist a continuous, positive definite function ˛ : R0 ! R0 , and positive definite, radially unbounded functions , ¯ such that S (iii) for all q 2 Q, all x 2 ˝q n i>q ˝ i0 ,
(jxjA ) Vq (x) ¯ (jxjA ) ;
(iv) for all q 2 Q, all x 2 ˝q n u q;x 2 U such that
S
i>q
˝ i0 , there exists
hrVq (x); f (x; u q;x )i ˛(jxjA ) ; S (v) for all q 2 Q, all x 2 @˝q n i>q ˝ i0 , the u q;x of (iii) can be chosen such that hn q (x); f (x; u q;x )i ˛(jxjA ); where n q (x) is the unit (outward) normal vector to ˝ q at x. Suppose that, for each x; v 2 Rn and c 2 R, the set fu 2 U j hv; f (x; u)i cg is convex, as always holds if f (x; u) is affine in u and U is convex. For each q 2 Q let [ ˝ i0 C q D ˝q n i>q
and
[
q D ˝q0 n
˝ i0 :
i2Q;i>q
It can be shown, in part via arguments similar to those one would use when constructing a feedback from a CLF, that for each q 2 Q there exists a continuous mapping kq : Cq ! U such that, for all x 2 C q , hrVq (x); f (x; k q (x))i
˛(jxjA ) ; 2
all maximal solutions to x˙ D f (x; k q (x)) are either complete or end in 0 1 [ @ i A [ Rn n C q ; i>q;i2Q
and no maximal solution starting in q reaches 1 0 [ Rn n @C q [ i A : i2Q;i>q
The feedbacks kq can now be combined in a hybrid feedback, by taking D q D ; for each q 2 Q, and following the construction of Subsect. “Supervisors of Hybrid Controllers”. Indeed, the properties of maximal solutions just mentioned ensure conditions I, II and III of that section; the choice of Cq and q also ensures that q C q and the union of q ’s covers Rn . Among other things, this construction illustrates that the idea of hybrid supervision of hybrid controllers applies also to combining standard, non hybrid controllers.
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Applications In this section, we make use of the following: For vectors in R2 , we will use the following multiplication rule, conjugate rule, and identity element:
z ˝ x :D
x1 z1 x 1 z2 x 2 ; x c :D ; z2 x 1 C z1 x 2 x2
1 1 :D : 0
The multiplication rule is commutative, associative, and distributive. Note that x D 1˝ x D x ˝ 1 and note that x c ˝ x D x ˝ x c D jxj2 1. Also, (z ˝ x)c D x c ˝ z c . For vectors in R4 , we will use the following multiplication rule, conjugate rule, and identity element (vectors are partitioned as x D [x1 x2T ]T where x2 2 R3 ):
z1 x1 z2T x2 ; z2 x 1 C z1 x 2 C z2 x 2
x1 ; xc D x2
1 1D : 0
z˝x D
The multiplication rule is associative and distributive but not necessarily commutative. Note that x D 1 ˝ x D x˝1 and x c ˝x D x˝x c D jxj2 1. Also, (z˝x)c D x c ˝ zc .
Overcoming Stabilization Obstructions Global Stabilization and Tracking on the Unit Circle In this section, we consider stabilization and tracking control of the constrained system ˙ D ˝ v(!) ;
v(!) :D
0 !
2 S1 ;
(24)
where S1 denotes the unit circle and ! 2 R is the control variable. Notice that S1 is invariant regardless of the choice of ! since h; ˝ v(!)i D 0 for all 2 S 1 and all ! 2 R. This model describes the evolution of orientation angle of a rigid body in the plane as a function of the angular velocity !, which is the control variable. We discuss robust, global asymptotic stabilization and tracking problems which cannot be solved with classical feedback control, even when discontinuous feedback laws are allowed, but can be solved with hybrid feedback control. Stabilization First, we consider the problem of stabilizing the point D 1. We note that the (classical) feedback control ! D 2 would almost solve this problem. We
would have ˙1 D 22 D 1 12 and the derivative of the energy function V() :D 1 1 would satisfy hrV(); ˝ v(!)i D 1 12 : We note that the energy will remain constant if starts at ˙1. Thus, since the goal was (robust) global asymptotic stability, this feedback does not achieve the desired goal. One could also consider the discontinuous feedback ! D sgn(2 ) where the function “sgn” is defined arbitrarily in the set f1; 1g when its argument is zero. This feedback is not robust to arbitrarily small measurement noise which can keep the trajectories of the system arbitrarily close to the point D 1 for all time. To visualize this, note that from points on the circle with 2 < 0 and close to D 1, this control law steers the trajectories towards D 1 counterclockwise, while from points on the circle with 2 > 0 and close to D 1, it steers the trajectories towards 1 clockwise. Then, from points on the circle arbitrarily close to D 1, one can generate an arbitrarily small measurement noise signal e that changes sign appropriately so that sgn(2 C e) is always pushing trajectories towards 1. In order to achieve a robust, global asymptotic stability result, we consider a hybrid controller that uses the controller ! D 2 when the state is not near 1 and uses a controller that drives the system away from 1 when it is near that point. One way to accomplish the second task is to build an almost global asymptotic stabilizer for a point different from 1 such that the basin of attraction contains all points in a neighborhood of 1. For example, consider the feedback controller ! D 1 D: 1 () which would almost globally asymptotically stabilize the point :D (0; 1) with the only point not in the basin of attraction being the point c . Strangely enough, each of the two controllers can be thought of as globally asymptotically stabilizing the point 1 if their domains are limited. In particular, let the domain of applicability for the controller ! D 1 be C1 :D S 1 \ f j1 1/3 g ; and let the domain of applicability for the controller ! D 2 be C2 : D S 1 \ f j1 2/3 g : Notice that C1 [C2 D S 1 D: . Thus, we are in a situation where a hybrid supervisor, as discussed in Subsect. “Supervisors of Hybrid Controllers”, may be able to give us a hybrid, global asymptotic stabilizer. (There is no state in the controllers we are working with here.) Let us take 1 :D C1 ;
2 :D S 1 nC1 :
Hybrid Control Systems
We have 1 [ 2 D S 1 . Next, we check the assumptions of Subsect. “Supervisors of Hybrid Controllers”. For each q 2 f1; 2g, the solutions of H q (the system we get by using ! D q () and restricting the flow to Cq ), are such that the point 1 is globally pre-asymptotically stable. For q D 1, this is because there are no complete solutions and 1 does not belong to C1 . For q D 2, this is because C2 is a subset of the basin of attraction for 1. We note that every maximal solution to H1 ends in 2 . Every maximal solution to H2 is complete and every maximal solution to H2 starting in 2 does not reach S 1 nC2 . Thus, the assumptions for a hybrid supervisor are in place. We follow the construction in Subsect. “Supervisors of Hybrid Controllers” to define the hybrid supervisor that combines the feedback laws 1 and 2 . We take ! :D q () C˜ q :D C q
˜ 1 :D 2 [ S 1 nC1 D S 1 nC1 D ˜ 2 :D S 1 nC2 : D For this particular problem, jumps toggle the mode q in the set f1; 2g. Thus, the jump map Gq can be simplified to G q :D 3 q. Tracking Let : R0 ! S 1 be continuously differentiable. Suppose we want to find a hybrid feedback controller so that the state of (24) tracks the signal . This problem can be reduced to the stabilization problem of the previous section. Indeed, first note that c ˝ D 1 and thus the following properties hold: ˙ c ˝ D c ˝ ˙
0 c ˙ ˝ D : 1 ˙2 2 ˙1
(25)
Then, with the coordinate transformation D z ˝ , we have: I.
By multiplying the coordinate transformation on the right by c we get z D ˝ c and z ˝ z c D ˝ c D 1, so that z 2 S 1 . II. D () z D 1. III. The derivative of z satisfies z˙ D ˙ ˝ c C ˝ ˙ c D ˝ v(!) ˝ c C ˝ ˙ c ˝ ˝ c h i D ˝ v(!) C ˙ c ˝ ˝ c h i D z ˝ ˝ v(!) c ˝ ˙ ˝ c :
Our desire is to pick ! so that we have z˙ D z ˝ v(˝) and then to choose ˝ to globally asymptotically stabilize the point z D 1. Due to (25) and the properties of multiplication, the vectors c ˝ ˙ and c ˝v(˝)˝ are in the range of v(!). So, we can pick v(!) D c ˝ ˙ C c ˝ v(˝) ˝ to achieve the robust, global tracking goal. In fact, since the multiplication operation is commutative in R2 , this is equivalent to the feedback v(!) D c ˝ ˙ C v(˝). Global Stabilization and Tracking for Unit Quaternions In this section, we consider stabilization and tracking control of the constrained system
0 (26) ˙ D ˝ v(!) ; v(!) :D 2 S3 ; ! where S3 denotes the hypersphere in R4 and ! 2 R3 is the control variable. Notice that S3 is invariant regardless of the choice of ! since h; ˝ v(!)i D 0 for all 2 S 3 and all ! 2 R3 . This model describes the evolution of orientation angle of a rigid body in space as a function of angular velocities !, which are the control variables. The state corresponds to a unit quaternion that can be used to characterize orientation. We discuss robust, global asymptotic stabilization and tracking problems which cannot be solved with classical feedback control, even when discontinuous feedback laws are allowed, but can be solved with hybrid feedback control. Stabilization First, we consider the problem of stabilizing the point D 1. We note that the (classical) feedback control ! :D 0 I v(2 ) D: 2 () (i. e., ! D 2 where 2 refers to the last three components of the vector ) would almost solve this problem. We would have ˙1 D 2T 2 D 1 12 and the derivative of the energy function V() :D 1 1 would satisfy hrV(); ˝ v(!)i D 1 12 : We note that the energy will remain constant if starts at ˙1. Thus, since the goal is (robust) global asymptotic stabilization, this feedback does not achieve the desired goal. One could also consider the discontinuous feedback ! D sgn(2 ) where the function “sgn” is the componentwise sign and each component is defined arbitrarily in the set f1; 1g when its argument is zero. This feedback is not robust to arbitrarily small measurement noise which can
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Hybrid Control Systems
keep the trajectories of the system arbitrarily close to the point 1. In order to achieve a robust, global asymptotic stability result, we consider a hybrid controller that uses the controller above when the state is not near 1 and uses a controller that drives the system away from 1 when it is near that point. One way to accomplish the second task is to build an almost global asymptotic stabilizer for a point different from 1 and so that the basin of attraction contains all points in a neighborhood of 1. For example, consider stabilizing the point : D (0; 1; 0; 0)T using the feedback controller (for more details see the next subsection) z D ˝ c ; ! D 0 I c ˝ v(z2 ) ˝ D: 1 () (i. e. ! D (1 ; 4 ; 3 )T where now the subscripts refer to the individual components of ). This feedback would almost globally asymptotically stabilize the point with the only point not in the basin of attraction being the point c . The two feedback laws, ! D 2 () and ! D 1 (), are combined into a single hybrid feedback law via the hybrid supervisor approach given in Subsect. “Supervisors of Hybrid Controllers”. In fact, the construction is just like the construction for the case of stabilization on a circle with the only difference being that S1 is replaced everywhere by S3 . Tracking Let : R0 ! S 3 be continuously differentiable. Suppose we want to find a hybrid feedback controller so that the state of (26) tracks the signal . This problem can be reduced to the stabilization problem of the previous section. Indeed, first note that c ˝ D 1 and thus the following properties hold: ˙ c ˝ D c ˝ ˙
0 c ˝ ˙ D : 1 ˙2 ˙1 2 2 ˙2
(27)
Then, with the coordinate transformation D z ˝ , we have: I.
By multiplying the coordinate transformation on the right by c we get z D ˝ c and z c ˝ z D c ˝ D 1 so that z 2 S 3 . II. D () z D 1. III. The derivative of z satisfies z˙ D ˙ ˝ c C ˝ ˙ c D ˝b ! ˝ c C ˝ ˙ c ˝ ˝ c h i D˝ b ! C ˙ c ˝ ˝ c h i D z˝ ˝ b ! c ˝ ˙ ˝ c :
Our desire is to pick ! so that we have z˙ D z ˝ v(˝) and then to choose ˝ to globally asymptotically stabilize the point z D 1. Due to (27) and the properties of multiplication, the vectors c ˝ ˙ and c ˝v(˝)˝ are in the range of v(!). So, we can pick v(!) D c ˝ ˙ C c ˝ v(˝) ˝ to achieve the robust, global tracking goal. Stabilization of a Mobile Robot Consider the global stabilization problem for a model of a unicycle or mobile robot, given as x˙ D # ˙ D ˝ v(!)
(28)
where x 2 R2 denotes planar position from a reference point (in meters), 2 S 1 denotes orientation, # 2 V :D [3; 30] denotes velocity (in meters per second), and ! 2 [4; 4] denotes angular velocity (in radians per second). Both # and ! are control inputs. Due to the specification of the set V , the vehicle is able to move more rapidly in the forward direction than in the backward direction. We define A0 to be the point (x; ) D (0; 1). The controllers below will all use a discrete state p 2 P :D f1; 1g. We take A :D A0 P and : D R2 S 1 P . This system also can be modeled as
x˙ D
cos( ) # sin( )
˙ D ! where D 0 corresponds to D 1 and > 0 is in the counterclockwise direction. The set A0 in these coordinates is given by the set f0g f j D 2k ; k 2 Zg . Even for the point (x; ) D (0; 0), this control system fails Brockett’s well-known condition for robust local asymptotic stabilization by classical (even discontinuous) timeinvariant feedback [9,26,52]. Nevertheless, for the control system (28), the point (0; 1) can be robustly, globally asymptotically stabilized by hybrid feedback. This is done by building three separate hybrid controllers and combining them with a supervisor. The three controllers are the following: The first hybrid controller, K1 , uses # D ProjV (k1 T x), where k1 < 0 and ProjV denotes the projection onto V , while the feedback for ! is given by the hybrid controller in Subsect. “Global Stabilization and Tracking on the Unit Circle” for tracking on the
Hybrid Control Systems
unit circle with reference signal for given by x/jxj. The two different values for q in that controller should be associated with the two values in P . The particular association does not matter. Note that the action of the tracking controller causes the vehicle eventually to use positive velocity to move x toward zero. The controller’s flow and jump sets are such that ˚ C1 [ D1 D x 2 R2 jjxj "11 S 1 P where "11 > 0, and C1 ; D1 are constructed from the hybrid controller in Subsect. “Global Stabilization and Tracking on the Unit Circle” for tracking on the unit circle. The second hybrid controller, K2 , uses # D ProjV (k2 T x), k2 0, while the feedback for ! is given as in Subsect. “Global Stabilization and Tracking on the Unit Circle” for stabilization of the point 1 on the unit circle. Again, the q values of that controller should be associated with the values in P and the particular association does not matter. The controller’s flow and jump sets are such that C2 [ D 2 D
x 2 R2 jjxj "21 S1 ˇ ˚ \ (x; ) ˇ1 1 "22 jxj2 P ;
˚
where "21 > "11 , "22 > 0, and C2 ; D2 are constructed from the hybrid controller in Subsect. “Global Stabilization and Tracking on the Unit Circle” for stabilization of the point 1 on the unit circle. The third hybrid controller, K3 , uses # D ProjV (k3 T x), k3 < 0, while the feedback for ! is hybrid as defined below. The controller’s flow and jump sets are designed so that C3 [ D3 D fx jjxj "31 g S 1 ˇ ˚ \ (x; ) ˇ1 1 "32 jxj2 P D: 3 ; where "31 > "21 and "32 > "22 . The control law for ! is given by ! D pk, where k > 0 and the discrete state p has dynamics given by p˙ D 0;
pC D p :
The flow and jump sets are given by
C3 :D3 \ f(p)2 0g ˚ [ (p)2 0; 1 1 "22 jxj2 D3 :D3 nC3 :
This design accomplishes the following: controller K1 makes track x/jxj as long as jxj is not too small, and thus the vehicle is driven towards x D 0 eventually using only positive velocity; controller K2 drives towards 1 to get the orientation of the vehicle correct; and controller K3 stabilizes to 1 in a persistently exciting manner so that # can be used to drive the vehicle to the origin. This control strategy is coordinated through a supervisor by defining 1 :DC1 [ D1 2 :D n1 \ (C2 [ D2 ) 3 :D fx jjxj "21 g S 1 ˇ ˚ \ (x; ) ˇ1 1 "22 jxj2 P : It can be verified that [q2Q q D and that the conditions in Subsect. “Supervisors of Hybrid Controllers” for a successful supervisor are satisfied. Figure 8 depicts simulation results of the mobile robot with the hybrid controller proposed above for global asymptotic stabilization of A Q. From the initial condition x(0; 0) D (10; 10) (in meters), (0; 0) corresponding to an angle of 4 radians, the mobile robot backs up using controller K1 until its orientation corresponds to about 3 4 radians, at which x is approximately (10; 9:5). The green ? denotes a jump of the hybrid controller K1 . From this configuration, the mobile robot is steered towards a neighborhood of the origin with orientation given by x/jxj. About a fifth of a meter away from it, a jump of the hybrid supervisor connects controller K3 to the vehicle input (the location at which the jump occurs is denoted by the red ?). Note that the trajectory is such that controller K2 is bypassed since at the jump of the hybrid supervisor, while jxj is small the orientation is such that the system state does not belong to 2 but to 3 ; that is, the orientation is already close enough to 1 at that jump. Figure 8a shows a zoomed version of the x trajectory in Fig. 8b. During this phase, controller K3 is in closed-loop with the mobile robot. The vehicle is steered to the origin with orientation close to 1 by a sequence of “parking” maneuvers. Note that after about seven of those maneuvers, the vehicle position is close to (0; 0:01) with almost the desired orientation. Source Localization Core of the Algorithm Consider the problem of programming an autonomous vehicle to find the location of a maximum for a continuously differentiable function by noting how the function values change as the vehicle
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Hybrid Control Systems
Hybrid Control Systems, Figure 8 Global stabilization of a mobile robot to the origin with orientation (1; 0). Vehicle starts at x(0; 0) D (10; 10) (in meters) and (0; 0) corresponding to an angle of 4 radians. a The vehicle is initially steered to a neighborhood of the origin with orientation x/jxj. At about 1/5 meters away from it, controller K3 is enabled to accomplish the stabilisation task. b Zoomed version of trajectory in a around the origin. Controller K3 steers the vehicle to x D (0; 0) and D 1 by a sequence of “parking” maneuvers
moves. Like before, the vehicle dynamics are given by x˙ D # ˙ D ˝ v(!) ;
2 S1 :
The vehicle is to search for the maximum of the function ' : R2 ! R. The function is assumed to be such that its maximum is unique, denoted x , that r'(x) D 0 if and only if x D x , and the union of its level sets over any interval of the form [c; 1), c 2 R, yields a compact set. For simplicity, we will assume that the sign of the derivative of the function ' in the direction x˙ D # is available as a measurement. We will discuss later how to approximate this quantity by considering the changes in the value of the function ' along solutions. We also assume that the vehicle’s angle, , can make jumps, according to C D ˝ ;
(; ) 2 S 1 S 1 :
We will discuss later how to solve the problem when the angle cannot change discontinuously. We propose the dynamic controller 9 # D #¯ > > = " # (x; ; z) 2 C z˙ D (z) > > ; ! ) " # zC 2 (z) (x; ; z) 2 D where #¯ is a positive constant, ˇ ˚ ¯ 0 ; 2 S1 ; z 2 C :D (x; ; z) ˇhr'(x); #i ˇ ˚ ¯ 0 ; 2 S1 ; z 2 ; D :D (x; ; z) ˇhr'(x); #i
(note that C and D use information about the sign of the derivative of ' in the direction of the flow of x) and the required properties for the set , the function , and the set-valued mapping are as follows: I. (a) The set is compact. (b) The maximal solutions of the continuous-time system
z˙ D (z) z 2 ! and the maximal solutions to the discrete-time system
C z 2 (z) z 2 are complete. II. There are no non-trivial solutions to the system 9 > x˙ D #¯ > > > ˙ D ˝ v(!) = (x; ; z) 2 C0
> > z˙ > > D (z) ; ! where
ˇ ˚ ¯ D 0 ; 2 S1 ; z 2 : C0 :D (x; ; z) ˇhr'(x); #i
III. The only complete solutions to the system 9 > xC D x > > > = C D˝ (x; ; z) 2 D
C > > z > 2 (z) > ; start from xı D x .
Hybrid Control Systems
The first assumption on (; ; ) above guarantees that the control algorithm can generate commands by either flowing exclusively or jumping exclusively. This permits arbitrary combinations of flows and jumps. Thus, since C [ D D R2 S 1 , all solutions to the closed-loop system are complete. Moreover, the assumption that is compact guarantees that the only way solutions can grow unbounded is if x grows unbounded. The second assumption on (; ; ) guarantees that closed-loop flows lead to an increase in the function '. One situation where the assumption is easy to check is when ! D 0 for all z 2 and the maxima of ' along each search direction are isolated. In other words, the maxima of the function ';x : R ! R given by 7! '(x C ) are isolated for each ( ; x) 2 S 1 R2 . In this case it is not possible to flow while keeping ' constant. The last assumption on (; ; ) guarantees that the discrete update algorithm is rich enough to be able to find eventually a direction of decrease for ' for every point x. (Clearly the only way this assumption can be satisfied is if r'(x) D 0 only if x D x , which is what we are assuming.) The assumption prevents the existence of discrete solutions at points where x ¤ x . One example of data (; ; ) satisfying the three conditions above is
D f0g ;
0 ; (z) D 0
3 z
(z) D 4 0 5 : 1 2
For this system, the state z does not change, the generated angular velocity ! is always zero, and the commanded rotation at each jump is /2 radians. Other more complicated orientation-generating algorithms that make use of the dynamic state z are also possible. For example, the algorithm in [42] uses the state variable z to generate conjugate directions at the update times. With the assumptions in place, the invariance principle of Subsect. “Invariance Principles” can be applied with the function ' to conclude that the closed-loop system has the compact set A :D fx gS 1 globally asymptotically stable. Moreover, because of the robustness of global asymptotic stability to small perturbations, the results that we obtain are robust, in a practical sense, to slow variations in the characterization of the function ', including the point where it obtains its maximum. Practical Modifications The assumptions of the previous section that we would like to relax are that can change discontinuously and that the derivative of ' is available as a measurement.
The first issue can be addressed by inserting a mode after every jump where the forward velocity is set to zero and a constant angular velocity is applied for the correct amount of time to drive to the value ˝ . If it is not possible to set the velocity to zero then some other openloop maneuver can be executed so that, after some time, the orientation has changed by the correct amount while the position has not changed. The second issue can be addressed by making sure that, after the direction is updated, values of the function ' along the solution are stored and compared to the current value of ' to determine the sign of the derivative. The comparison should not take place until after a sufficient amount of time has elapsed, to enhance robustness to measurement noise. The robustness of the nominal algorithm to temporal regularization and other perturbations permits such a practical implementation. Discussion and Final Remarks We have presented one viewpoint on hybrid control systems, but the field is still developing and other authors will give a different emphasis. For starters, we have stressed a dynamical systems view of hybrid systems, but authors with a computer science background typically will emphasize a hybrid automaton point of view that separates discrete-valued variables from continuous-valued variables. This decomposition can be found in the early work [61] and [59], and in the more recent work [2,8,58]. An introduction to this modeling approach is given in [7]. Impulsive systems, as described in [4], are closely related to hybrid systems but rely on ordinary time domains and usually do not consider the case where the flow set and jump set overlap. The work in [18] on “left-continuous systems” is closely linked to such systems. Passing to hybrid time domains (under different names) can be seen in [20,24,39,40]; other generalized concepts of time domains can be found in [43] and in the literature on dynamical systems on time scales, see [41] for an introduction. For simplicity, we have taken the flow map to be a function rather than a set-valued mapping. A motivation for set-valued mappings satisfying basic conditions is given in [56] where the notion of generalized solutions is developed and shown to be equivalent to solutions in the presence of vanishing perturbations or noise. Set-valued dynamics, with some consideration of the regularity of the mappings defining them, can be found in [1,2]. Implications of data regularity on the basic structural properties of the set of solutions to a system were outlined concurrently in [20,24]. The work in [24] emphasized the impli-
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cations for robustness of stability theory. The latter work preceded the rigorous derivations in [23] where the proofs of statements in the Subsect. “Conditions for Well-Posedness” can be found. To derive stronger results on continuous dependence of solutions to initial conditions, extra assumptions must be added. An early result in this direction appeared in [59]; see also [11,20,40]. The work in [11] exhibited a continuous selection of solutions, continuous dependence in the set-valued sense was addressed in [17]. Sufficient Lyapunov stability conditions, for hybrid or switching systems, and relying on various concepts of a solution, appeared in [6,21,40,62]. The results in Subsect. “Lyapunov Stability Theorem” are contained in [54] which develops a general invariance principle for hybrid systems. A part of the latter result is quoted in Subsect. “Invariance Principles”. Other invariance results for hybrid or switching systems have appeared in [3,18,28,30,40]. Some early converse results for hybrid systems, relying on nonsmooth and possibly discontinuous Lyapunov functions, were given in [62]. The results quoted in Subsect. “Converse Lyapunov Theorems and Robustness” come from [15]. The development of hybrid control theory is still in its formative stages. This article has focused on the development of supervisors of hybrid controllers and related topics, as well as to applications where hybrid control gives solutions that overcome obstacles faced by classical control. For hybrid supervisors used in the context of adaptive control, see [63] and the references therein. Other results related to supervisors include [47], which considers the problem discussed at the end of Subsect. “Supervisors of Hybrid Controllers”, and [57]. The field of hybrid control systems is moving in the direction of systematic design tools, but the capabilities of hybrid control have been recognized for some time. Many of the early observations were in the context of nonholonomic systems, like the result for mobile robots we have presented as an application of supervisors. For example, see [29,31,37,44,49]. More recently, in [48] and [50] it has been established that every asymptotically controllable nonlinear system can be robustly asymptotically stabilized using logic-based hybrid feedback. Other recent results include the work in [25] and its predecessor [12], and the related work on linear reset control systems as considered in [5,45] and the references therein. This article did not have much to do with the control of hybrid systems, other than the discussion of the juggling problem in the introduction and the structure of hybrid controllers for hybrid systems in Subsect. “Hybrid Controllers for Hybrid Systems”. A significant amount of work on the control of hybrid systems has been done, although
typically not in the framework proposed here. Notable references include [10,46,51] and the references therein. Other interesting topics and open questions in the area of hybrid control systems are developed in [38]. Future Directions What does the future hold for hybrid control systems? With a framework in place that mimics the framework of ordinary differential and difference equations, it appears that many new results will become available in directions that parallel results available for nonlinear control systems. These include certain types of separation principles, control algorithms based on zero dynamics (results in this direction can be found in [60] and [13]), and results based on interconnections and time-scale separation. Surely there will be unexpected results that are enabled by the fundamentally different nature of hybrid control as well. The theory behind the control of systems with impacts will continue to develop and lead to interesting applications. It is also reasonable to anticipate further developments related to the construction of robust, embedded hybrid control systems and robust networked control systems. Likely the research community will also be inspired by hybrid control systems discovered in nature. The future is bright for hybrid control systems design and it will be exciting to see the progress that is made over the next decade and beyond. Bibliography Primary Literature 1. Aubin JP, Haddad G (2001) Cadenced runs of impulse and hybrid control systems. Internat J Robust Nonlinear Control 11(5):401–415 2. Aubin JP, Lygeros J, Quincampoix M, Sastry SS, Seube N (2002) Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans Automat Control 47(1):2–20 3. Bacciotti A, Mazzi L (2005) An invariance principle for nonlinear switched systems. Syst Control Lett 54:1109–1119 4. Bainov DD, Simeonov P (1989) Systems with impulse effect: stability, theory, and applications. Ellis Horwood, Chichester; Halsted Press, New York 5. Beker O, Hollot C, Chait Y, Han H (2004) Fundamental properties of reset control systems. Automatica 40(6):905–915 6. Branicky M (1998) Multiple Lyapunov functions and other analysis tools for switched hybrid systems. IEEE Trans Automat Control 43(4):475–482 7. Branicky M (2005) Introduction to hybrid systems. In: Levine WS, Hristu-Varsakelis D (eds) Handbook of networked and embedded control systems. Birkhäuser, Boston, pp 91–116 8. Branicky M, Borkar VS, Mitter SK (1998) A unified framework for hybrid control: Model and optimal control theory. IEEE Trans Automat Control 43(1):31–45
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9. Brockett RW (1983.) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control Theory. Birkhauser, Boston, MA, pp 181–191 10. Brogliato B (1996) Nonsmooth mechanics models, dynamics and control. Springer, London 11. Broucke M, Arapostathis A (2002) Continuous selections of trajectories of hybrid systems. Syst Control Lett 47:149–157 12. Bupp RT, Bernstein DS, Chellaboina VS, Haddad WM (2000) Resetting virtual absorbers for vibration control. J Vib Control 6:61 13. Cai C, Goebel R, Sanfelice R, Teel AR (2008) Hybrid systems: limit sets and zero dynamics with a view toward output regulation. In: Astolfi A, Marconi L (eds) Analysis and design of nonlinear control systems – In Honor of Alberto Isidori. Springer, pp 241–261 http://www.springer.com/west/home/ generic/search/results?SGWID=4-40109-22-173754110-0 14. Cai C, Teel AR, Goebel R (2007) Results on existence of smooth Lyapunov functions for asymptotically stable hybrid systems with nonopen basin of attraction. In: Proc. 26th American Control Conference, pp 3456–3461, http://www.ccec.ece.ucsb. edu/~cai/ 15. Cai C, Teel AR, Goebel R (2007) Smooth Lyapunov functions for hybrid systems - Part I: Existence is equivalent to robustness. IEEE Trans Automat Control 52(7):1264–1277 16. Cai C, Teel AR, Goebel R (2008) Smooth Lyapunov functions for hybrid systems - Part II: (Pre-)asymptotically stable compact sets. IEEE Trans Automat Control 53(3):734–748. See also [14] 17. Cai C, Goebel R, Teel A (2008) Relaxation results for hybrid inclusions. Set-Valued Analysis (To appear) 18. Chellaboina V, Bhat S, Haddad W (2003) An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlin Anal 53:527–550 19. Clegg JC (1958) A nonlinear integrator for servomechanisms. Transactions AIEE 77(Part II):41–42 20. Collins P (2004) A trajectory-space approach to hybrid systems. In: 16th International Symposium on Mathematical Theory of Networks and Systems, CD-ROM 21. DeCarlo R, Branicky M, Pettersson S, Lennartson B (2000) Perspectives and results on the stability and stabilizability of hybrid systems. Proc of IEEE 88(7):1069–1082 22. Filippov A (1988) Differential equations with discontinuous right-hand sides. Kluwer, Dordrecht 23. Goebel R, Teel A (2006) Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 42(4):573–587 24. Goebel R, Hespanha J, Teel A, Cai C, Sanfelice R (2004) Hybrid systems: Generalized solutions and robust stability. In: Proc. 6th IFAC Symposium in Nonlinear Control Systems, pp 1–12, http://www-ccec.ece.ucsb.edu/7Ersanfelice/ Preprints/final_nolcos.pdf 25. Haddad WM, Chellaboina V, Hui Q, Nersesov SG (2007) Energy- and entropy-based stabilization for lossless dynamical systems via hybrid controllers. IEEE Trans Automat Control 52(9):1604–1614, http://ieeexplore.ieee.org/iel5/ 9/4303218/04303228.pdf 26. Hájek O (1979) Discontinuous differential equations, I. J Diff Eq 32:149–170 27. Hermes H (1967) Discontinuous vector fields and feedback control. In: Differential Equations and Dynamical Systems, Academic Press, New York, pp 155–165
28. Hespanha J (2004) Uniform stability of switched linear systems: Extensions of LaSalle’s invariance principle. IEEE Trans Automat Control 49(4):470–482 29. Hespanha J, Morse A (1999) Stabilization of nonholonomic integrators via logic-based switching. Automatica 35(3): 385–393 30. Hespanha J, Liberzon D, Angeli D, Sontag E (2005) Nonlinear norm-observability notions and stability of switched systems. IEEE Trans Automat Control 50(2):154–168 31. Hespanha JP, Liberzon D, Morse AS (1999) Logic-based switching control of a nonholonomic system with parametric modeling uncertainty. Syst Control Lett 38:167–177 32. Johansson K, Egerstedt M, Lygeros J, Sastry S (1999) On the regularization of zeno hybrid automata. Syst Control Lett 38(3):141–150 33. Kellet CM, Teel AR (2004) Smooth Lyapunov functions and robustness of stability for differential inclusions. Syst Control Lett 52:395–405 34. Krasovskii N (1970) Game-Theoretic Problems of capture. Nauka, Moscow 35. LaSalle JP (1967) An invariance principle in the theory of stability. In: Hale JK, LaSalle JP (eds) Differential equations and dynamical systems. Academic Press, New York 36. LaSalle J (1976) The stability of dynamical systems. SIAM’s Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia 37. Lucibello P, Oriolo G (1995) Stabilization via iterative state steering with application to chained-form systems. In: Proc. 35th IEEE Conference on Decision and Control, pp 2614–2619 38. Lygeros J (2005) An overview of hybrid systems control. In: Levine WS, Hristu-Varsakelis D (eds) Handbook of Networked and Embedded Control Systems. Birkhäuser, Boston, pp 519–538 39. Lygeros J, Johansson K, Sastry S, Egerstedt M (1999) On the existence of executions of hybrid automata. In: Proc. 38th IEEE Conference on Decision and Control, pp 2249–2254 40. Lygeros J, Johansson K, Simi´c S, Zhang J, Sastry SS (2003) Dynamical properties of hybrid automata. IEEE Trans Automat Control 48(1):2–17 41. M Bohner M, Peterson A (2001) Dynamic equations on time scales. An introduction with applications. Birkhäuser, Boston 42. Mayhew CG, Sanfelice RG, Teel AR (2007) Robust source seeking hybrid controllers for autonomous vehicles. In: Proc. 26th American Control Conference, pp 1185–1190 43. Michel A (1999) Recent trends in the stability analysis of hybrid dynamical systems. IEEE Trans Circuits Syst – I Fund Theory Appl 45(1):120–134 44. Morin P, Samson C (2000) Robust stabilization of driftless systems with hybrid open-loop/feedback control. In: Proc. 19th American Control Conference, pp 3929–3933 45. Nesic D, Zaccarian L, Teel A (2008) Stability properties of reset systems. Automatica 44(8):2019–2026 46. Plestan F, Grizzle J, Westervelt E, Abba G (2003) Stable walking of a 7-dof biped robot. IEEE Trans Robot Automat 19(4): 653–668 47. Prieur C (2001) Uniting local and global controllers with robustness to vanishing noise. Math Contr, Sig Syst 14(2): 143–172 48. Prieur C (2005) Asymptotic controllability and robust asymptotic stabilizability. SIAM J Control Opt 43:1888–1912
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49. Prieur C, Astolfi A (2003) Robust stabilization of chained systems via hybrid control. IEEE Trans Automat Control 48(10):1768–1772 50. Prieur C, Goebel R, Teel A (2007) Hybrid feedback control and robust stabilization of nonlinear systems. IEEE Trans Automat Control 52(11):2103–2117 51. Ronsse R, Lefèvre P, Sepulchre R (2007) Rhythmic feedback control of a blind planar juggler. IEEE Transactions on Robotics 23(4):790–802, http://www.montefiore.ulg.ac.be/ services/stochastic/pubs/2007/RLS07 52. Ryan E (1994) On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J Control Optim 32(6):1597–1604 53. Sanfelice R, Goebel R, Teel A (2006) A feedback control motivation for generalized solutions to hybrid systems. In: Hespanha JP, Tiwari A (eds) Hybrid Systems: Computation and Control: 9th International Workshop, vol LNCS, vol 3927. Springer, Berlin, pp 522–536 54. Sanfelice R, Goebel R, Teel A (2007) Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Trans Automat Control 52(12):2282– 2297 55. Sanfelice R, Teel AR, Sepulchre R (2007) A hybrid systems approach to trajectory tracking control for juggling systems. In: Proc. 46th IEEE Conference on Decision and Control, pp 5282–5287 56. Sanfelice R, Goebel R, Teel A (2008) Generalized solutions to hybrid dynamical systems. ESAIM: Control, Optimisation and Calculus of Variations 14(4):699–724 57. Sanfelice RG, Teel AR (2007) A “throw-and-catch” hybrid control strategy for robust global stabilization of nonlinear systems. In: Proc. 26th American Control Conference, pp 3470– 3475
58. van der Schaft A, Schumacher H (2000) An introduction to hybrid dynamical systems. Lecture notes in control and information sciences. Springer, London 59. Tavernini L (1987) Differential automata and their discrete simulators. Nonlinear Anal 11(6):665–683 60. Westervelt E, Grizzle J, Koditschek D (2003) Hybrid zero dynamics of planar biped walkers. IEEE Trans Automat Control 48(1):42–56 61. Witsenhausen HS (1966) A class of hybridstate continuoustime dynamic systems. IEEE Trans Automat Control 11(2):161– 167 62. Ye H, Mitchel A, Hou L (1998) Stability theory for hybrid dynamical systems. IEEE Trans Automat Control 43(4):461–474 63. Yoon TW, Kim JS, Morse A (2007) Supervisory control using a new control-relevant switching. Automatica 43(10):1791– 1798
Books and Reviews Aubin JP, Cellina A (1984) Differential inclusions. Springer, Berlin Haddad WM, Chellaboina V, Nersesov SG (2006) Impulsive and hybrid dynamical systems: stability, dissipativity, and control. Princeton University Press, Princeton Levine WS, Hristu-Varsakelis D (2005) Handbook of networked and embedded control systems. Birkhäuser, Boston Liberzon D (2003) Switching in systems and control. Systems and control: Foundations and applications. Birkhäuser, Boston Matveev AS, Savkin AV (2000) Qualitative theory of hybrid dynamical systems. Birkhäuser, Boston Michel AN, Wang L, Hu B (2001) Qualitative theory of dynamical systems. Dekker Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin
Hyperbolic Conservation Laws
Hyperbolic Conservation Laws ALBERTO BRESSAN Department of Mathematics, Penn State University, University Park, USA Article Outline Glossary Definition of the Subject Introduction Examples of Conservation Laws Shocks and Weak Solutions Hyperbolic Systems in One Space Dimension Entropy Admissibility Conditions The Riemann Problem Global Solutions Hyperbolic Systems in Several Space Dimensions Numerical Methods Future Directions Bibliography Glossary Conservation law Several physical laws state that certain basic quantities such as mass, energy, or electric charge, are globally conserved. A conservation law is a mathematical equation describing how the density of a conserved quantity varies in time. It is formulated as a partial differential equation having divergence form. Flux function The flux of a conserved quantity is a vector field, describing how much of the given quantity moves across any surface, at a given time. Shock Solutions to conservation laws often develop shocks, i. e. surfaces across which the basic physical fields are discontinuous. Knowing the two limiting values of a field on opposite sides of a shock, one can determine the speed of propagation of a shock in terms of the Rankine–Hugoniot equations. Entropy An entropy is an additional quantity which is globally conserved for every smooth solution to a system of conservation laws. In general, however, entropies are not conserved by solutions containing shocks. Imposing that certain entropies increase (or decrease) in correspondence to a shock, one can determine a unique physically admissible solution to the mathematical equations. Definition of the Subject According to some fundamental laws of continuum physics, certain basic quantities such as mass, momen-
tum, energy, electric charge. . . , are globally conserved. As time progresses, the evolution of these quantities can be described by a particular type of mathematical equations, called conservation laws. Gas dynamics, magneto-hydrodynamics, electromagnetism, motion of elastic materials, car traffic on a highway, flow in oil reservoirs, can all be modeled in terms of conservation laws. Understanding, predicting and controlling these various phenomena is the eventual goal of the mathematical theory of hyperbolic conservation laws. Introduction Let u D u(x; t) denote the density of a physical quantity, say, the density of mass. Here t denotes time, while x D (x1 ; x2 ; x3 ) 2 R3 is a three-dimensional space variable. A conservation law is a partial differential equation of the form @ u C div f D 0 : @t
(1)
which describes how the density u changes in time. The vector field f D ( f1 ; f2 ; f3 ) is called the flux of the conserved quantity. We recall that the divergence of f is div f D
@ f1 @ f2 @ f3 C C : @x1 @x2 @x3
To appreciate the meaning of the above Eq. (1), consider a fixed region ˝ R3 of the space. The total amount of mass contained inside ˝ at time t is computed as Z u(x; t)dx : ˝
This integral may well change in time. Using the conservation law (1) and then the divergence theorem, one obtains d dt
Z
Z ˝
u(x; t)dx D
˝
Z @ div fdx u(x; t)dx D @t Z ˝ D f nd˙ : ˙
Here ˙ denotes the boundary of ˝, while the integrand f n denotes the inner product of the vector f with the unit outer normal n to the surface ˙ . According to the above identities, no mass is created or destroyed. The total amount of mass contained inside the region ˝ changes in time only because some of the mass flows in or out across the boundary ˙ . Assuming that the flux f can be expressed as a function of the density u alone, one obtains a closed equation. If the initial density u¯ at time t D 0 is known, then the values of
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Hyperbolic Conservation Laws
the function u D u(x; t) at all future times t > 0 can be found by solving the initial-value problem u t C div f(u) D 0
¯ u(0; x) D u(x) :
More generally, a system of balance laws is a set of partial differential equations of the form 8@ ˆ < @t u1 C div f1 (u1 ; : : : ; u n ) D 1 ; (2) ˆ :@ u C div f (u ; : : : ; u ) D : n 1 n n @t n Here u1 ; : : : ; u n are the conserved quantities, f1 ; : : : ; fn are the corresponding fluxes, while the functions i D i (t; x; u1 ; : : : ; u n ) represent the source terms. In the case where all i vanish identically, we refer to (2) as a system of conservation laws. Systems of this type express the fundamental balance equations of continuum physics, when small dissipation effects are neglected. A basic example is provided by the equations of non-viscous gases, accounting for the conservation of mass, momentum and energy. This subject is thus very classical, having a long tradition which can be traced back to Euler [21] and includes contributions by Stokes, Riemann, Weyl and von Neumann, among several others. In spite of continuing efforts, the mathematical theory of conservation laws is still largely incomplete. Most of the literature has been concerned with two main cases: (i) a single conservation law in several space dimensions, and (ii) systems of conservation laws in one space dimension. For systems of conservation laws in several space dimensions, not even the global-in-time existence of solutions is presently known, in any significant degree of generality. Several mathematical studies are focused on particular solutions, such as traveling waves, multi-dimensional Riemann problems, shock reflection past a wedge, etc. . . Toward a rigorous mathematical analysis of solutions, the main difficulty that one encounters is the lack of regularity. Due to the strong nonlinearity of the equations and the absence of dissipation terms with regularizing effect, solutions which are initially smooth may become discontinuous within finite time. In the presence of discontinuities, most of the classical tools of differential calculus do not apply. Moreover, the Eqs. (2) must be suitably reinterpreted, since a discontinuous function does not admit derivatives in a classical sense. Topics which have been more extensively investigated in the mathematical literature are the following: Existence and uniqueness of solutions to the initialvalue problem. Continuous dependence of the solutions on the initial data [8,10,24,29,36,49,57].
Admissibility conditions for solutions with shocks, characterizing the physically relevant ones [23,38,39, 46]. Stability of special solutions, such as traveling waves, w.r.t. small perturbations [34,47,49,50,62]. Relations between the solutions of a hyperbolic system of conservation laws and the solutions of various approximating systems, modeling more complex physical phenomena. In particular: vanishing viscosity approximations [6,20,28], relaxations [5,33,48], kinetic models [44,53]. Numerical algorithms for the efficient computation of solutions [32,41,41,56,58]. Some of these aspects of conservation laws will be outlined in the following sections. Examples of Conservation Laws We review here some of the most common examples of conservation laws. Throughout the sequel, subscripts such as u t ; f x will denote partial derivatives. Example 1 (Traffic flow) Let u(x, t) be the density of cars on a highway, at the point x at time t. This can be measured as the number of cars per kilometer (see Fig. 1). In first approximation, following [43] we shall assume that u is continuous and that the speed s of the cars depends only on their density, say s D s(u). Given any two points a, b on the highway, the number of cars between a and b therefore varies according to Z
b
Z d b u(x; t)dx dt a D [inflow at x D a] [outflow at x D b]
u t (x; t)dx D a
D s(u(a; t)) u(a; t) s(u(b; t)) u(b; t) Z b D [s(u)u]x dx : a
Since the above equalities hold for all a, b, one obtains the conservation law in one space dimension u t C [s(u)u]x D 0 ;
(3)
where u is the conserved quantity and f (u) D s(u)u is the flux function. Based on experimental data, an appropriate flux function has the form a 2 u (0 < u a2 ) ; f (u) D a1 ln u for suitable constants a1 ; ~a2 .
Hyperbolic Conservation Laws
A typical choice here is p(v) D kv , with 2 [1; 3]. In particular 1:4 for air. In general, p is a decreasing function of v. Near a constant state v0 , one can approximate p by a linear function, say p(v) p(v0 ) c 2 (v v0 ). Here c 2 D p0 (v0 ). In this case the Eq. (4) reduces to the familiar wave equation Hyperbolic Conservation Laws, Figure 1 Modelling the density of cars by a conservation law
Example 2 (Gas dynamics) The a compressible, non-viscous gas in take the form 8 @ ˆ
C div( v) ˆ @t ˆ ˆ ˆ ˆ (conservation of mass) ; ˆ ˆ ˆ < @ ( v ) C div( v v) C @ p @t
i
i
@x i
v t t c2 vx x D 0 :
Euler equations for Eulerian coordinates
i D 1; 2; 3;
ˆ (conservation of momentum) ˆ ˆ ˆ ˆ @ ˆ E C div((E C p)v) D0 ˆ @t ˆ ˆ : (conservation of energy) :
jvj2 C e ; 2
where the first term accounts for the kinetic energy while e is the internal energy density (related to the temperature). The system is closed by an additional equation p D p( ; e), called the equation of state, depending on the particular gas under consideration [18]. Notice that here we are neglecting small viscous forces, as well as heat conductivity. Calling fe1 ; e2 ; e3 g the standard basis of unit vectors in R3 , one has @p/@x i D div(pe i ). Hence all of the above equations can be written in the standard divergence form (1).
(5)
Here u is the conserved quantity while f is the flux. As long as the function u is continuously differentiable, using the chain rule the equation can be rewritten as u t C f 0 (u)u x D 0 ;
Here is the mass density, v D (v1 ; v2 ; v3 ) is the velocity vector, E is the energy density, and p the pressure. In turn, the energy can be represented as a sum ED
A single conservation law in one space dimension is a first order partial differential equation of the form u t C f (u)x D 0 :
D0 D0
Shocks and Weak Solutions
(6)
According to (6), in the x–t-plane, the directional derivative of the function u in the direction of the vector v D ( f 0 (u); 1) vanishes. ¯ At time t D 0, let an initial condition u(x; 0) D u(x) be given. As long as the solution u remains smooth, it can be uniquely determined by the so-called method of characteristics. For every point x0 , consider the straight line (see Fig. 2) ¯ 0) t : x D x0 C f 0 u(x On this line, by (6) the value of u is constant. Hence ¯ 0 ) t ; t D u(x ¯ 0) u x0 C f 0 u(x
Example 3 (Isentropic gas dynamics in Lagrangian variables) Consider a gas in a tube. Particles of the gas will be labeled by a one-dimensional variable y determined by their position in a reference configuration with constant unit density. Using this Lagrangian coordinate, we denote by u(y, t) the velocity of the particle y at time t, and by v(y; t) D 1 (y; t) its specific volume. The so-called p-system of isentropic gas dynamics [57] consists of the two conservation laws vt ux D 0 ;
ut C px D 0 :
(4)
The system is closed by an equation of state p D p(v) expressing the pressure as a function of the specific volume.
Hyperbolic Conservation Laws, Figure 2 Solving a conservation law by the method of characteristics. The function u D u(x; t) is constant along each characteristic line
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Hyperbolic Conservation Laws
Hyperbolic Conservation Laws, Figure 3 Shock formation
for all t 0. This allows to construct a solution up to the first time T where two or more characteristic lines meet. Beyond this time, the solution becomes discontinuous. Figure 3 shows the graph of a typical solution at three different times. Points on the graph of u move horizontally with speed f 0 (u). If this speed is not constant, the shape of the graph will change in time. In particular, there will be an instant T at which one of the tangent lines becomes vertical. For t > T, the solution u(; t) contains a shock. The position y(t) of this shock can be determined by imposing that the total area of the region below the graph of u remains constant in time. In order to give a meaning to the conservation law (6) when u D u(x; t) is discontinuous, one can multiply both sides of the equation by a test function ' and integrate by parts. Assuming that ' is continuously differentiable and vanishes outside a bounded set, one formally obtains “ fu' t C f (u)'x gdxdt D 0 : (7)
u D (u1 ; : : : ; u n ) is called a weak solution to the system of conservation laws (8) if the integral identity (7) holds true, for every continuously differentiable test function ' vanishing outside a bounded set. Consider the n n Jacobian matrix of partial derivatives f at the point u: 1 0 @ f1 /@u1 @ f1 /@u n : A: A(u) D D f (u) D @ @ f n /@u1 @ f n /@u n Using the chain rule, the system (8) can be written in the quasilinear form u t C A(u)u x D 0 :
(9)
We say that the above system is strictly hyperbolic if every matrix A(u) has n real, distinct eigenvalues, say 1 (u) < < n (u). In this case, one can find a basis of right eigenvectors of A(u), denoted by r1 (u); : : : ; r n (u), such that, for i D 1; : : : ; n, A(u)r i (u) D i (u)r i (u) ;
jr i (u)j D 1 :
Example 4 Assume that the flux function is linear: f (u) D Au for some constant matrix A 2 Rnn . If 1 < 2 < < n are the eigenvalues of A and r1 ; : : : ; r n are the corresponding eigenvectors, then any vector function of the form u(x; t) D
n X
g i (x i t)r i
iD1
Since the left hand side of the above equation does not involve partial derivatives of u, it remains meaningful for a discontinuous function u. A locally integrable function u is defined to be a weak solution of the conservation law (6) if the integral identity (7) holds true for every test function ', continuously differentiable and vanishing outside a bounded set. Hyperbolic Systems in One Space Dimension A system of n conservation laws in one space dimension can be written as 8@ @ ˆ < @t u1 C @x [ f1 (u1 ; : : : ; u n )] D 0 ; (8) ˆ :@ @ u C [ f (u ; : : : ; u )] D 0 : n 1 n @t n @x For convenience, this can still be written in the form (5), but keeping in mind that now u D (u1 ; : : : ; u n ) 2 Rn is a vector and that f D ( f1 ; : : : ; f n ) is a vector-valued function. As in the case of a single equation, a vector function
provides a weak solution to the system (8). Here it is enough to assume that the functions g i are locally integrable, not necessarily continuous. Example 3 (continued) For the system (4) describing isentropic gas dynamics, the Jacobian matrix of partial derivatives of the flux is 0 1 @u/@v @u/@u D : Df D p0 (v) 0 @p/@v @p/@u Assuming that p0 (v) < p 0, this matrix has the two real distinct eigenvalues ˙ p0 (v). Therefore, the system is strictly hyperbolic. Entropy Admissibility Conditions Given two states u ; uC 2 Rn and a speed , consider the piecewise constant function defined as ( u if x < t ; u(x; t) D uC if x > t :
Hyperbolic Conservation Laws
Then one can show that the discontinuous function u is a weak solution of the hyperbolic system (8) if and only if it satisfies the Rankine–Hugoniot equations (uC u ) D f (uC ) f (u ) :
(10)
More generally, consider a function u D u(x; t) which is piecewise smooth in the x–t-plane. Assume that these discontinuities are located along finitely many curves x D
˛ (t), ˛ D 1; : : : ; N, and consider the left and right limits u˛ (t) D u˛C (t) D
lim
u(x; t) ;
lim
u(x; t) :
x!˛ (t) x!˛ (t)C
Then u is a weak solution of the system of conservation laws if and only if it satisfies the quasilinear system (9) together with the Rankine–Hugoniot equations
˛0 (t) u˛C (t) u˛ (t) D f u˛C (t) f u˛ (t) along each shock curve. Here ˛0 D d ˛ /dt. ¯ Given an initial condition u(x; 0) D u(x) containing jumps, however, it is well known that the Rankine– Hugoniot conditions do not determine a unique weak solution. Several “admissibility conditions” have thus been proposed in the literature, in order to single out a unique physically relevant solution. A basic criterion relies on the concept of entropy: A continuously differentiable function : Rn 7! R is called an entropy for the system of conservation laws (8), with entropy flux q : Rn 7! R if D(u) D f (u) D Dq(u) : If u D (u1 ; : : : ; u n ) is a smooth solution of (8), not only the quantities u1 ; : : : ; u n are conserved, but the additional conservation law (u) t C q(u)x D 0 holds as well. Indeed, D(u)u t C Dq(u)u x D D(u)[D f (u)u x ] C Dq(u)u x D 0 :
On the other hand, if the solution u is not smooth but contains shocks, the quantity D (u) may no longer be conserved. The admissibility of a shock can now be characterized by requiring that certain entropies be increasing (or decreasing) in time. More precisely, a weak solution u of (8) is said to be entropy-admissible if the inequality (u) t C q(u)x 0 holds in the sense of distributions, for every pair (; q), where is a convex entropy and q is the corresponding flux. Calling u ; uC the states to the left and right of the shock, and its speed, the above condition implies [(uC ) (u )] q(uC ) q(u ) : Various alternative conditions have been studied in the literature, in order to characterize the physically admissible shocks. For these we refer to Lax [39], or Liu [46]. The Riemann Problem Toward the construction of general solutions for the system of conservation laws (8), a basic building block is the so-called Riemann problem [55]. This amounts to choosing a piecewise constant initial data with a single jump at the origin: ( u if x < 0 ; u(x; 0) D (11) uC if x > 0 : In the special case where the system is linear, i. e. f (u) D Au, the solution is piecewise constant in the x– t-plane. It contains n C 1 constant states u D !0 ; !1 ; : : : ; !n D uC (see Fig. 4, left). Each jump ! i ! i1 is an eigenvector of the matrix A, and is located along the line x D i t, whose speed equals the corresponding eigenvalue i . For nonlinear hyperbolic systems of n conservation laws, assuming that the amplitude juC u j of the jump is
Hyperbolic Conservation Laws, Figure 4 Solutions of a Riemann problem. Left: the linear case. Right: a nonlinear example
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sufficiently small, the general solution was constructed in a classical paper of Lax [38], under the additional hypothesis (H) For each i D 1; : : : ; n, the ith field is either genuinely nonlinear, so that D i (u) r i (u) > 0 for all u, or linearly degenerate, with D i (u) r i (u) D 0 for all u. The solution is self-similar: u(x; t) D U(x/t). It still consists of n C 1 constant states !0 D u , !1 ; : : : ; !n D uC (see Fig. 4, right). Each couple of adjacent states ! i1 , ! i is separated either by a shock satisfying the Rankine Hugoniot equations, or else by a centered rarefaction. In this second case, the solution u varies continuously between ! i1 and ! i in a sector of the t–x-plane where the gradient ux coincides with an i-eigenvector of the matrix A(u). Further extensions, removing the technical assumption (H), were obtained by T. P. Liu [46] and by S. Bianchini [3]. Global Solutions Approximate solutions to a more general Cauchy problem can be constructed by patching together several solutions of Riemann problems. In the Glimm scheme [24], one works with a fixed grid in the x–t plane, with mesh sizes x, t. At time t D 0 the initial data is approximated by a piecewise constant function, with jumps at grid points (see Fig. 5, left). Solving the corresponding Riemann problems, a solution is constructed up to a time t sufficiently small so that waves generated by different Riemann problems do not interact. By a random sampling procedure, the solution u(t; ) is then approximated by a piecewise constant function having jumps only at grid points. Solving the new Riemann problems at every one of these points, one can prolong the solution to the next time interval [t; 2t], etc. . . An alternative technique for constructing approximate solutions is by wave-front tracking (Fig. 5, right). This method was introduced by Dafermos [17] in the scalar case and later developed by various authors [7,19,29]. It now
Hyperbolic Conservation Laws, Figure 5 Left: the Glimm scheme. Right: a front tracking approximation
provides an efficient tool in the study of general n n systems of conservation laws, both for theoretical and numerical purposes. The initial data is here approximated with a piecewise constant function, and each Riemann problem is solved approximately, within the class of piecewise constant functions. In particular, if the exact solution contains a centered rarefaction, this must be approximated by a rarefaction fan, containing several small jumps. At the first time t1 where two fronts interact, the new Riemann problem is again approximately solved by a piecewise constant function. The solution is then prolonged up to the second interaction time t2 , where the new Riemann problem is solved, etc. . . The main difference is that with the Glimm scheme one specifies a priori the nodal points where the the Riemann problems are to be solved. On the other hand, in a solution constructed by wave-front tracking the locations of the jumps and of the interaction points depend on the solution itself. Moreover, no restarting procedure is needed. In the end, both algorithms produce a sequence of approximate solutions, whose total variation remains uniformly bounded. We recall here that the total variation of a function u : R 7! Rn is defined as N
X : Tot.Var. fug D sup ju(x i ) u(x i1 )j ; iD1
where the supremum is taken over all N 1 and all N-tuples of points x0 < x1 < < x N . For functions of several variables, a more general definition can be found in [22]. Relying on a compactness argument, one can then show that these approximations converge to a weak solution to the system of conservation laws. Namely, one has: Theorem 1 Let the system of conservation laws (8) be strictly hyperbolic. Then, for every initial data u¯ with sufficiently small total variation, the initial value problem u t C f (u)x D 0
¯ u(x; 0) D u(x)
(12)
Hyperbolic Conservation Laws
has a unique entropy-admissible weak solution, defined for all times t 0. The existence part was first proved in the famous paper of Glimm [24], under the additional hypothesis (H), later removed by Liu [46]. The uniqueness of the solution was proved more recently, in a series of papers by the present author and collaborators, assuming that all shocks satisfy suitable admissibility conditions [8,10]. All proofs are based on careful analysis of solutions of the Riemann problem and on the use of a quadratic interaction functional to control the formation of new waves. These techniques also provided the basis for further investigations of Glimm and Lax [25] and Liu [45] on the asymptotic behavior of weak solutions as t ! 1. It is also interesting to compare solutions with different initial data. In this direction, we observe that a function of two variables u(x, t) can be regarded as a map t 7! u(; t) from a time interval [0; T] into a space L1 (R) of integrable functions. Always assuming that the total variation remains small, the distance between two solutions u; v at any time t > 0 can be estimated as ku(t) v(t)kL1 L ku(0) v(0)kL1 ; where L is a constant independent of time. Estimates on the rate of convergence of Glimm approximations to the unique exact solutions are available. For every fixed time T 0, letting the grid size x; x tend to zero keeping the ratio t/x constant, one has the error estimate [11] Glimm u (T; ) uexact (T; )L1 D0: p lim x!0 x j ln xj An alternative approximation procedure involves the addition of a small viscosity. For " > 0 small, one considers the viscous initial value problem u"t C f (u" )x D " u"x x ;
¯ u" (x; 0) D u(x) :
(13)
For initial data u¯ with small total variation, the analysis in [6] has shown that the solutions u" have small total variation for all times t > 0, and converge to the unique weak solution of (12) as " ! 0. Hyperbolic Systems in Several Space Dimensions In several space dimensions there is still no comprehensive theory for systems of conservation laws. Much of the literature has been concerned with three main topics: (i) Global solutions to a single conservation law. (ii) Smooth solutions to a hyperbolic system, locally in time. (iii) Particular solutions to initial or initial-boundary value problems.
Scalar Conservation Laws The single conservation law on Rm u t C div f(u) D 0 has been extensively studied. The fundamental works of Volpert [59] and Kruzhkov [36] have established the global existence of a unique, entropy-admissible solution to the initial value problem, for any initial data u(x; 0) D ¯ u(x) measurable and globally bounded. This solution can be obtained as the unique limit of vanishing viscosity approximations, solving u"t C div f(u" ) D "u" ; u" (x; 0) D u¯ : As in the one-dimensional case, solutions which are initially smooth may develop shocks and become discontinuous in finite time. Given any two solutions u; v, the following key properties remain valid also in the presence of shocks: (i) If at the initial time t D 0 one has u(x; 0) v(x; 0) for all x 2 Rm , then u(x; t) v(x; t) for all x and all t 0. (ii) The L1 distance between any two solutions does not increase in time. Namely, for any 0 s t one has ku(t) v(t)kL1 (Rm ) ku(s) v(s)kL1 (Rm ) : Alternative approaches to the analysis of scalar conservation laws were developed by Crandall [16] using nonlinear semigroup theory, and by Lions, Perthame and Tadmor [44] using a kinetic formulation. Regularity results can be found in [30]. Smooth Solutions to Hyperbolic Systems Using the chain rule, one can rewrite the system of conservation laws (2) in the quasi-linear form ut C
m X
A˛ (u)u x ˛ D 0 :
(14)
˛D1
Various definitions of hyperbolicity can be found in the literature. Motivated by several examples from mathematical physics, the system (14) is said to be symmetrizable hyperbolic if there exists a positive definite symmetric matrix S D S(u) such that all matrices S ˛ (u) D SA˛ are symmetric. In particular, this condition implies that each n n matrix A˛ (u) has real eigenvalues and admits a basis of linearly independent eigenvectors. As shown in [23], if a system of conservation laws admits a strictly convex entropy (u), such that the Hessian matrix of second deriva-
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tives D2u (u) is positive definite at every point u, then the system is symmetrizable. A classical theorem states that, for a symmetrizable hyperbolic system with smooth initial data, the initial value problem has a unique smooth solution, locally in time. This solution can be prolonged in time up to the first time where the spatial gradient becomes unbounded at one or more points. In this general setting, however, it is not known whether the solution can be extended beyond this time of shock formation. Special Solutions In two space dimensions, one can study special solutions which are independent of time, so that u(x1 ; x2 ; t) D U(x1 ; x2 ). In certain cases, one can regard one of the variables, say x1 as a new time and derive a one-dimensional hyperbolic system of equations for U involving the remaining one-dimensional space variable x2 . Another important class of solutions relates to two-dimensional Riemann problems. Here the initial data, assigned on the x1 -x2 plane, is assumed to be constant along rays through the origin. Taking advantage of this self-similarity, the solution can be written in the form u(x1 ; x2 ; t) D U(x1 /t; x2 /t). This again reduces the problem to an equation in two independent variables [35]. Even for the equation of gas dynamics, a complete solution to the Riemann problem is not available. Several particular cases are analyzed in [42,61]. Several other examples, in specific geometries have been analyzed. A famous problem is the reflection of a shock hitting a wedge-shaped rigid obstacle [15,51]. Numerical Methods Generally speaking, there are three major classes of numerical methods suitable for partial differential equations: finite difference methods (FDM), finite volume methods (FVM) and finite element methods (FEM). For conservation laws, one also has semi-discrete methods, such as the method of lines, and conservative front tracking methods. The presence of shocks and the rich structure of shock interactions cause the main difficulties in numerical computations. To illustrate the main idea, consider a uniform grid in the x–t-plane, with step sizes x and t. Consider the times t n D nt and let I i D [x i1/2 ; x iC1/2 ] be a cell. We wish to compute an approximate value for the cell averages u¯ i over I i . Integrating the conservation law over the rectangle I i [t n ; t nC1 ] and dividing by x, one obtains D u¯ ni C u¯ nC1 i
t [F i1/2 F iC1/2] ; x
where F iC1/2 is the average flux Z t nC1 1 f (u(x iC1/2 ))dt : F iC1/2 D t t n FVM methods seek a suitable approximation to this average flux FiC1/2 . First order methods, based on piecewise constant approximations, are usually stable, but contain large numerical diffusion which smears out the shock profile. High order methods are achieved by using polynomials of higher degree, but this produces numerical oscillations around the shock. The basic problem is how to accurately capture the approximated solution near shocks, and at the same time retain stability of the numerical scheme. A common technique is to use a high order scheme on regions where the solution is smooth, and switch to a lower order method near a discontinuity. Well-known methods of this type include the Godunov methods and the MUSCL schemes, wave propagation methods [40,41], the central difference schemes [52,58] and the ENO/WENO schemes [56]. The conservative front tracking methods combine the FDM/FVM with the standard front tracking [26,27]. Based on a high order FDM/FVM, the methods in addition track the location and the strength of the discontinuities, and treat them as moving boundaries. The complexity increases with the number of fronts. In the FEM setting, the discontinuous Galerkin’s methods are widely used [13,14]. The method uses finite element discretization in space, with piecewise polynomials approximation, but allows the approximation to be discontinuous at cell boundaries. Some numerical methods can be directly extended to the multi-dimensional case, but others need to use a dimensional splitting technique, which introduces additional diffusion. The performance of numerical algorithms is usually tested with some benchmark problem, and little is known theoretically, apart from the case of a scalar conservation law. Moreover, it remains a challenging problem to construct efficient high order numerical methods for systems of conservation laws, both in one and in several space dimensions. Future Directions In spite of extensive research efforts, the mathematical theory of hyperbolic conservation laws is still largely incomplete. For hyperbolic systems in one space dimension, a major challenge is to study the existence and uniqueness of solutions to problems with large initial data. In this direction, some counterexamples show that, for particular sys-
Hyperbolic Conservation Laws
tems, solutions can become unbounded in finite time [31]. However, it is conjectured that for many physical systems, endowed with a strictly convex entropy, such pathological behavior should not occur. In particular, the so-called “p-system” describing isentropic gas dynamics (4) should have global solutions with bounded variation, for arbitrarily large initial data [60]. It is worth mentioning that, for large initial data, the global existence of solutions is known mainly in the scalar case [36,59]. For hyperbolic systems of two conservation laws, global existence can still be proved, relying on a compensated compactness argument [20]. This approach, however, does not provide information on the uniqueness of solutions, or on their continuous dependence on the initial data. Another major open problem is to theoretically analyze the convergence of numerical approximations. Error bounds on discrete approximations are presently available only in the scalar case [37]. For solutions to hyperbolic systems of n conservation laws, proofs of the convergence of viscous approximations [6], semidiscrete schemes [4], or relaxation schemes [5] have always relied on a priori bounds on the total variation. On the other hand, the counterexample in [2] shows that in general one cannot have any a priori bounds on the total variation of approximate solutions constructed by fully discrete numerical schemes. Understanding the convergence of these discrete approximations will likely require a new approach. At present, the most outstanding theoretical open problem is to develop a fundamental existence and uniqueness theory for hyperbolic systems in several space dimensions. In order to achieve an existence proof, a key step is to identify the appropriate functional space where to construct solutions. In the one-dimensional case, solutions are found in the space BV of functions with bounded variation. In several space dimensions, however, it is known that the total variation of an arbitrary small solution can become unbounded almost immediately [54]. Hence the space BV does not provide a suitable framework to study the problem. For a special class of systems, a positive result and a counterexample, concerning global existence and continuous dependence on initial data can be found in [1] and in [9], respectively. Bibliography Primary Literature 1. Ambrosio L, Bouchut F, De Lellis C (2004) Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm Part Diff Equat 29:1635–1651
2. Baiti P, Bressan A, Jenssen HK (2006) An instability of the Godunov scheme. Comm Pure Appl Math 59:1604–1638 3. Bianchini S (2003) On the Riemann problem for non-conservative hyperbolic systems. Arch Rat Mach Anal 166:1–26 4. Bianchini S (2003) BV solutions of the semidiscrete upwind scheme. Arch Ration Mech Anal 167:1–81 5. Bianchini S (2006) Hyperbolic limit of the Jin-Xin relaxation model. Comm Pure Appl Math 59:688–753 6. Bianchini S, Bressan A (2005) Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann Math 161:223–342 7. Bressan A (1992) Global solutions to systems of conservation laws by wave-front tracking. J Math Anal Appl 170:414–432 8. Bressan A (2000) Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford 9. Bressan A (2003) An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend Sem Mat Univ Padova 110:103–117 10. Bressan A, Liu TP, Yang T (1999) L1 stability estimates for n n conservation laws. Arch Ration Mech Anal 149:1–22 11. Bressan A, Marson A (1998) Error bounds for a deterministic version of the Glimm scheme. Arch Rat Mech Anal 142: 155–176 12. Chen GQ, Zhang Y, Zhu D (2006) Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Ration Mech Anal 181:261–310 13. Cockburn B, Shu CW (1998) The local discontinuous Galerkin finite element method for convection diffusion systems. SIAM J Numer Anal 35:2440–2463 14. Cockburn B, Hou S, Shu C-W (1990) The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math Comput 54:545–581 15. Courant R, Friedrichs KO (1948) Supersonic Flow and Shock Waves. Wiley Interscience, New York 16. Crandall MG (1972) The semigroup approach to first-order quasilinear equations in several space variables. Israel J Math 12:108–132 17. Dafermos C (1972) Polygonal approximations of solutions of the initial value problem for a conservation law. J Math Anal Appl 38:33–41 18. Dafermos C (2005) Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin, pp 325–626 19. DiPerna RJ (1976) Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J Differ Equ 20: 187–212 20. DiPerna R (1983) Convergence of approximate solutions to conservation laws. Arch Ration Mech Anal 82:27–70 21. Euler L (1755) Principes généraux du mouvement des fluides. Mém Acad Sci Berlin 11:274–315 22. Evans LC, Gariepy RF (1992) Measure Theory and Fine Properties of Functions. C.R.C. Press, Boca Raton, pp viii–268 23. Friedrichs KO, Lax P (1971) Systems of conservation laws with a convex extension. Proc Nat Acad Sci USA 68:1686–1688 24. Glimm J (1965) Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math 18:697–715 25. Glimm J, Lax P (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am Math Soc Memoir 101:xvii–112
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26. Glimm J, Li X, Liu Y (2001) Conservative Front Tracking in One Space Dimension. In: Fluid flow and transport in porous media: mathematical and numerical treatment, (South Hadley, 2001), pp 253–264. Contemp Math 295, Amer Math Soc, Providence 27. Glimm J, Grove JW, Li XL, Shyue KM, Zeng Y, Zhang Q (1998) Three dimensional front tracking. SIAM J Sci Comput 19: 703–727 28. Goodman J, Xin Z (1992) Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch Ration Mech Anal 121:235–265 29. Holden H, Risebro NH (2002) Front tracking for hyperbolic conservation laws. Springer, New York 30. Jabin P, Perthame B (2002) Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim Calc Var 8: 761–774 31. Jenssen HK (2000) Blowup for systems of conservation laws. SIAM J Math Anal 31:894–908 32. Jiang G-S, Levy D, Lin C-T, Osher S, Tadmor E (1998) High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws. SIAM J Numer Anal 35:2147–2168 33. Jin S, Xin ZP (1995) The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm Pure Appl Math 48:235–276 34. Kawashima S, Matsumura A (1994) Stability of shock profiles in viscoelasticity with non-convex constitutive relations. Comm Pure Appl Math 47:1547–1569 35. Keyfitz B (2004) Self-similar solutions of two-dimensional conservation laws. J Hyperbolic Differ Equ 1:445–492 36. Kruzhkov S (1970) First-order quasilinear equations with several space variables. Math USSR Sb 10:217–273 37. Kuznetsov NN (1976) Accuracy of some approximate methods for computing the weak solution of a first order quasilinear equation. USSR Comp Math Math Phys 16:105–119 38. Lax P (1957) Hyperbolic systems of conservation laws II. Comm Pure Appl Math 10:537–566 39. Lax P (1971) Shock waves and entropy. In: Zarantonello E (ed) Contributions to Nonlinear Functional Analysis. Academic Press, New York, pp 603–634 40. Leveque RJ (1990) Numerical methods for conservation laws. Lectures in Mathematics. Birkhäuser, Basel 41. Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge 42. Li J, Zhang T, Yang S (1998) The two dimensional problem in gas dynamics. Pitman, Longman Essex 43. Lighthill MJ, Whitham GB (1955) On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc Roy Soc Lond A 229:317–345 44. Lions PL, Perthame E, Tadmor E (1994) A kinetic formulation of multidimensional scalar conservation laws and related equations. JAMS 7:169–191 45. Liu TP (1977) Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Comm Pure Appl Math 30:767–796 46. Liu TP (1981) Admissible solutions of hyperbolic conservation laws. Mem Am Math Soc 30(240):iv–78 47. Liu TP (1985) Nonlinear stability of shock waves for viscous conservation laws. Mem Am Math Soc 56(328):v–108 48. Liu TP (1987) Hyperbolic conservation laws with relaxation. Comm Math Pys 108:153–175
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Hyperbolic Dynamical Systems VITOR ARAÚJO1,2 , MARCELO VIANA 3 1 CMUP, Porto, Portugal 2 IM-UFRJ, Rio de Janeiro, Brazil 3 IMPA, Rio de Janeiro, Brazil Article Outline Glossary Definition Introduction Linear Systems Local Theory Hyperbolic Behavior: Examples Hyperbolic Sets Uniformly Hyperbolic Systems Attractors and Physical Measures Obstructions to Hyperbolicity Partial Hyperbolicity Non-Uniform Hyperbolicity – Linear Theory Non-Uniformly Hyperbolic Systems Future Directions Bibliography Glossary Homeomorphism, diffeomorphism A homeomorphism is a continuous map f : M ! N which is one-toone and onto, and whose inverse f 1 : N ! M is also continuous. It may be seen as a global continuous change of coordinates. We call f a diffeomorphism if, in addition, both it and its inverse are smooth. When M D N, the iterated n-fold composition f ı : n: : ı f is denoted by f n . By convention, f 0 is the identity map, and f n D ( f n )1 D ( f 1 )n for n 0. Smooth flow A flow f t : M ! M is a family of diffeomorphisms depending in a smooth fashion on a parameter t 2 R and satisfying f sCt D f s ı f t for all s, t 2 R. This property implies that f 0 is the identity map. Flows usually arise as solutions of autonomous differential equations: let t 7! t (v) denote the solution of X˙ D F(X) ; X(0) D v ;
(1)
and assume solutions are defined for all times; then the family t thus defined is a flow (at least as smooth as the vector field F itself). The vector field may be recovered from the flow, through the relation F(X) D @ t t (X) j tD0 .
Ck topology Two maps admitting continuous derivatives are said to be C1 -close if they are uniformly close, and so are their derivatives. More generally, given any k 1, we say that two maps are Ck -close if they admit continuous derivatives up to order k, and their derivatives of order i are uniformly close, for every i D 0; 1; : : : ; k. This defines a topology in the space of maps of class Ck . Foliation A foliation is a partition of a subset of the ambient space into smooth submanifolds, that one calls leaves of the foliation, all with the same dimension and varying continuously from one point to the other. For instance, the trajectories of a vector field F, that is, the solutions of Eq. (1), form a 1-dimensional foliation (the leaves are curves) of the complement of the set of zeros of F. The main examples of foliations in the context of this work are the families of stable and unstable manifolds of hyperbolic sets. Attractor A subset of the ambient space M is invariant under a transformation f if f 1 () D , that is, a point is in if and only if its image is. is invariant under a flow if it is invariant under f t for all t 2 R. An attractor is a compact invariant subset such that the trajectories of all points in a neighborhood U converge to as times goes to infinity, and is dynamically indecomposable (or transitive): there is some trajectory dense in . Sometimes one asks convergence only for points in some “large” subset of a neighborhood U of , and dynamical indecomposability can also be defined in somewhat different ways. However, the formulations we just gave are fine in the uniformly hyperbolic context. Limit sets The !-limit set of a trajectory f n (x); n 2 Z is the set !(x) of all accumulation points of the trajectory as time n goes to C1. The ˛-limit set is defined analogously, with n ! 1. The corresponding notions for continuous time systems (flows) are defined analogously. The limit set L(f ) (or L( f t ), in the flow case) is the closure of the union of all 0 !-limit and all ˛-limit sets. The non-wandering set ˝( f ) (or ˝( f t ), in the flow case) is that set of points such that every neighborhood U contains some point whose orbit returns to U in future time (then some point returns to U in past time as well). When the ambient space is compact all these sets are non-empty. Moreover, the limit set is contained in the non-wandering set. Invariant measure A probability measure in the ambient space M is invariant under a transformation f if ( f 1 (A)) D (A) for all measurable subsets A. This means that the “events” x 2 A and f (x) 2 A have equally probable. We say is invariant under a flow if it is invariant under f t for all t. An invariant proba-
Hyperbolic Dynamical Systems
bility measure is ergodic if every invariant set A has either zero or full measure. An equivalently condition is that can not be decomposed as a convex combination of invariant probability measures, that is, one can not have D a1 C (1 a)2 with 0 < a < 1 and 1 ; 2 invariant. Definition In general terms, a smooth dynamical system is called hyperbolic if the tangent space over the asymptotic part of the phase space splits into two complementary directions, one which is contracted and the other which is expanded under the action of the system. In the classical, socalled uniformly hyperbolic case, the asymptotic part of the phase space is embodied by the limit set and, most crucially, one requires the expansion and contraction rates to be uniform. Uniformly hyperbolic systems are now fairly well understood. They may exhibit very complex behavior which, nevertheless, admits a very precise description. Moreover, uniform hyperbolicity is the main ingredient for characterizing structural stability of a dynamical system. Over the years the notion of hyperbolicity was broadened (non-uniform hyperbolicity) and relaxed (partial hyperbolicity, dominated splitting) to encompass a much larger class of systems, and has become a paradigm for complex dynamical evolution. Introduction The theory of uniformly hyperbolic dynamical systems was initiated in the 1960s (though its roots stretch far back into the 19th century) by S. Smale, his students and collaborators, in the west, and D. Anosov, Ya. Sinai, V. Arnold, in the former Soviet Union. It came to encompass a detailed description of a large class of systems, often with very complex evolution. Moreover, it provided a very precise characterization of structurally stable dynamics, which was one of its original main goals. The early developments were motivated by the problem of characterizing structural stability of dynamical systems, a notion that had been introduced in the 1930s by A. Andronov and L. Pontryagin. Inspired by the pioneering work of M. Peixoto on circle maps and surface flows, Smale introduced a class of gradient-like systems, having a finite number of periodic orbits, which should be structurally stable and, moreover, should constitute the majority (an open and dense subset) of all dynamical systems. Stability and openness were eventually established, in the thesis of J. Palis. However, contemporary results of M. Levinson, based on previous work by M. Cartwright and J. Littlewood, provided examples of open subsets of dy-
namical systems all of which have an infinite number of periodic orbits. In order to try and understand such phenomenon, Smale introduced a simple geometric model, the now famous “horseshoe map”, for which infinitely many periodic orbits exist in a robust way. Another important example of structurally stable system which is not gradient like was R. Thom’s so-called “cat map”. The crucial common feature of these models is hyperbolicity: the tangent space at each point splits into two complementary directions such that the derivative contracts one of these directions and expands the other, at uniform rates. In global terms, a dynamical system is called uniformly hyperbolic, or Axiom A, if its limit set has this hyperbolicity property we have just described. The mathematical theory of such systems, which is the main topic of this paper, is now well developed and constitutes a main paradigm for the behavior of “chaotic” systems. In our presentation we go from local aspects (linear systems, local behavior, specific examples) to the global theory (hyperbolic sets, stability, ergodic theory). In the final sections we discuss several important extensions (strange attractors, partial hyperbolicity, non-uniform hyperbolicity) that have much broadened the scope of the theory. Linear Systems Let us start by introducing the phenomenon of hyperbolicity in the simplest possible setting, that of linear transformations and linear flows. Most of what we are going to say applies to both discrete time and continuous time systems in a fairly analogous way, and so at each point we refer to either one setting or the other. In depth presentations can be found in e. g. [6,8]. The general solution of a system of linear ordinary differential equations X˙ D AX ; X(0) D v ; where A is a constant n n real matrix and v 2 Rn is fixed, is given by X(t) D e tA v ; t 2R; P n where e tA D 1 nD0 (tA) /n!. The linear flow is called hyperbolic if A has no eigenvalues on the imaginary axis. Then the exponential matrix e A has no eigenvalues with norm 1. This property is very important for a number of reasons.
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Stable and Unstable Spaces
Hyperbolic Linear Systems
For one thing it implies that all solutions have well-defined asymptotic behavior: they either converge to zero or diverge to infinity as time t goes to ˙1. More precisely, let
There is a corresponding notion of hyperbolicity for discrete time linear systems
Es (stable subspace) be the subspace of Rn spanned by the generalized eigenvector associated to eigenvalues of A with negative real part. Eu (unstable subspace) be the subspace of Rn spanned by the generalized eigenvector associated to eigenvalues of A with positive real part Then these subspaces are complementary, meaning that Rn D E s ˚ E u , and every solution e tA v with v 62 E s [ E u diverges to infinity both in the future and in the past. The solutions with v 2 E s converge to zero as t ! C1 and go to infinity as t ! 1, and analogously when v 2 E u , reversing the direction of time.
X nC1 D CX n ; X0 D v ; with C a n n real matrix. Namely, we say the system is hyperbolic if C has no eigenvalue in the unit circle. Thus a matrix A is hyperbolic in the sense of continuous time systems if and only if its exponential C D eA is hyperbolic in the sense of discrete time systems. The previous observations (well-defined behavior, robustness, denseness and stability) remain true in discrete time. Two hyperbolic matrices are conjugate by a homeomorphism if and only if they have the same index, that is, the same number of eigenvalues with norm less than 1, and they both either preserve or reverse orientation.
Robustness and Density Another crucial feature of hyperbolicity is robustness: any matrix that is close to a hyperbolic one, in the sense that corresponding coefficients are close, is also hyperbolic. The stable and unstable subspaces need not coincide, of course, but the dimensions remain the same. In addition, hyperbolicity if dense: any matrix is close to a hyperbolic one. That is because, up to arbitrarily small modifications of the coefficients, one may force all eigenvalues to move out of the imaginary axis. Stability, Index of a Fixed Point In addition to robustness, hyperbolicity also implies stability: if B is close to a hyperbolic matrix A, in the sense we have just described, then the solutions of X˙ D BX have essentially the same behavior as the solutions of X˙ D AX. What we mean by “essentially the same behavior” is that there exists a global continuous change of coordinates, that is, a homeomorphism h : Rn ! Rn , that maps solutions of one system to solutions of the other, preserving the time parametrization: h e tA v D e tB h(v) for all t 2 R :
Local Theory Now we move on to discuss the behavior of non-linear systems close to fixed or, more generally, periodic trajectories. By non-linear system we understand the iteration of a diffeomorphism f , or the evolution of a smooth flow f t , on some manifold M. The general philosophy is that the behavior of the system close to a hyperbolic fixed point very much resembles the dynamics of its linear part. A fixed point p 2 M of a diffeomorphism f : M ! M is called hyperbolic if the linear part D f p : Tp M ! Tp M is a hyperbolic linear map, that is, if Df p has no eigenvalue with norm 1. Similarly, an equilibrium point p of a smooth vector field F is hyperbolic if the derivative DF(p) has no pure imaginary eigenvalues. Hartman–Grobman Theorem This theorem asserts that if p is a hyperbolic fixed point of f : M ! M then there are neighborhoods U of p in M and V of 0 in the tangent space Tp M such that we can find a homeomorphism h : U ! V such that h ı f D D fp ı h ;
More generally, two hyperbolic linear flows are conjugated by a homeomorphism h if and only if they have the same index, that is, the same number of eigenvalues with negative real part. In general, h can not be taken to be a diffeomorphism: this is possible if and only if the two matrices A and B are obtained from one another via a change of basis. Notice that in this case they must have the same eigenvalues, with the same multiplicities.
whenever the composition is defined. This property means that h maps orbits of Df (p) close to zero to orbits of f close to p. We say that h is a (local) conjugacy between the nonlinear system f and its linear part Df p . There is a corresponding similar theorem for flows near a hyperbolic equilibrium. In either case, in general h can not be taken to be a diffeomorphism.
Hyperbolic Dynamical Systems
Stable Sets
Hyperbolic Behavior: Examples
The stable set of the hyperbolic fixed point p is defined by ˚ W s (p) D x 2 M : d( f n (x); f n (p)) ! 0 :
Now let us review some key examples of (semi)global hyperbolic dynamics. Thorough descriptions are available in e. g. [6,8,9].
Given ˇ > 0 we also consider the local stable set of size ˇ > 0, defined by
A Linear Torus Automorphism
n!C1
˚
Wˇs (p) D x 2 M : d( f n (x); f n (p)) ˇ
for all n 0 :
The image of Wˇs under the conjugacy h is a neighborhood of the origin inside Es . It follows that the local stable set is an embedded topological disk, with the same dimension as Es . Moreover, the orbits of the points in Wˇs (p) actually converges to the fixed point as time goes to infinity. Therefore, z 2 W s (p) , f n (z) 2 Wˇs (p) for some n 0 : Stable Manifold Theorem The stable manifold theorem asserts that Wˇs (p) is actually a smooth embedded disk, with the same order of differentiability as f itself, and it is tangent to Es at the point p. It follows that W s (p) is a smooth submanifold, injectively immersed in M. In general, W s (p) is not embedded in M: in many cases it has self-accumulation points. For these reasons one also refers to W s (p) and Wˇs (p) as stable manifolds of p. Unstable manifolds are defined analogously, replacing the transformation by its inverse. Local Stability We call index of a diffeomorphism f at a hyperbolic fixed point p the index of the linear part, that is, the number of eigenvalues of Df p with negative real part. By the Hartman–Grobman theorem and previous comments on linear systems, two diffeomorphisms are locally conjugate near hyperbolic fixed points if and only if the stable indices and they both preserve/reverse orientation. In other words, the index together with the sign of the Jacobian determinant form a complete set of invariants for local topological conjugacy. Let g be any diffeomorphism C1 -close to f . Then g has a unique fixed point pg close to p, and this fixed point is still hyperbolic. Moreover, the stable indices and the orientations of the two diffeomorphisms at the corresponding fixed points coincide, and so they are locally conjugate. This is called local stability near of diffeomorphisms hyperbolic fixed points. The same kind of result holds for flows near hyperbolic equilibria.
Consider the linear transformation A: R2 ! R2 given by the following matrix, relative to the canonical base of the plane: 2 1 : 1 1 The 2-dimensional torus T 2 is the quotient R2 /Z2 of the plane by the equivalence relation (X1 ; y1 ) (x2 ; y2 )
,
(x1 x2 ; y1 y2 ) 2 Z2 :
Since A preserves the lattice Z2 of integer vectors, that is, since A(Z2 ) D Z2 , the linear transformation defines an invertible map f A : T 2 ! T 2 in the quotient space, which is an example of linear automorphism of T 2 . We call affine line in T 2 the projection under the quotient map of any affine line in the plane. The linear transformation A is hyperbolic, with eigenvalues 0 < 1 < 1 < 2 , and the corresponding eigenspaces E1 and E2 have irrational slope. For each point z 2 T 2 , let W i (z) denote the affine line through z and having the direction of Ei , for i D 1; 2: distances along W1 (z) are multiplied by 1 < 1 under forward iteration of f A distances along W2 (z) are multiplied by 1/2 < 1 under backward iteration of f A . Thus we call W1 (z) stable manifold and W2 (z) unstable manifold of z (notice we are not assuming z to be periodic). Since the slopes are irrational, stable and unstable manifolds are dense in the whole torus. From this fact one can deduce that the periodic points of f A form a dense subset of the torus, and that there exist points whose trajectories are dense in T 2 . The latter property is called transitivity. An important feature of this systems is that its behavior is (globally) stable under small perturbations: given any diffeomorphism g : T 2 ! T 2 sufficiently C1 -close to f A , there exists a homeomorphism h : T 2 ! T 2 such that h ı g D f A ı h. In particular, g is also transitive and its periodic points form a dense subset of T 2 . The Smale Horseshoe Consider a stadium shaped region D in the plane divided into three subregions, as depicted in Fig. 1: two half disks,
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Hyperbolic Dynamical Systems, Figure 2 The solenoid attractor
Hyperbolic Dynamical Systems, Figure 1 Horseshoe map
A and C, and a square, B. Next, consider a map f : D ! D mapping D back inside itself as described in Fig. 1: the intersection between B and f (B) consists of two rectangles, R0 and R1 , and f is affine on the pre-image of these rectangles, contracting the horizontal direction and expanding the vertical direction. The set D \n2Z f n (B), formed by all the points whose orbits never leave the square B is totally disconnected, in fact, it is the product of two Cantor sets. A description of the dynamics on may be obtained through the following coding of orbits. For each point z 2 and every time n 2 Z the iterate f n (z) must belong to either R0 or R1 . We call itinerary of z the sequence fs n gn2Z with values in the set f0; 1g defined by f n (z) 2 Rs n for all n 2 Z. The itinerary map ! f0; 1gZ ; z 7! fs n gn2Z is a homeomorphism, and conjugates f restricted to to the so-called shift map defined on the space of sequences by f0; 1gZ ! f0; 1gZ ; fs n gn2Z 7! fs nC1 gn2Z : Since the shift map is transitive, and its periodic points form a dense subset of the domain, it follows that the same is true for the horseshoe map on . From the definition of f we get that distances along horizontal line segments through points of are contracted at a uniform rate under forward iteration and, dually, distances along vertical line segments through points of are contracted at a uniform rate under backward iteration. Thus, horizontal line segments are local stable sets and vertical line segments are local unstable sets for the points of .
A striking feature of this system is the stability of its dynamics: given any diffeomorphism g sufficiently C1 -close to f , its restriction to the set g D \n2Z g n (B) is conjugate to the restriction of f to the set D f (and, consequently, is conjugate to the shift map). In addition, each point of g has local stable and unstable sets which are smooth curve segments, respectively, approximately horizontal and approximately vertical. The Solenoid Attractor The solid torus is the product space S1 D, where S1 D R/Z is the circle and D D fz 2 C : jzj < 1g is the unit disk in the complex plane. Consider the map f : S1 D ! S1 D given by ( ; z) 7! (2 ; ˛z C ˇe i /2 ) ; 2 R/Z and ˛; ˇ 2 R with ˛ C ˇ < 1. The latter condition ensures that the image f (S1 D) is strictly contained in S1 D. Geometrically, the image is a long thin domain going around the solid torus twice, as described in Fig. 2. Then, for any n 1, the corresponding iterate f n (S1 D) is an increasingly thinner and longer domain that winds 2k times around S1 D. The maximal invariant set D \n0 f n (S1 D) is called solenoid attractor. Notice that the forward orbit under f of every point in S1 D accumulates on . One can also check that the restriction of f to the attractor is transitive, and the set of periodic points of f is dense in . In addition has a dense subset of periodic orbits and also a dense orbit. Moreover every point in a neighborhood of converges to and this is why this set is called an attractor. Hyperbolic Sets The notion we are now going to introduce distillates the crucial feature common to the examples presented pre-
Hyperbolic Dynamical Systems
viously. A detailed presentation is given in e. g. [8,10]. Let f : M ! M be a diffeomorphism on a manifold M. A compact invariant set M is a hyperbolic set for f if the tangent bundle over admits a decomposition T M D E u ˚ E s ; invariant under the derivative and such that kD f 1 j E u k < and kD f j E s k < for some constant < 1 and some choice of a Riemannian metric on the manifold. When it exists, such a decomposition is necessarily unique and continuous. We call Es the stable bundle and Eu the unstable bundle of f on the set . The definition of hyperbolicity for an invariant set of a smooth flow containing no equilibria is similar, except that one asks for an invariant decomposition T M D E u ˚ E 0 ˚ E s , where Eu and Es are as before and E0 is a line bundle tangent to the flow lines. An invariant set that contains equilibria is hyperbolic if and only it consists of a finite number of points, all of them hyperbolic equilibria. Cone Fields The definition of hyperbolic set is difficult to use in concrete situations, because, in most cases, one does not know the stable and unstable bundles explicitly. Fortunately, to prove that an invariant set is hyperbolic it suffices to have some approximate knowledge of these invariant subbundles. That is the contents of the invariant cone field criterion: a compact invariant set is hyperbolic if and only if there exists some continuous (not necessarily invariant) decomposition T M D E 1 ˚ E 2 of the tangent bundle, some constant < 1, and some cone field around E1
ously to some small neighborhood U of , and then so do the cone fields. By continuity, conditions (a) and (b) above remain valid on U, possibly for a slightly larger constant . Most important, they also remain valid when f is replaced by any other diffeomorphism g which is sufficiently C1 close to it. Thus, using the cone field criterion once more, every compact set K U which is invariant under g is a hyperbolic set for g. Stable Manifold Theorem Let be a hyperbolic set for a diffeomorphism f : M ! M. Assume f is of class Ck . Then there exist "0 > 0 and 0 < < 1 and, for each 0 < " "0 and x 2 , the local stable manifold of size " W"s (x) D fy 2 M : dist( f n (y); f n (x)) " for all n 0g; and the local unstable manifold of size " W"u (x) D fy 2 M : dist( f n (y); f n (x)) " for all n 0g are Ck embedded disks, tangent at x to E sx and E ux , respectively, and satisfying f (W"s (x)) W"s ( f (x)) and f 1 (W"u (x))
W"u ( f 1 (x)); dist( f (x); f (y)) dist(x; y) for all y 2 W"s (x) dist( f 1 (x); f 1 (y)) dist(x; y) for all y 2 W"u (x) W"s (x) and W"u (x) vary continuously with the point x, in the Ck topology. Then, the global stable and unstable manifolds of x, W s (x) D
C 1a (x)
D fv D v1 C v2 2
E 1x
˚
E 2x
: kv2 k akv1 kg ; x 2 ;
which is
[
f n W"s ( f n (x))
and
n0
W u (x) D
[
f n W"u ( f n (x)) ;
n0
D f x (C 1a (x))
(a) forward invariant: (b) expanded by forward for every v 2 C 1a (x)
1 ( f (x)) and
Ca iteration: kD f x (v)k 1 kvk
and there exists a cone field Cb2 (x) around E2 which is backward invariant and expanded by backward iteration.
are smoothly immersed submanifolds of M, and they are characterized by W s (x) D fy 2 M : dist( f n (y); f n (x)) ! 0 as n ! 1g W u (x) D fy 2 M : dist( f n (y); f n (x)) ! 0 as n ! 1g :
Robustness An easy, yet very important consequence is that hyperbolic sets are robust under small modifications of the dynamics. Indeed, suppose is a hyperbolic set for f : M ! M, and let C 1a (x) and Cb2 (x) be invariant cone fields as above. The (non-invariant) decomposition E 1 ˚ E 2 extends continu-
Shadowing Property This crucial property of hyperbolic sets means that possible small “errors” in the iteration of the map close to the set are, in some sense, unimportant: to the resulting “wrong” trajectory, there corresponds a nearby genuine orbit of the
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map. Let us give the formal statement. Recall that a hyperbolic set is compact, by definition. Given ı > 0, a ı-pseudo-orbit of f : M ! M is a sequence fx n gn2Z such that dist(x nC1 ; f (x n )) ı
for all n 2 Z :
Given " > 0, one says that a pseudo-orbit is "-shadowed by the orbit of a point z 2 M if dist( f n (z); x n ) " for all n 2 Z. The shadowing lemma says that for any " > 0 one can find ı > 0 and a neighborhood U of the hyperbolic set such that every ı-pseudo-orbit in U is "-shadowed by some orbit in U. Assuming " is sufficiently small, the shadowing orbit is actually unique. Local Product Structure In general, these shadowing orbits need not be inside th hyperbolic set . However, that is indeed the case if is a maximal invariant set, that is, if it admits some neighborhood U such that coincides with the set of points whose orbits never leave U: \ D f n (U) : n2Z
A hyperbolic set is a maximal invariant set if and only if it has the local product structure property stated in the next paragraph. Let be a hyperbolic set and " be small. If x and y are nearby points in then the local stable manifold of x intersects the local unstable manifold of y at a unique point, denoted [x; y], and this intersection is transverse. This is because the local stable manifold and the local unstable manifold of every point are transverse, and these local invariant manifolds vary continuously with the point. We say that has local product structure if there exists ı > 0 such that [x; y] belongs to for every x; y 2 with dist(x; y) < ı. Stability The shadowing property may also be used to prove that hyperbolic sets are stable under small perturbations of the dynamics: if is a hyperbolic set for f then for any C1 close diffeomorphism g there exists a hyperbolic set g close to and carrying the same dynamical behavior. The key observation is that every orbit f n (x) of f inside is a ı-pseudo-orbits for g in a neighborhood U, where ı is small if g is close to f and, hence, it is shadowed by some orbit g n (z) of g. The correspondence h(x) D z thus defined is injective and continuous. For any diffeomorphism g close enough to f , the orbits of x in the maximal g-invariant set g (U) inside U
are pseudo-orbits for f . Therefore the shadowing property above enables one to bijectively associate g-orbits of g (U) to f -orbits in . This provides a homeomorphism h : g (U) ! which conjugates g and f on the respective hyperbolic sets: f ı h D h ı g. Thus hyperbolic maximal sets are structurally stable: the persistent dynamics in a neighborhood of these sets is the same for all nearby maps. If is a hyperbolic maximal invariant set for f then its hyperbolic continuation for any nearby diffeomorphism g is also a maximal invariant set for g. Symbolic Dynamics The dynamics of hyperbolic sets can be described through a symbolic coding obtained from a convenient discretization of the phase space. In a few words, one partitions the set into a finite number of subsets and assigns to a generic point in the hyperbolic set its itinerary with respect to this partition. Dynamical properties can then be read out from a shift map in the space of (admissible) itineraries. The precise notion involved is that of Markov partition. A set R is a rectangle if [x; y] 2 R for each x; y 2 R. A rectangle is proper if it is the closure of its interior relative to . A Markov partition of a hyperbolic set is a cover R D fR1 ; : : : ; R m g of by proper rectangles with pairwise disjoint interiors, relative to , and such W u ( f (x)) \ R j f (W u (x) \ R i ) and f (W s (x) \ R i ) W s ( f (x)) \ R j for every x 2 int (R i ) with f (x) 2 int (R j ). The key fact is that any maximal hyperbolic set admits Markov partitions with arbitrarily small diameter. Given a Markov partition R with sufficiently small diameter, and a sequence j D ( j n )n2Z in f1; : : : ; mg, there exists at most one point x D h(j) such that f n (x) 2 R j n
for each n 2 Z :
We say that j is admissible if such a point x does exist and, in this case, we say x admits j as an itinerary. It is clear that f ı h D h ı , where is the shift (left-translation) in the space of admissible itineraries. The map h is continuous and surjective, and it is injective on the residual set of points whose orbits never hit the boundaries (relative to ) of the Markov rectangles. Uniformly Hyperbolic Systems A diffeomorphism f : M ! M is uniformly hyperbolic, or satisfies the Axiom A, if the non-wandering set ˝( f )
Hyperbolic Dynamical Systems
is a hyperbolic set for f and the set Per( f ) of periodic points is dense in ˝( f ). There is an analogous definition for smooth flows f t : M ! M; t 2 R. The reader can find the technical details in e. g. [6,8,10]. Dynamical Decomposition The so-called “spectral” decomposition theorem of Smale allows for the global dynamics of a hyperbolic diffeomorphism to be decomposed into elementary building blocks. It asserts that the non-wandering set splits into a finite number of pairwise disjoint basic pieces that are compact, invariant, and dynamically indecomposable. More precisely, the non-wandering set ˝( f ) of a uniformly hyperbolic diffeomorphism f is a finite pairwise disjoint union ˝( f ) D 1 [ [ N of f -invariant, transitive sets i , that are compact and maximal invariant sets. Moreover, the ˛-limit set of every orbit is contained in some i and so is the !-limit set. Geodesic Flows on Surfaces with Negative Curvature Historically, the first important example of uniform hyperbolicity was the geodesic flow Gt on Riemannian manifolds of negative curvature M. This is defined as follows. Let M be a compact Riemannian manifold. Given any tangent vector v, let v : R ! TM be the geodesic with initial condition v D v (0). We denote by ˙v (t) the velocity vector at time t. Since k ˙v (t)k D kvk for all t, it is no restriction to consider only unit vectors. There is an important volume form on the unit tangent bundle, given by the product of the volume element on the manifold by the volume element induced on each fiber by the Riemannian metric. By integration of this form, one obtains the Liouville measure on the unit tangent bundle, which is a finite measure if the manifold itself has finite volume (including the compact case). The geodesic flow is the flow G t : T 1 M ! T 1 M on the unit tangent bundle T 1 M of the manifold, defined by G t (v) D ˙v (t) : An important feature is that this flow leaves invariant the Liouville measure. By Poincaré recurrence, this implies that ˝(G) D T 1 M. A major classical result in Dynamics, due to Anosov, states that if M has negative sectional curvature then this measure is ergodic for the flow. That is, any invariant set has zero or full Liouville measure. The special case when M is a surface, had been dealt before by Hedlund and Hopf.
The key ingredient to this theorem is to prove that the geodesic flow is uniformly hyperbolic, in the sense we have just described, when the sectional curvature is negative. In the surface case, the stable and unstable invariant subbundles are differentiable, which is no longer the case in general in higher dimensions. This formidable obstacle was overcome by Anosov through showing that the corresponding invariant foliations retain, nevertheless, a weaker form of regularity property, that suffices for the proof. Let us explain this. Absolute Continuity of Foliations The invariant spaces E sx and E ux of a hyperbolic system depend continuously, and even Hölder continuously, on the base point x. However, in general this dependence is not differentiable, and this fact is at the origin of several important difficulties. Related to this, the families of stable and unstable manifolds are, usually, not differentiable foliations: although the leaves themselves are as smooth as the dynamical system itself, the holonomy maps often fail to be differentiable. By holonomy maps we mean the projections along the leaves between two given cross-sections to the foliation. However, Anosov and Sinai observed that if the system is at least twice differentiable then these foliations are absolutely continuous: their holonomy maps send zero Lebesgue measure sets of one cross-section to zero Lebesgue measure sets of the other cross-section. This property is crucial for proving that any smooth measure which is invariant under a twice differentiable hyperbolic system is ergodic. For dynamical systems that are only once differentiable the invariant foliations may fail to be absolutely continuous. Ergodicity still is an open problem. Structural Stability A dynamical system is structurally stable if it is equivalent to any other system in a C1 neighborhood, meaning that there exists a global homeomorphism sending orbits of one to orbits of the other and preserving the direction of time. More generally, replacing C1 by Cr neighborhoods, any r 1, one obtains the notion of Cr structural stability. Notice that, in principle, this property gets weaker as r increases. The Stability Conjecture of Palis–Smale proposed a complete geometric characterization of this notion: for any r 1; C r structurally stable systems should coincide with the hyperbolic systems having the property of strong transversality, that is, such that the stable and unstable manifolds of any points in the non-wandering set are transversal. In particular, this would imply that the prop-
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erty of Cr structural stability does not really depend on the value of r. That hyperbolicity and strong transversality suffice for structural stability was proved in the 1970s by Robbin, de Melo, Robinson. It is comparatively easy to prove that strong transversality is also necessary. Thus, the heart of the conjecture is to prove that structurally stable systems must be hyperbolic. This was achieved by Mañé in the 1980s, for C1 diffeomorphisms, and extended about ten years later by Hayashi for C1 flows. Thus a C1 diffeomorphism, or flow, on a compact manifold is structurally stable if and only if it is uniformly hyperbolic and satisfies the strong transversality condition. ˝-stability A weaker property, called ˝-stability is defined requiring equivalence only restricted to the non-wandering set. The ˝-Stability Conjecture of Palis–Smale claims that, for any r 1; ˝-stable systems should coincide with the hyperbolic systems with no cycles, that is, such that no basic pieces in the spectral decomposition are cyclically related by intersections of the corresponding stable and unstable sets. The ˝-stability theorem of Smale states that these properties are sufficient for C r ˝stability. Palis showed that the no-cycles condition is also necessary. Much later, based on Mañé’s aforementioned result, he also proved that for C1 diffeomorphisms hyperbolicity is necessary for ˝-stability. This was extended to C1 flows by Hayashi in the 1990s. Attractors and Physical Measures A hyperbolic basic piece i is a hyperbolic attractor if the stable set W s ( i ) D fx 2 M : !(x) i g contains a neighborhood of i . In this case we call W s ( i ) the basin of the attractor i , and denote it B( i ). When the uniformly hyperbolic system is of class C2 , a basic piece is an attractor if and only if its stable set has positive Lebesgue measure. Thus, the union of the basins of all attractors is a full Lebesgue measure subset of M. This remains true for a residual (dense Gı ) subset of C1 uniformly hyperbolic diffeomorphisms and flows. The following fundamental result, due to Sinai, Ruelle, Bowen shows that, no matter how complicated it may be, the behavior of typical orbits in the basin of a hyperbolic attractor is well-defined at the statistical level: any hyperbolic attractor of a C2 diffeomorphism (or flow) supports
a unique invariant probability measure such that Z n1 1X j lim '( f (z)) D ' d n!1 n
(2)
jD0
for every continuous function ' and Lebesgue almost every point x 2 B(). The standard reference here is [3]. Property (2) also means that the Sinai–Ruelle–Bowen measure may be “observed”: the weights of subsets may be found with any degree of precision, as the sojourn-time of any orbit picked “at random” in the basin of attraction: (V ) D fraction of time the orbit of z spends in V for typical subsets V of M (the boundary of V should have zero -measure), and for Lebesgue almost any point z 2 B(). For this reason is called a physical measure. It also follows from the construction of these physical measures on hyperbolic attractors that they depend continuously on the diffeomorphism (or the flow). This statistical stability is another sense in which the asymptotic behavior is stable under perturbations of the system, distinct from structural stability. There is another sense in which this measure is “physical” and that is that is the zero-noise limit of the stationary measures associated to the stochastic processes obtained by adding small random noise to the system. The idea is to replace genuine trajectories by “random orbits” (z n )n , where each z n C 1 is chosen "-close to f (zn ). We speak of stochastic stability if, for any continuous function ', the random time average lim
n!1
n1 1X '(z j ) n jD0
R
is close to ' d for almost all choices of the random orbit. One way to construct such random orbits is through randomly perturbed iterations, as follows. Consider a family of probability measures " in the space of diffeomorphisms, such that each " is supported in the "-neighborhood of f . Then, for each initial state z0 define z nC1 D f nC1 (z n ), where the diffeomorphisms f n are independent random variables with distribution law " . A probability measure " on the basin B() is stationary if it satisfies Z " (E) D " (g 1 (E)) d" (g) : Stationary measures always exist, and they are often unique for each small " > 0. Then stochastic stability corresponds to having " converging weakly to when the noise level " goes to zero.
Hyperbolic Dynamical Systems
Hyperbolic Dynamical Systems, Figure 3 A planar flow with divergent time averages
The notion of stochastic stability goes back to Kolmogorov and Sinai. The first results, showing that uniformly hyperbolic systems are stochastically stable, on the basin of each attractor, were proved in the 1980s by Kifer and Young. Let us point out that physical measures need not exist for general systems. A simple counter-example, attributed to Bowen, is described in Fig. 3: time averages diverge over any of the spiraling orbits in the region bounded by the saddle connections. Notice that the saddle connections are easily broken by arbitrarily small perturbations of the flow. Indeed, no robust examples are known of systems whose time-averages diverge on positive volume sets. Obstructions to Hyperbolicity Although uniform hyperbolicity was originally intended to encompass a residual or, at least, dense subset of all dynamical systems, it was soon realized that this is not the case: many important examples fall outside its realm. There are two main mechanisms that yield robustly nonhyperbolic behavior, that is, whole open sets of non-hyperbolic systems. Heterodimensional Cycles Historically, the first such mechanism was the coexistence of periodic points with different Morse indices (dimensions of the unstable manifolds) inside the same transitive set. See Fig. 4. This is how the first examples of C1 -open subsets of non-hyperbolic diffeomorphisms were obtained by Abraham, Smale on manifolds of dimension d 3. It was also the key in the constructions by Shub and Mañé of non-hyperbolic, yet robustly transitive diffeomorphisms, that is, such that every diffeomorphism in a C1 neighborhood has dense orbits. For flows, this mechanism may assume a novel form, because of the interplay between regular orbits and singularities (equilibrium points). That is, robust non-hyperbolicity may stem from the coexistence of regular and singular orbits in the same transitive set. The first, and
Hyperbolic Dynamical Systems, Figure 4 A heterodimensional cycle
very striking example was the geometric Lorenz attractor proposed by Afraimovich, Bykov, Shil’nikov and Guckenheimer, Williams to model the behavior of the Lorenz equations, that we shall discuss later. Homoclinic Tangencies Of course, heterodimensional cycles may exist only in dimension 3 or higher. The first robust examples of nonhyperbolic diffeomorphisms on surfaces were constructed by Newhouse, exploiting the second of these two mechanisms: homoclinic tangencies, or non-transverse intersections between the stable and the unstable manifold of the same periodic point. See Fig. 5. It is important to observe that individual homoclinic tangencies are easily destroyed by small perturbations of the invariant manifolds. To construct open examples of surface diffeomorphisms with some tangency, Newhouse started from systems where the tangency is associated to a periodic point inside an invariant hyperbolic set with rich geometric structure. This is illustrated on the right hand side of Fig. 5. His argument requires a very delicate control of distortion, as well as of the dependence of the fractal dimension on the dynamics. Actually, for this reason, his construction is restricted to the Cr topology for
Hyperbolic Dynamical Systems, Figure 5 Homoclinic tangencies
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r 2. A very striking consequence of this construction is that these open sets exhibit coexistence of infinitely many periodic attractors, for each diffeomorphism on a residual subset. A detailed presentation of his result and consequences is given in [9]. Newhouse’s conclusions have been extended in two ways. First, by Palis, Viana, for diffeomorphisms in any dimension, still in the Cr topology with r 2. Then, by Bonatti, Díaz, for C1 diffeomorphisms in any dimension larger or equal than 3. The case of C1 diffeomorphisms on surfaces remains open. As a matter of fact, in this setting it is still unknown whether uniform hyperbolicity is dense in the space of all diffeomorphisms. Partial Hyperbolicity Several extensions of the theory of uniform hyperbolicity have been proposed, allowing for more flexibility, while keeping the core idea: splitting of the tangent bundle into invariant subbundles. We are going to discuss more closely two such extensions. On the one hand, one may allow for one or more invariant subbundles along which the derivative exhibits mixed contracting/neutral/expanding behavior. This is generically referred to as partial hyperbolicity, and a standard reference is the book [5]. On the other hand, while requiring all invariant subbundles to be either expanding or contraction, one may relax the requirement of uniform rates of expansion and contraction. This is usually called non-uniform hyperbolicity. A detailed presentation of the fundamental results about this notion is available e. g. in [6]. In this section we discuss the first type of condition. The second one will be dealt with later. Dominated Splittings Let f : M ! M be a diffeomorphism on a closed manifold M and K be any f -invariant set. A continuous splitting Tx M D E1 (x) ˚ ˚ E k (x); x 2 K of the tangent bundle over K is dominated if it is invariant under the derivative Df and there exists ` 2 N such that for every i < j, every x 2 K, and every pair of unit vectors u 2 E i (x) and v 2 E j (x), one has 1 kD f x` uk < ; 2 kD f x` vk
one has kD f xn uk < Cn kD f xn vk
for all n 1 :
Let f be a diffeomorphism and K be an f -invariant set having a dominated splitting TK M D E1 ˚ ˚ E k . We say that the splitting and the set K are partially hyperbolic the derivative either contracts uniformly E1 or expands uniformly Ek : there exists ` 2 N such that either kD f ` j E1 k
0 are all negative then the trivial solution v(t) 0 is asymptotically stable, and even exponentially stable. The stability theorem of A. M. Lyapunov asserts that, under an additional regularity condition, stability is still valid for nonlinear perturbations w(t) D B(t) w C F(t; w) ; > 0. That is, the trivwith kF(t; w)k ial solution w(t) 0 is still exponentially asymptotically stable. The regularity condition means, essentially, that the limit in (5) does exist, even if one replaces vectors v by elements v1 ^ ^ v l of any lth exterior power of Rd ; 1 l d. By definition, the norm of an l-vector v1 ^ ^ v l is the volume of the parallelepiped determined by the vectors v1 ; : : : ; v k . This condition is usually tricky to check in specific situations. However, the multiplicative ergodic theorem of V. I. Oseledets asserts that, for very general matrix-valued stationary random processes, regularity is an almost sure property. Multiplicative Ergodic Theorem Let f : M ! M be a measurable transformation, preserving some measure , and let A: M ! GL(d; R) be any measurable function such that log kA(x)k is -integrable. The Oseledets theorem states that Lyapunov exponents exist for the sequence An (x) D A( f n1 (x)) : : : A( f (x))A(x) for -almost every x 2 M. More precisely, for -almost every x 2 M there exists k D k(x) 2 f1; : : : ; dg, a filtration
Fx1
Fxk1
Fxk
DR ; d
and numbers 1 (x) < < k (x) such that lim
n!1
1 log kAn (x) vk D i (x) ; n
A(x) Fxi D F fi (x) : It is in the nature of things that, usually, these objects are not defined everywhere and they depend discontinuously on the base point x. When the transformation f is invertible one obtains a stronger conclusion, by applying the previous result also to the inverse automorphism: assuming that log kA(x)1 k is also in L1 (), one gets that there exists a decomposition Vx D E 1x ˚ ˚ E xk ;
const kwk1Cc ; c
Fx0
for all v 2 Fxi n Fxi1 and i 2 f1; : : : ; kg. More generally, this conclusion holds for any vector bundle automorphism V ! V over the transformation f , with A x : Vx ! V f (x) denoting the action of the automorphism on the fiber of x. The Lyapunov exponents i (x), and their number k(x), are measurable functions of x and they are constant on orbits of the transformation f . In particular, if the measure is ergodic then k and the i are constant on a full -measure set of points. The subspaces F ix also depend measurably on the point x and are invariant under the automorphism:
defined at almost every point and such that A(x) E xi D E if (x) and lim
n!˙1
1 log kAn (x) vk D i (x) ; n
for all v 2 E xi different from zero and all i 2 f1; : : : ; kg. These Oseledets subspaces Eix are related to the subspaces F ix through j
j
Fx D ˚ iD1 E xi : Hence, dim E xi D dim Fxi dim Fxi1 is the multiplicity of the Lyapunov exponent i (x). The angles between any two Oseledets subspaces decay sub-exponentially along orbits of f : ! M j M 1 i E f n (x) ; E f n (x) D 0 ; log angle lim n!˙1 n i2I
j…I
for any I f1; : : : ; kg and almost every point. These facts imply the regularity condition mentioned previously and, in particular, k
lim
n!˙1
X 1 i (x) dim E xi : log j det An (x)j D n iD1
Consequently, if det A(x) D 1 at every point then the sum of all Lyapunov exponents, counted with multiplicity, is identically zero.
Hyperbolic Dynamical Systems
Non-Uniformly Hyperbolic Systems The Oseledets theorem applies, in particular, when f : M ! M is a C1 diffeomorphism on some compact manifold and A(x) D D f x . Notice that the integrability conditions are automatically satisfied, for any f -invariant probability measure , since the derivative of f and its inverse are bounded in norm. Lyapunov exponents yield deep geometric information on the dynamics of the diffeomorphism, especially when they do not vanish. We call a hyperbolic measure if all Lyapunov exponents are non-zero at -almost every point. By non-uniformly hyperbolic system we shall mean a diffeomorphism f : M ! M together with some invariant hyperbolic measure. A theory initiated by Pesin provides fundamental geometric information on this class of systems, especially existence of stable and unstable manifolds at almost every point which form absolutely continuous invariant laminations. For most results, one needs the derivative Df to be Hölder continuous: there exists c > 0 such that
surably on x. Another key difference with respect to the uniformly hyperbolic setting is that the numbers Cx and x can not be taken independent of the point, in general. Likewise, one defines local and global unstable manifolds, tangent to E ux at almost every point. Most important for the applications, both foliations, stable and unstable, are absolutely continuous. In the remaining sections we briefly present three major results in the theory of non-uniform hyperbolicity: the entropy formula, abundance of periodic orbits, and exact dimensionality of hyperbolic measures. The Entropy Formula The entropy of a partition P of M is defined by h ( f ; P ) D lim
n!1
where P n is the partition into sets of the form P D P0 \ f 1 (P1 ) \ \ f n (Pn ) with Pj 2 P and X H (P n ) D (P) log (P) :
kD f x D f y k const d(x; y)c : These notions extend to the context of flows essentially without change, except that one disregards the invariant line bundle given by the flow direction (whose Lyapunov exponent is always zero). A detailed presentation can be found in e. g. [6]. Stable Manifolds An essential tool is the existence of invariant families of local stable sets and local unstable sets, defined at -almost every point. Assume is a hyperbolic measure. Let Eux and Esx be the sums of all Oseledets subspaces corresponding to positive, respectively negative, Lyapunov exponents, and let x > 0 be a lower bound for the norm of every Lyapunov exponent at x. Pesin’s stable manifold theorem states that, for -als (x) most every x 2 M, there exists a C1 embedded disk Wloc s tangent to E x at x and there exists C x > 0 such that dist( f n (y); f n (x)) C x e nx dist(y; x) s for all y 2 Wloc (x) : s (x)g is invariant, in the sense Moreover, the family fWloc s s that f (Wloc (x)) Wloc ( f (x)) for -almost every x. Thus, one may define global stable manifolds
W s (x) D
1 [
s f n Wloc (x)
for -almost every x :
nD0
In general, the local stable disks W s (x) depend only mea-
1 H (P n ) ; n
P2P n
The Kolmogorov–Sinai entropy h ( f ) of the system is the supremum of h ( f ; P ) over all partitions P with finite entropy. The Ruelle–Margulis inequality says that h ( f ) is bounded above by the averaged sum of the positive Lyapunov exponents. A major result of the theorem, Pesin’s entropy formula, asserts that if the invariant measure is smooth (for instance, a volume element) then the entropy actually coincides with the averaged sum of the positive Lyapunov exponents ! Z X k h ( f ) D maxf0; j g d : jD1
A complete characterization of the invariant measures for which the entropy formula is true was given by F. Ledrappier and L. S. Young. Periodic Orbits and Entropy It was proved by A. Katok that periodic motions are always dense in the support of any hyperbolic measure. More than that, assuming the measure is non-atomic, there exist Smale horseshoes H n with topological entropy arbitrarily close to the entropy h ( f ) of the system. In this context, the topological entropy h( f ; H n ) may be defined as the exponential rate of growth lim
k!1
1 log #fx 2 H n : f k (x) D xg : k
of the number of periodic points on H n .
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Dimension of Hyperbolic Measures Another remarkable feature of hyperbolic measures is that they are exact dimensional: the pointwise dimension
The reader is referred to the bibliography, especially the book [2] for a review of much recent progress. Bibliography
log (Br (x)) r!0 log r
d(x) D lim
exists at almost every point, where Br (x) is the neighborhood of radius r around x. This fact was proved by L. Barreira, Ya. Pesin, and J. Schmeling. Note that this means that the measure (Br (x)) of neighborhoods scales as r d(x) when the radius r is small.
Future Directions The theory of uniform hyperbolicity showed that dynamical systems with very complex behavior may be amenable to a very precise description of their evolution, especially in probabilistic terms. It was most successful in characterizing structural stability, and also established a paradigm of how general “chaotic” systems might be approached. A vast research program has been going on in the last couple of decades or so, to try and build such a global theory of complex dynamical evolution, where notions such as partial and non-uniform hyperbolicity play a central part.
1. Araujo V, Pacifico MJ (2007) Three Dimensional Flows. In: XXV Brazillian Mathematical Colloquium. IMPA, Rio de Janeiro 2. Bonatti C, Díaz LJ, Viana M (2005) Dynamics beyond uniform hyperbolicity. In: Encyclopaedia of Mathematical Sciences, vol 102. Springer, Berlin 3. Bowen R (1975) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. In: Lecture Notes in Mathematics, vol 470. Springer, Berlin 4. Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic theory. In: Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol 245. Springer, New York 5. Hirsch M, Pugh C, Shub M (1977) Invariant manifolds. In: Lecture Notes in Mathematics, vol 583. Springer, Berlin 6. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge 7. Mañé R (1987) Ergodic theory and differentiable dynamics. Springer, Berlin 8. Palis J, de Melo W (1982) Geometric theory of dynamical systems. An introduction. Springer, New York 9. Palis J, Takens F (1993) Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations. Cambridge University Press, Cambridge 10. Shub M (1987) Global stability of dynamical systems. Springer, New York
Infinite Dimensional Controllability
Infinite Dimensional Controllability OLIVIER GLASS Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France
Article Outline Glossary Definition of the Subject Introduction First Definitions and Examples Linear Systems Nonlinear Systems Some Other Problems Future Directions Bibliography Glossary Infinite dimensional control system A infinite dimensional control system is a dynamical system whose state lies in an infinite dimensional vector space—typically a Partial Differential Equation (PDE)—and depending on some parameter to be chosen, called the control. Exact controllability The exact controllability property is the possibility to steer the state of the system from any initial data to any target by choosing the control as a function of time in an appropriate way. Approximate controllability The approximate controllability property is the possibility to steer the state of the system from any initial data to a state arbitrarily close to a target by choosing a suitable control. Controllability to trajectories The controllability to trajectories is the possibility to make the state of the system join some prescribed trajectory by choosing a suitable control. Definition of the Subject Controllability is a mathematical problem, which consists of determining the targets to which one can drive the state of some dynamical system, by means of a control parameter present in the equation. Many physical systems such as quantum systems, fluid mechanical systems, wave propagation, diffusion phenomena, etc., are represented by an infinite number of degrees of freedom, and their evolution follows some partial differential equation (PDE). Finding active controls in order to properly influence the dynamics
of these systems generate highly involved problems. The control theory for PDEs, and among this theory, controllability problems, is a mathematical description of such situations. Any dynamical system represented by a PDE, and on which an external influence can be described, can be the object of a study from this point of view. Introduction The problem of controllability is a mathematical description of the general following situation. We are given an evolution system (typically a physical one), on which we can exert a certain influence. Is it possible to use this influence to make the system reach a certain state? More precisely, the system takes generally the following form: y˙ D F(t; y; u) ;
(1)
where y is a description of the state of the system, y˙ denotes its derivative with respect to the time t, and u is the control, that is, a parameter which can be chosen in a suitable range. The standard problem of controllability is the following. Given a time T > 0, an initial state y0 and a target y1 , is it possible to find a control function u (depending on the time), such that the solution of the system, starting from y0 and provided with this function u reaches the state y1 at time T? If the state of the system can be described by a finite number of degrees of freedom (typically, if it belongs to an Euclidean space or to a manifold), we call the problem finite dimensional. The present article deals with the case where y belongs to an infinite-dimensional space, typically a Banach space or a Hilbert space. Hence the systems described here have an infinite number of degrees of freedom. The potential range of the applications of the theory is extremely wide: the models from fluid dynamics (see for instance [52]) to quantum systems (see [6,54]), networks of structures (see [23,44]), wave propagation, etc., are countless. In the infinite dimensional setting, the Eq. (1) is typically a partial differential equation, where F acts as a differential operator on the function y, and the influence of u can take multiple different forms: typically, u can be an additional (force) term in the right-hand side of the equation, localized in a part of the domain; it can also appear in the boundary conditions; but other situations can clearly be envisaged (we will describe some of them). Of course the possibilities to introduce a control problem for partial differential equations are virtually infinite. The number of results since the beginning of the theory in the 1960s has been constantly growing and has reached
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huge dimensions: it is, in our opinion, hopeless to give a fair general view of all the activity in the theory. This paper will only try to present some basic techniques connected to the problem of infinite-dimensional controllability. As in the finite dimensional setting, one can distinguish between the linear systems, where the partial differential equation under view is linear (as well as the action of the control), and the nonlinear ones. Structure of the Paper In Sect. “First Definitions and Examples”, we will introduce the problems that are under view and give some examples. The main parts of this paper are Sects. “Linear Systems” and “Nonlinear Systems”, where we consider linear and nonlinear systems, respectively. First Definitions and Examples General Framework We define an infinite-dimensional control system as the following data: 1. an evolution system (typically a PDE)
amples below concern the wave and the heat equations with Dirichlet boundary conditions: these classical equations are reversible and irreversible, respectively, which, as we will see, is of high importance when considering controllability problems. Example 1 Distributed control for the wave/heat equation with Dirichlet boundary conditions. We consider: ˝ a regular domain in Rn , which is in general required to be bounded, ! a nonempty open subdomain in ˝, the wave/heat equation is posed in [0; T]˝, with a localized source term in !:
wave equation v :D v D 1! u ; vj@˝ D 0 ; @2t t v
heat equation @ t v v D 1! u ; vj@˝ D 0 :
In the first case, the state y of the system is given by the couple (v(t; ); @ t v(t; )), for instance considered in the space H01 (˝) L2 (˝) or in L2 (˝) H 1 (˝). In the second case, the state y of the system is given by the function v(t; ), for instance in the space L2 (˝). In both cases, the control is the function u, for instance considered in L2 ([0; T]; L2 (!)).
y˙ D F(t; y; u) ; 2. the unknown y is the state of the system, which is a function depending on time: t 2 [0; T] 7! y(t) 2 Y , where the set Y is a functional space (for instance a Banach or a Hilbert space), or a part of a functional space, 3. a parameter u called the control, which is a time-dependent function t 2 [0; T] 7! u(t) 2 U, where the set U of admissible controls is again some part of a functional space. As a general rule, one expects that for any initial data yjtD0 and any appropriate control function u there exists a unique solution of the system (at least locally in time). In some particular cases, one can find problems “of controllability” type for stationary problems (such as elliptic equations), see for instance [56].
Example 2 Boundary control of the wave/heat equation with Dirichlet boundary conditions. We consider: ˝ a regular domain in Rn , typically a bounded one, ˙ an open nonempty subset of the boundary @˝, the heat/wave equation in [0; T] ˝, with nonhomogeneous boundary conditions inside ˙ : wave equation
v :D @2t t v v D 0 ; vj@˝ D 1˙ u ;
heat equation @ t v v D 0; vj@˝ D 1˙ u :
The states are the same as in the previous example, but here the control u is imposed on a part of the boundary. One can for instance consider the set of controls as L2 ([0; T]; L2 (˙ )) in the first case, as C01 ((0; T) ˙ ) in the second case.
Examples Let us give two classical examples of the situation. These are two types of acting control frequently considered in the literature: in one case, the control acts as a localized source term in the equation, while in the second one, the control acts on a part of the boundary conditions. The ex-
Needless to say, one can consider other boundary conditions than Dirichlet’s. Let us emphasize that, while these two types of control are very frequent, these are not by far the only ones: for instance, one can consider the following “affine control”: the heat equation with a right hand side
Infinite Dimensional Controllability
u(t)g(x) where the control u depends only on the time, and g is a fixed function:
@ t v v D u(t)g(x) ; vj@˝ D 0 ; see an example of this below. Also, one could for instance consider the “bilinear control,” which takes the form:
@ t v v D g(x)u(t)v ; vj@˝ D 0 : Main Problems Now let us give some definitions of the typical controllability problems, associated to a control system. Definition 1 A control system is said to be exactly controllable in time T > 0 if and only if, for all y0 and y1 in Y , there is some control function u : [0; T] ! U such that the unique solution of the system
y˙ D F(t; y; u) (2) yjtD0 D y0 ; satisfies yjtDT D y1 : Definition 2 We suppose that the space Y is endowed with a metric d. The control system is said to be approximately controllable in time T > 0 if and only if, for all y0 and y1 in Y , for any " > 0, there exists a control function u : [0; T] ! U such that the unique solution of the system (2) satisfies d(yjtDT ; y1 ) < " : Definition 3 We consider a particular element 0 of Y . A control system is said to be zero-controllable in time T > 0 if and only if, for all y0 in Y , there exists a control function u : [0; T] ! U such that the unique solution of the system (2) satisfies
Definition 5 All the above properties are said to be fulfilled locally, if they are proved for y0 sufficiently close to the target y1 or to the starting point of the trajectory y(0); they are said to be fulfilled globally if the property is established without such limitations. Remarks We can already make some remarks concerning the problems that we described above. 1. The different problems of controllability should be distinguished from the problems of optimal control, which give another viewpoint on control theory. In general, problems of optimal control look for a control u minimizing some functional J(u; y(u)) ; where y(u) is the trajectory associated to the control u. 2. It is important to notice that the above properties of controllability depend in a crucial way on the choice of the functional spaces Y ; U. The approximate controllability in some space may be the exact controllability in another space. In the same way, we did not specify the regularity in time of the control functions in the above definitions: it should be specified for each problem. 3. A very important fact for controllability problems is that when a problem of controllability has a solution, it is almost never unique. For instance, if a time-invariant system is controllable regardless of the time T, it is clear that one can choose u arbitrarily in some interval [0; T/2], and then choose an appropriate control (for instance driving the system to 0) during the interval [T/2; T]. In such a way, one has constructed a new control which fulfills the required task. The number of controls that one can construct in this way is clearly infinite. This is of course already true for finite dimensional systems. 4. Formally, the problem of interior control when the control zone ! is the whole domain ˝ is not very difficult, since it suffices to consider the trajectory
yjtDT D 0 : v(t) :D v0 C Definition 4 A control system is said to be controllable to trajectories in time T > 0 if and only if, for all y0 in Y and any trajectory y of the system (typically but not necessarily satisfying (2) with u D 0), there exists a control function u : [0; T] ! U such that the unique solution of the system (2) satisfies yjtDT D y(T) :
t (v1 v0 ) ; T
to compute the left-hand side of the equation with this trajectory, and to choose it as the control. However, by doing so, one obtains in general a control with very low regularity. Note also that, on the contrary, as long as the boundary control problem is concerned, the case when ˙ is equal to the whole boundary @˝ is not that simple.
757
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Infinite Dimensional Controllability
5. Let us also point out a “principle” which shows that interior and boundary control problems are not very different. We give the main ideas. Suppose for instance that controllability holds for any domain and subdomain ˝ and !. When considering the controllability problem on ˝ via boundary control localized on ˙ , one may introduce an extension ˝˜ of ˝, obtained by gluing along ˙ an “additional” open set ˝ 2 , so that ˝˜ D ˝ [ ˝2 ; ˝ \ ˝2 ˙ and ˝˜ is regular : Consider now ! ˝2 , and obtain a controllability result on ˝˜ via interior control located in ! (one has, of ˜ course, to extend initial and final states from ˝ to ˝). Consider y a solution of this problem, driving the system from y0 to y1 , in the case of exact controllability, for instance. Then one gets a solution of the boundary controllability problem on ˝, by taking the restriction of y on ˝, and by fixing the trace of y on ˙ as the corresponding control (in the case of Dirichlet boundary conditions), the normal derivative in the case of Neumann boundary conditions, etc. Conversely, when one has some boundary controllability result, one can obtain an interior control result in the following way. Consider the problem in ˝ with interior control distributed in !. Solve the boundary control problem in ˝ n ! via boundary controls in @!. Consider y the solution of this problem. Extend properly the solution y to ˝, and as previously, compute the left hand side for the extension, and consider it as a control (it is automatically distributed in !). Of course, in both situations, the regularity of the control that we obtain has to be checked, and this might need a further treatment. 6. Let us also remark that for linear systems, there is no difference between controllability to zero and controllability to trajectories. In that case it is indeed equivalent to bring y0 to y(T) or to bring y0 y(0) to zero. Note that even for linear systems, on the contrary, approximate controllability and exact controllability differ: an affine subspace of an infinite dimensional space can be dense without filling all the space.
Linear Systems In this section, we will briefly describe the theory of controllability for linear systems. The main tool here is the duality between controllability and observability of the adjoint system, see in particular the works by Lions, Russell, and Dolecki and Russell [24,50,51,63,65]. This duality
is also of primary importance for finite-dimensional systems. Let us first describe informally the method of duality for partial differential equations in two cases given in the examples above. Two Examples The two examples that we wish to discuss are the boundary controllability of the wave equation and the interior controllability of the heat equation. The complete answers to these problems have been given by Bardos, Lebeau and Rauch [9] for the wave equation (see also Burq and Gérard [14] and Burq [13] for another proof and a generalization), and independently by Lebeau and Robbiano [48] and Fursikov and Imanuvilov, see [35] for the heat equation. The complete proofs of these deep results are clearly out of the reach of this short presentation, but one can explain rather easily with these examples how the corresponding controllability problems can be transformed into some observability problems. These observability problems consist of proving a certain inequality. We refer for instance to Lions [51] or Zuazua [76] for a more complete introduction to these problems. First, we notice an important difference between the two equations, which will clearly have consequences concerning the controllability problems. It is indeed wellknown that, while the wave equation is a reversible equation, the heat equation is on the contrary irreversible and has a strong regularizing effect. From the latter property, one sees that it is not possible to expect an exact controllability result for the heat equation: outside the control zone !, the state u(T) will be smooth, and in particular one cannot attain an arbitrary state. As a consequence, while it is natural to seek the exact controllability for the wave equation, it will be natural to look either for approximate controllability or controllability to zero as long as the heat equation is concerned. For both systems we will introduce the adjoint system (typically obtained via integration by parts): it is central in the resolution of the control problems of linear equations. In both cases, the adjoint system is written backward in time form. We consider our two examples separately. Wave Equation We first consider the case of the wave equation with boundary control on ˙ and Dirichlet boundary conditions on the rest of the boundary: 8 2 < @ t t v v D 0 ; v D 1˙ u ; : j@˝ (v; v t )jtD0 D (v0 ; v00 ) :
Infinite Dimensional Controllability
The problem considered is the exact controllability in L2 (˝) H 1 (˝) (recall that the state of the system is (v; v t )), by means of boundary controls in L2 ((0; T) ˙ ). For this system, the adjoint system reads: 8 2 ˆ < @ t t j@˝
ˆ :
( ;
˝
Z
(T; x)v 0T dx
˝
D0; t )jtDT
Z
D0; (3) D(
T;
0 T)
:
Notice that this adjoint equation is well-posed: here it is trivial since the equation is reversible. The key argument which connects the controllability problem of the equation with the study of the properties of the adjoint system is the following duality formula. It is formally easily obtained by multiplying the equation with the adjoint state and in integrating by parts. One obtains
Z
From (4), we see that reaching (v T ; v 0T ) from (0; 0) will be achieved if and only if the relation
Z (; x)v t (; x)dx ˝ “ D
T t (; x)v(; x)dx
(0;T)˙
0
@ 1˙ udt d : @n
(4)
In other words, this central formula describes in a simple manner the jump in the evolution of the state of the system in terms of the control, when measured against the dual state. To make the above computation more rigorous, one can consider for dual state the “classical” solutions in C 0 ([0; T]; H01 (˝)) \ C 1 ([0; T]; L2 (˝)) (these solutions are typically obtained by using a diagonalizing basis for the Dirichlet laplacian, or by using evolution semi-group theory), while the solutions of the direct problem for (v0 ; v1 ; u) 2 L2 (˝) H 1 (˝) L2 (˙ ) are defined in C 0 ([0; T]; L2 (˝)) \ C 1 ([0; T]; H 1 (˝)) via the transposition method; for more details we refer to the book by Lions [51]. Now, due to the linearity of the system and because we consider the problem of exact controllability (hence things will be different for what concerns the heat equation), it is not difficult to see that it is not restrictive to consider the problem of controllability starting from 0 (that is the problem of reaching any y1 when starting from y0 :D (v0 ; v00 ) D 0). Denote indeed R(T; y0 ) the affine subspace made of states that can be reached from y0 at time T for some control. Then calling y(T) the final state of the system for yjtD0 and u D 0, then one has R(T; y0 ) D y(T) C R(T; 0). Hence R(T; y0 ) D Y , R(T; 0) D Y.
0
(T; x)v T dx “ @ D 1˙ udt d (0;T)˙ @n ˝
(5)
is satisfied for all choices of ( T ; T0 ). On the left-hand side, we have a linear form on ( T ; 0 ) in H 1 (˝) L2 (˝), while on the right hand side, we 0 T have a bilinear form on (( T ; T0 ); u). Suppose that we make the choice of looking for a control in the form uD
@ 1˙ ; @n
(6)
for some solution of (3). Then one sees, using Riesz’ theorem, that to solve this problem for (v T ; v 0T ) 2 L2 (˝) H 1 (˝), it is sufficient to prove that the map ( T ; T0 ) 7! k@ /(@n) 1˙ kL 2 ((0;T)˙ ) is a norm equivalent to the H01 (˝) L2 (˝) one: for some C > 0, k(
T;
0 T )k H 10 (˝)L 2 (˝)
Ck
@ 1˙ kL 2 ((0;T)˙ ) : (7) @n
This is the observability inequality to be proved to prove the controllability of the wave equation. Let us mention that the inequality in the other sense, that is, the fact that the linear map ( T ; T0 ) 7! @ /(@n) 1˙ is well-defined and continuous from H01 (˝) L2 (˝) to L2 (@˝) is true but not trivial: it is a “hidden” regularity result, see [66]. When this inequality is proved, a constructive way to 0 select the control is to determine a minimum ( T ; T ) of the functional (
T;
0 T)
“ ˇ ˇ ˇ @ ˇ2 ˇ dt d ˇ ˙ @n ˝ ˛ C h T ; v1 iH 1 H 1 T0 ; v0 L 2 L 2 ;
7! J(
T;
0 T)
:D 0
0
1 2
(8)
then to associate to ( T ; T ) the solution of (5), and finally to set u as in (6). The way described above to determine a particular control—it is clear that not all controls are in the form (6)—is referred as Lions’s HUM method (see [51]). This particular control can be proved to be optimal in the L2 norm, that is, any other control answering to the controllability problem has a larger norm in L2 ((0; T) ˙ ). As a matter of fact, looking for the L2 optimal control among those which answer to the problem is a way to justify the choice (6), see [51].
759
760
Infinite Dimensional Controllability
Heat Equation Now let us consider the heat equation with Dirichlet boundary conditions and localized distributed control: 8 < @ t v v D 1! u ; D0; v : j@˝ vjtD0 D v0 : In this case, we consider in the same way the dual problem: 8 ˆ < @ t D 0 ; j@˝ D 0 ; (9) ˆ : (T) D T : Notice that this adjoint equation is well-posed. Here it is very important that the problem is formulated in a backward way: the backward in time setting compensates the opposite sign before the time derivative. It is clear that the above equation is ill-posed when considering initial data at t D 0. In the same way as for the wave equation, multiplying the equation by the adjoint state and integrating by parts yields, at least formally: “ Z Z T vjtDT dx (0)v0 dx D udtdx: (10) ˝
˝
results; however it requires the analyticity of the coefficients of the operator, and in many situations one cannot use it directly. Let us also mention that as for the exact controllability of the wave equation above, and the zero-controllability for the heat equation below, one can single out a control for the approximate controllability by using a variational approach consisting of minimizing some functional as in (8): see [26,53]. Controllability to zero. Now considering the problem of controllability to zero, we see that, in order that the control u brings the system to 0, it is necessary and sufficient that for all choice of T 2 L2 (˝), we have “
Z
˝
(0)v0 dx D
Here one would like to reason as for the wave equation, that is, make the choice to look for u in the form u :D 1! ; for some solution of the adjoint system. But here the application
(0;T)!
Note that standard methods yield regular solutions for both direct and adjoint equations when v0 and T belong to L2 (˝), and u belongs to L2 ((0; T) !). Now let us discuss the approximate controllability and the controllability to zero problems separately. Approximate controllability. Because of linearity, the approximate controllability is equivalent to the approximate controllability starting from 0. Now the set R(0; T) of all final states v(T) which can be reached from 0, via controls u 2 L2 ((0; T) !), is a vector subspace of L2 (˝). The density of R(0; T) in L2 (˝) amounts to the existence of a nontrivial element in (R(0; T))? . Considering T such an element, and introducing it in (10) together with v0 D 0, we see that this involves the existence of a nontrivial solution of (9), satisfying j(0;T)! D 0 : Hence to prove the approximate controllability, we have to prove that there is no such nontrivial solution, that is, we have to establish a unique continuation result. In this case, this can be proved by using Holmgren’s unique continuation principle (see for instance [40]), which establishes the result. Note in passing that Holmgren’s theorem is a very general and important tool to prove unique continuation
udtdx : (0;T)!
2
!1
“
2
2
N : T 2 L (˝) 7!
dtdx
:
(0;T)!
determines a norm (as seen from the above unique continuation result), but this norm is no longer equivalent to the usual L2 (˝) norm (if it was, one could establish an exact controllability result!). A way to overcome the problem is to introduce the Hilbert space X obtained by completing L2 (˝) for the norm N. In this way, we see that to solve the zero-controllability problem, it is sufficient to prove that the linear mapping T 7! (0) is continuous with respect to the norm N: for some C > 0, k(0)kL 2 (˝) Ck 1! kL 2 ((0;T)!) :
(11)
This is precisely the observability inequality which one has to prove to establish the zero controllability of the heat equation. It is weaker than the observability inequality when the left-hand side is k(T)kL 2 (˝) (which as we noticed is false). When this inequality is proven, a constructive way to determine a suitable control is to determine a minimum T in X of the functional T 7!
1 2
“ (0;T)!
jj2 dtdx C
Z ˝
(0)v0 dx ;
Infinite Dimensional Controllability
then to associate to T the solution of (3), and finally to set u :D 1! : (That the mapping T 7! 1! can be extended to a mapping from X to L2 (!) comes from the definition of X.) So in both situations of exact controllability and controllability to zero, one has to establish a certain inequality in order to get the result; and to prove approximate controllability, one has to establish a unique continuation result. This turns out to be very general, as described in Paragraph Subsect. “Abstract Approach”. Remarks Let us mention that concerning the heat equation, the zero-controllability holds for any time T > 0 and for any nontrivial control zone !, as shown by means of a spectral method—see Lebeau and Robbiano [48], or by using a global Carleman estimate in order to prove the observability property—see Fursikov and Imanuvilov [35]. That the zero-controllability property does not require the time T or the control zone ! to be large enough is natural since parabolic equations have an infinite speed of propagation; hence one should not require much time for the information to propagate from the control zone to the whole domain. The zero controllability of the heat equation can be extended to a very wide class of parabolic equations; for such results global Carleman estimates play a central role, see for instance the reference book by Fursikov and Imanuvilov [35] and the review article by Fernández-Cara and Guerrero [28]. Note also that Carleman estimates are not used only in the context of parabolic equations: see for instance [35] and Zhang [70]. Let us also mention two other approaches for the controllability of the one-dimensional heat equation: the method of moments of Fattorini and Russell (see [27] and a brief description below), and the method of Laroche, Martin and Rouchon [45] to get an explicit approximate control, based on the idea of “flatness” (as introduced by Fliess, Lévine, Rouchon and Martin [33]). On the other hand, the controllability for the wave equation does not hold for any T or any ˙ . Roughly speaking, the result of Bardos, Lebeau and Rauch [9] states that the controllability property holds if and only if every ray of the geometric optics in the domain (reflecting on the boundary) meets the control zone during the time interval [0; T]. In this case, it is also natural that the time should be large enough, because of the finite speed of propagation of the equation: one has to wait for the information coming from the control zone to influence the whole domain. Let us emphasize that in some cases, the geometry
of the domain and the control zone is such that the controllability property does not hold, no matter how long the time T is. An example of this is for instance a circle in which some antipodal regions both belong to the uncontrolled part of the boundary. This is due to the existence of Gaussian beams, that is, solutions that are concentrated along some ray of the geometrical optics (and decay exponentially away from it); when this ray does not meet the control zone, this contradicts in particular (7). The result of [9] relies on microlocal analysis. Note that another important tool used for proving observability inequalities is a multiplier method (see in particular [42,51,60]): however in the case of the wave equation, this can be done only in particular geometric situations. Abstract Approach The duality between the controllability of a system and the observability of its adjoint system turns out to be very general, and can be described in an abstract form due to Dolecki and Russell [24]. For more recent references see [46,69]. Here we will only describe a particular simplified form of the general setting of [24]. Consider the following system y˙ D Ay C Bu ;
(12)
where the state y belongs to a certain Hilbert space H, on which A, which is densely defined and closed, generates a strongly continuous semi-group of bounded operators. The operator B belongs to L(U; H), where U is also a Hilbert space. The solutions of (12) are given by the method of variation of constants so that Z T e(T)A Bu()d : y(T) D e tA y(0) C 0
We naturally associate with the control equation (12), the following observation system:
z˙ D A z ; z(T) D z T ; (13) c :D B z : The dual operator A generates a strongly continuous semi-group for negative times, and c in L2 (0; T; U) is obtained by
c(t) D B e(Tt)A z T : In the above system, the dynamics of the (adjoint) state z is free, and c is called the observed quantity of the system. The core of the method is to connect the controllability properties of (12) with observability properties of (13) such as described below.
761
762
Infinite Dimensional Controllability
Definition 6 The system (13) is said to satisfy the unique continuation property if and only if the following implication holds true: c D 0 in [0; T] H) z D 0 in [0; T] : Definition 7 The system (13) is said to be observable if and only if there exists C > 0 such that the following inequality is valid for all solutions z of (13): kz(T)kH CkckL 2 (0;T;U ) :
(14)
Hence the equivalences 1 and 3 are derived from a classical result from functional analysis (see for instance the book of Brezis [12]): the range of S is dense if and only if S is one-to-one; S is surjective if and only if S satisfies for some C > 0: kz T k CkS z T k ; that is, when (14) is valid. 2. In this case, still due to linearity, the zero-controllability for system (12) is equivalent to the following inclusion: Range (e TA ) Range (S) :
Definition 8 The system (13) is said to be observable at time 0 if and only if there exists C > 0 such that the following inequality is valid for all solutions z of (13): kz(0)kH CkckL 2 (0;T;U ) :
(15)
k(e TA ) hk CkS hk ;
The main property is the following. Duality property 1. The exact controllability of (12) is equivalent to the observability of (13). 2. The zero controllability of (12) is equivalent to the observability at time 0 of (13). 3. The approximate controllability of (12) in H is equivalent to the unique continuation property for (13). Brief explanation of the duality property. The main fact is the following duality formula hy(T); z(T)i H hy(0); z(0)i H Z TD E u(); B e(T)A z(T) d ; D U
0
(16)
which in fact can be used to define the solutions of (12) by transposition. 1 and 3. Now it is rather clear that by linearity, we can reduce the problems of exact and approximate controllability to the ones when y(0) D 0. Now the property of exact controllability for system (12) is equivalent to the surjectivity of the operator S : u 2 L2 (0; T; U) 7!
Z
T
e(T)A Bu() d 2 H ;
0
and the approximate controllability is equivalent to the density of its range. By (16) its adjoint operator is S : z T 2 H 7!
Z 0
T
In this case there is also a functional analysis result (generalizing the one cited above) which asserts the equivalence of this property with the existence of C > 0 such that
B e(T)A z T d 2 L2 (0; T; U) :
see Dolecki and Russell [24] and Douglas [25] for more general results. It follows that in many situations, the exact controllability of a system is proved by establishing an observability inequality on the adjoint system. But this general method is not the final point of the theory: not only is this only valid for linear systems, but importantly as well, it turns out that these types of inequalities are in general very difficult to establish. In general when a result of exact controllability is established by using this duality, the largest part of the proof is devoted to establishing the observability inequality. Another information given by the observability is an estimate of the size of the control. Indeed, following the lines of the proof of the correspondence between observability and controllability, one can see that one can find a control u which satisfies: Case of exact controllability: kukL 2 (0;T;U) Cobs ky1 e TA y0 kH ; Case of zero controllability: kukL 2 (0;T;U) Cobs ky0 kH ; where in both cases Cobs determines a constant for which the corresponding observability inequality is true. (Obviously, not all the controls answering to the question satisfy this estimate). This gives an upper bound on the size of a possible control. This can give some precise estimates on the cost of the control in terms of the parameters of the problem, see for instance the works of Fernández-Cara and Zuazua [30] and Miller [58] and Seidman [66] concerning the heat equation.
Infinite Dimensional Controllability
Some Different Methods Herein we will discuss two methods that do not rely on controllability/observability duality. This does not pretend to give a general vision of all the techniques that can be used in problems of controllability of linear partial differential equations. We just mention them in order to show that in some situations, duality may not be the only tool available. Characteristics First, let us mention that in certain situations, one may use a characteristics method (see e. g. [22]). A very simple example is the first-dimensional wave equation, 8 < v t t vx x D 0 ; v D0; : xjxD0 v xjxD1 D u(t) ;
v t C A(x)v x C B(x)v D 0 ;
L2 (0; T).
where the control is u(t) 2 This is taken from Russell [62]. The problem is transformed into @w D @t
0 1
1 0
@w @x
w :D
with
w1 w2
w2 (t; 0) D 0
:D
vt vx
;
and w2 (t; 0) D u(t) :
Of course, this means that w1 w2 and w1 Cw2 are constant along characteristics, which are straight lines of slope 1 and 1, respectively (this is d’Alembert’s decomposition). Now one can deduce an explicit appropriate control for the controllability problem in [0; 1] for T > 2, by constructing the solution w of the problem directly (and one takes the values of w2 at x D 1 as the “resulting” control u). The function w is completely determined from the initial and final values in the domains of determinacy D1 and D2 : D1 :D f(t; x) 2 [0; T] [0; 1] / x C t 1g
and
D2 :D f(t; x) 2 [0; T] [0; 1] / t x T 1g : That T > 2 means that these two domains do not intersect. Now it suffices to complete the solution w in D3 :D [0; T] [0; 1] n (D1 [ D2 ). For that, one chooses w1 (0; t) arbitrarily in the part ` of the axis f0g [0; 1] located between D1 and D2 , that is for xD0
the symmetric role of x and t, one considers x as the time. Then the initial condition is prescribed on `, as well as the boundary conditions on the two characteristic lines x C t D 1 and x t D T 1. One can solve elementarily this problem by using the characteristics, and this finishes the construction of w, and hence of u. Note that as a matter of fact, the observability inequality in this (one-dimensional) case is also elementary to establish, by relying on Fourier series or on d’Alembert’s decomposition, see for instance [23]. The method of characteristics described above can be generalized in broader situations, including for instance the problem of boundary controllability of one-dimensional linear hyperbolic systems
and 1 t T 1 :
Once this choice is made, it is not difficult to solve the Goursat problem consisting of extending w in D3 : using
where A is a real symmetric matrix with eigenvalues bounded away from zero, and A and B are smooth; see for instance Russell [65]. As a matter of fact, in some cases this method can be used to establish the observability inequality from the controllability result (while in most cases the other implication in the equivalence is used). Of course, the method of characteristics may be very useful for transport equations @f C v:r f D g @t
or
@f C div (v f ) D g : @t
An example of this is the controllability of the Vlasov– Poisson equation, see [36]. Let us finally mention that this method is found also to tackle directly several nonlinear problems, as we will see later. Moments Another method which we would like to briefly discuss is the method of moments (see for instance Avdonin and Ivanov [4] for a general reference, see also Russell [65]), which can appear in many situations; in particular this method was used by Fattorini and Russell to prove the controllability of the heat equation in one space dimension, see [27]. Consider for instance the problem of controllability of the one-dimensional heat equation
v t v x x D g(x)u(t) ; vj[0;T]f0;1g D 0 :
Actually in [27], much more general situations are considered: in particular it concerns more general parabolic equations and boundary conditions, and boundary controls can also be included in the discussion. It is elementary to develop the solution in the L2 (0; 1) orthonormal basis (sin(k x)) k2N . One obtains that the
763
764
Infinite Dimensional Controllability
state zero is reached at time T > 0 if and only if X
2
ek T v k sin(k x)
k>0
D
XZ
T
ek
2 (Tt)
g k u(t) sin(k x)dt ;
k>0 0
where vk and g k are the coordinates of v0 and g in the basis. Clearly, this means that we have to find u such that for all k 2 N, Z
T
ek
2 (Tt)
2
g k u(t)dt D ek T v k :
0
The classical Muntz–Szász theorem states that for an increasing family (n )n2N of positive numbers, the family fen t ; n 2 N g is dense in L2 (0; T) if and only if X 1 D C1 ; n0 n and in the opposite case, the family is independent and spans a proper closed subspace of L2 (0; T). Here the exponential family which we consider is n D n2 and we are in the second situation. The same method applies for other problems in which n cannot be completely computed but is a perturbation of n2 ; this allows us to treat a wider class of problems. Now in this situation (see e. g. [65]), one can construct in L2 (0; T) a biorthogonal family to fen t ; n 2 N g, that is a family (p n (t))n2N 2 (L2 (0; T))N satisfying Z
T
2
p n (t)ek t dt D ı kn :
0
Once such a family is obtained, one has formally the following solution, under the natural assumption that jg k j ck ˛ : u(t) D
X k2N
ek T v k p k (t) : gk
which allows us to conclude.
The most frequent method for dealing with control problems of nonlinear systems is the natural one (as for the Cauchy problem): one has to linearize (at least in some sense) the equation, try to prove some controllability result on the linear equation (for instance by using the duality principle), and then try to pass to the nonlinear system via typical methods such as inverse mapping theorem, fixed point theory, iterative schemes. . . As for the usual inverse mapping theorem, it is natural to hope for a local result from the controllability of the linear problem. Here we find an important difference between linear and nonlinear systems: while for linear systems, no distinction has to be made between local and global results, concerning the nonlinear systems, the two problems are really of different nature. One should probably not expect a very general result indicating that the controllability of the linearized equation involves the local controllability of the nonlinear system. The linearization principle is a general approach which one has to adapt to the different situations that one can meet. We give below some examples that present to the reader some existing approaches. Some Linearization Situations Let us first discuss some typical situations where the linearized equation has good controllability properties, and one can hope to get a result (in general local) from this information. In some situations where the nonlinearity is not too strong, one can hope to get global results. As previously, the situation where the underlying linear equation is reversible and the situation when it is not have to be distinguished. We briefly describe this in two different examples. Semilinear Wave Equation Let us discuss first an example with the wave equation. This is borrowed from the work of Zuazua [71,72], where the equation considered is
To actually get a control in L2 (0; T), one has to estimate kp k kL 2 , in order that the above sum is well-defined in L2 . In [27] it is proven that one can construct pk in such a way that kp k kL 2 (0;T) K0 exp(K1 ! k ) ;
Nonlinear Systems
! k :D
p k ;
v t t v x x C f (v) D u 1! in [0; 1] ; vjxD0 D 0; vjxD1 D 0 ;
where ! :D (l1 ; l2 ) is the control zone and the nonlinearity f 2 C 1 (R; R) is at most linear at infinity (see however the remark below) in the sense that for some C > 0, j f (x)j C(1 C jxj) on R :
(17)
The global exact controllability in H01 L2 (recall that the state of the system is (v; v t )) by means of a control in
Infinite Dimensional Controllability
L2 ((0; T) !) is proved by using the following linearization technique. In [72], it is proven that the following linearized equation:
v t t v x x C a(x)v D u 1! in [0; 1] ; vjxD0 D 0; vjxD1 D 0 :
is controllable in H01 (0; 1) L2 (0; 1) through u(t) 2 L2 ((0; T) !), for times T > 2 max(l1 ; 1 l2 ), for any a 2 L1 ((0; T) (0; 1)). The corresponding observability inequality is
Using the above information, one can deduce estimates for vˆ in C 0 ([0; T]; L2 (0; 1)) \ L2 (0; T; H01 ) independently of v, and then show by Schauder’s fixed point theorem that the above process has a fixed point, which shows a global controllability result for this semilinear wave equation. Remark 1 As a matter of fact, [72] proves the global exact controllability for f satisfying the weaker assumption lim
jxj!C1
k(
T;
0 T )k L 2 (˝)H 1 (˝)
Cobs k 1! kL 2 ((0;T)!) :
Moreover, the observability constant that one can obtain can be bounded in the following way:
lim inf
f (x) D f (0) C x g(x) ;
vˆt t vˆx x C vˆ g(v) D f (0) C u 1! in [0; 1] ; vˆjxD0 D 0; vˆjxD1 D 0 :
(Note that the “drift” term f (0) on the right hand side can be integrated in the final state—just consider the solution of the above system with u D 0 and vˆjtD0 D 0 and withdraw it—or integrated in the formulation (5)). Here an issue is that, as we recalled earlier, a controllability problem has almost never a unique solution. The method in [72] consists of selecting a particular control, which is the one of smallest L2 -norm. Taking the fact that g is bounded and (18) into account, this particular control satisfies kukL 2 (0;T) C(kv0 kH 1 C kv00 kL 2 C kv1 kH 1 C kv10 kL 2 C j f (0)j); for some C > 0 independent of v.
for some p > 2 (and ! 6D (0; 1)) then the system is not globally controllable due to blow-up phenomena. To obtain this result one has to use Leray–Schauder’s degree theory instead of Schauder’s fixed point theorem. For analogous conditions on the semilinear heat equation, see Fernández-Cara and Zuazua [31]. Burgers Equation Now let us discuss a parabolic example, namely the local controllability to trajectories of the viscous Burgers equation:
where g is continuous and bounded. The idea is to associate to any v 2 L1 ((0; T) (0; 1)) a solution of the linear control problem
0
j f (x)j >0 (1 C jxj) log p (jxj)
(18)
where ˛ is nondecreasing in the second variable. One would like to describe a fixed-point scheme as follows. Given v, one considers the linearized problem around v, solves this problem and deduces a solution vˆ of the problem of controllability from (v0 ; v00 ) to (v1 ; v10 ) in time T. More precisely, the scheme is constructed in the following way. We write
D0:
This is optimal since it is also proven in [72] that if
jxj!C1
Cobs ˛(T; kak1 ) ;
j f (x)j (1 C jxj) log2 (jxj)
0
v t C (v 2 )x v x x D 0 in (0; T) (0; 1) ; vjxD0 D u0 (t) and vjxD1 D u1 (t) ;
(19)
controlled on the boundary via u0 and u1 . This is taken from Fursikov and Imanuvilov [34]. Now the linear result concerns the zero-controllability via boundary controls of the system
v t C (zv)x v x x D 0 in (0; T) (0; 1) ; vjxD0 D u0 (t) and vjxD1 D u1 (t) :
(20)
Consider X the Hilbert space composed of functions z in L2 (0; T; H 2 (0; 1)) such that z t 2 L2 (0; 1; L2 (0; 1)). In [34] it is proved that given v0 2 H 1 (0; 1) and T > 0, one can construct a map which to any z in X, associates v 2 X such that v is a solution of (20) which drives v0 to 0 during time interval [0; T], and moreover this map is compact from X to X. As in the previous situation, the particular control (u0 ; u1 ) has to be singled out in order for the above mapping to be single-valued (and compact). But here, the criterion is not quite the optimality in L2 -norm of the control. The idea is the following: first, one transforms the controllability problem for (20) from y0 to 0 into
765
766
Infinite Dimensional Controllability
a problem of “driving” 0 to 0 for a problem with righthand side:
w t C (zw)x w x x D f0 in (0; T) (0; 1) ; (21) wjxD0 D u0 (t) and wjxD1 D u1 (t) ; for some f 0 supported in (T/3; 2T/3) (0; 1). For this, one introduces 2 C 1 ([0; T]; R) such that D 1 during [0; T/3] and D 0 during [2T/3; T], and vˆ the solution of (20) starting from v0 with u0 D u1 D 0, and considers w :D v ˆv . Now the operator mentioned above is the one which to z associates the solution of the controllability problem which minimizes the L2 -norm of w among all the solutions of this controllability problem. The optimality criterion yields a certain form for the solution. That the corresponding control exists (and is unique) relies on a Carleman estimate, see [34] (moreover this allows estimates on the size of w). As a matter of fact, to obtain the compactness of the operator, one extends the domain, solves the above problem in this extended domain, and then uses an interior parabolic regularity result to have bounds in smaller spaces, we refer to [34] for more details. Once the operator is obtained, the local controllability to trajectories is obtained as follows. One considers v a trajectory of the system (19), belonging to X. Withdrawing (19) for v to (19) for the unknown, we see that the problem to solve through boundary controls is
y t y x x C [(2v C y)y]x D f0 in (0; T) (0; 1) ; yjxD0 D v0 v(0) and yjtDT D 0 :
Now consider v0 2 H 1 (0; 1) such that kv0 v(0)kH 1 (0;1) < r ; for r > 0 to be chosen. To any y 2 X, one associates the solution of the controllability problem (21) constructed above, driving v0 vˆ(0) to 0, for z :D (2ˆv C y). The estimates on the solution of the control problem allow one to establish that the unit ball of X is stable by this process provided r is small enough. The compactness of the process is already proved, so Schauder’s theorem allows one to conclude. Note that it is also proved in [34] that the global approximate controllability does not hold. Some Other Examples Let us also mention two other approaches that may be useful in this type of situation. The first is the use of Kakutani–Tikhonov fixed-point theorem for multivalued maps (cf. for instance [68]), see in particular [38,39] and [26]. Such a technique is particularly
useful, because it avoids the selection process of a particular control. One associates to v the set T(v) of all vˆ solving the controllability problem for the equation linearized around v, with all possible controls (in a suitable class). Then under appropriate conditions, one can find a fixed point in the sense that v 2 T(v). Another approach that is very promising is the use of a Nash–Moser process, see in particular the work [10] by Beauchard. In this paper the controllability of a Schrödinger equation via a bilinear control is considered. In that case, one can solve some particular linearized equation (as a matter of fact, the return method described in the next paragraph is used), but with a loss of derivative; as a consequence the approaches described above fail, but the use of Nash–Moser’s theorem allows one to get a result. Note finally that in certain other functional settings, the controllability of this system fails, as shown by using a general result on bilinear control by Ball, Marsden and Slemrod [5]. The Return Method It occurs that in some situations, the linearized equation is not systematically controllable, and one cannot hope by applying directly the above process to get even local exact controllability. The return method was introduced by Coron to deal with such situations (see in particular [18]). As a matter of fact, this method can be useful even when the linearized equation is controllable. The principle of the method is the following: find a particular trajectory y of the nonlinear system, starting at some base point (typically 0) and returning to it, such that the linearized equation around this is controllable. In that case, one can hope to find a solution of the nonlinear local controllability problem close to y. A typical situation of this is the two-dimensional Euler equation for incompressible inviscid fluids (see Coron [16]), which reads 8 < @ t y C (y:r)y D r p in ˝ ; divy D 0 in ˝ ; (22) : y:n D 0 on @˝ n ˙ ; where the unknown is the velocity field y : ˝ ! R2 (the pressure p can be eliminated from the equation), ˝ is a regular bounded domain (simply connected to simplify), n is the unit outward normal on the boundary, and ˙ @˝ is the control zone. On ˙ , the natural control which can be assigned is the normal velocity y:n and the vorticity ! :D curly :D @1 y 2 @2 y 1 at “entering” points, that is points where y:n < 0. Let us discuss this example in an informal way. As noticed by J.-L. Lions, the linearized equation around the null
Infinite Dimensional Controllability
state 8 < @ t y D r p in ˝ ; divy D 0 in ˝; : y:n D 0 on @˝ n ˙ ; is trivially not controllable (even approximately). Now the main goal is to find the trajectory y such that the linearized equation near y is controllable. It will be easier to work with the vorticity formulation
@ t ! C (y:r)! D 0 in ˝ ; curl y D !; divy D 0 :
(23)
In fact, one can show that assigning y(T) is equivalent to assigning both !(T) in ˝ and y(T):n on ˙ , and since this latter is a part of the control, it is sufficient to know how to assign the vorticity of the final state. We can linearize (23) in the following way: to y one associates yˆ through
@ t ! C (y:r)! D 0 in ˝ ; curl yˆ D !; div yˆ D 0 :
(24)
Considering Eq. (24), we see that if the flow of y is such that any point in ˝ at time T “comes from” the control zone ˙ , then one can assign easily ! through a method of characteristics. Hence one has to find a solution y of the system, starting and ending at 0, and such that in its flow, all points in ˝ at time T, come from ˙ at some stage between times 0 and T. Then a simple Gronwall-type argument shows that this property holds for y in a neighborhood of y, hence Eq. (24) is controllable in a neighborhood of y. Then a fixed-point scheme allows one to prove a controllability result locally around 0. But the Euler equation has some time-scale invariance: y(t; x) is a solution on [0; T] ) y (t; x) :D 1 y(1 t; x) is a solution on [0; T] : Hence given y0 and y1 , one can solve the problem of driving y0 to y1 for small enough. Changing the variables, one see that it is possible to drive y0 to y1 in time T, that is, in smaller times. Hence one deduces a global controllability result from the above local result. As a consequence, the central part of the proof is to find the function y. This is done by considering a special type of solution of the Euler equation, namely the potential solution: any y :D r (t; x) with regular satisfying x (t; x) D 0 in ˝;
8t 2 [0; T];
@n D 0 on @˝;
8t 2 [0; T] ;
satisfies (22). In [16] it is proven that there exists some satisfying the above equation and whose flow makes all points at time T come from ˙ . This concludes the argument. This method has been used in various situations; see [18] and references therein. Let us underline that this method can be of great interest even in the cases where the linearized equation is actually controllable. An important example obtained by Coron concerns the Navier–Stokes equation and is given in [17] (see also [19]): here the return method is used to prove some global approximate result, while the linearized equation is actually controllable but yields in general a local controllability result (see in particular [29,35,41]). Some Other Methods Let us finally briefly mention that linearizing the equation (whether using the standard approach or the return method) is not systematically the only approach to the controllability of a nonlinear system. Sometimes, one can “work at the nonlinear level.” An important example is the control of one-dimensional hyperbolic systems: v t C A(v)v x D F(v); v : [0; T] [0; 1] ! Rn ;
(25)
via the boundary controls. In (25), A satisfies the hyperbolicity property that it possesses at every point n real eigenvalues; these are moreover supposed to be strictly separated from 0. In the case of regular C1 solutions, this was approached by a method of characteristics to give general local results, see in particular the works by Cirinà [15] and Li and Rao [49]. Interestingly enough, the linear tool of duality between observability and controllability has some counterpart in this particular nonlinear setting, see Li [55]. In the context of weak entropy solutions, some results for particular systems have been obtained via the ad hoc method of front-tracking, see in particular Ancona, Bressan and Coclite [2], the author [37] and references therein. Other nonlinear tools can be found in Coron’s book [18]. Let us mention two of them. The first one is power series expansion. It consists of considering, instead of the linearization of the system, the development to higher order of the nonlinearity. In such a way, one can hope to attain the directions which are unreachable for the linearized system. This has for instance applications for the Korteweg–de-Vries equation, see Coron and Crépeau [20] and the earlier work by Rosier [61]. The other one is quasistatic deformations. The general idea is to find an (explicit) “almost trajectory” (y("t); u("t)) during [0; T/"] of
767
768
Infinite Dimensional Controllability
the control system y˙ D f (y; u), in the sense that d y("t) f (y("t); u("t)) dt is of order " (due to the “slow” motion). Typically, the trajectory (y; u) is composed of equilibrium states (that is f (y(); u()) D 0). In such a way, one can hope to connect y(0) to a state close to y(T), and then to exactly y(T) via a local result. This was used for instance by Coron and Trélat in the context of a semilinear heat equation, see [21]. Finally, let us cite a recent approach by Agrachev and Sarychev [1] (see also [67]), which uses a generalization of the “Lie bracket” approach (a standard approach from finite-dimensional nonlinear systems), to obtain some global approximate results on the Navier–Stokes equation with a finite-dimensional (low modes) affine control.
Future Directions There are many future challenging problems for control theory. Controllability is one of the possible approaches to try to construct strategies for managing complex systems. On the road towards systems with increasing complexity, many additional difficulties have to be considered in the design of the control law: one should expect the control to take into account the possible errors of modelization, of measurement of the state, of the control device, etc. All of these should be included in the model so some robustness of the control can be expected, and to make it closer to the real world. Of course, numerics should play an important role in the direction of applications. Moreover, one should expect more and more complex systems, such as environmental or biological systems (how are regulation mechanisms designed in a natural organism?), to be approached from this point of view. We refer for instance to [32,59] for some discussions on some perspectives of the theory.
Some Other Problems As we mentioned earlier, many aspects of the theory have not been referred to herein. Let us cite some of them. Concerning the connection between the problem of controllability and the problem of stabilization, we refer to Lions [50], Russell [63,64,65] and Lasiecka and Triggiani [47]. A very important problem which has recently attracted great interest is the problem of numerics and discretization of distributed systems (see in particular [74,75] and references therein). The main difficulty here comes from the fact that the operations of discretizing the equation and controlling it do not commute. Other important questions considered in particular in the second volume of Lions’s book [51] are the problems of singular perturbations, homogenization, thin domains in the context controllability problems. Here the questions are the following: considering a “perturbed” system, for which we have some controllability result, how does the solution of the controllability problem (for instance associated to the L2 -optimal control) behave as the system converges to its limit? This kind of question is the source of numerous new problems. Another subject is the controllability of equations with some stochastic terms (see for instance [8]). One can also consider systems with memory (see again [51] or [7]). Finally, let us mention that many partial differential equations widely studied from the point of view of Cauchy theory, still have not been studied from the point of view of controllability. The reader looking for more discussion on the subject can consider the references below.
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Notes in Pure Appl Math, vol 10. Dekker, New York, pp 291– 319 Russell DL (1978) Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev 20:639–739 Seidman TI (1988) How violent are fast controls? Math Control Signal Syst 1(1):89–95 Shirikyan A (2006) Approximate control lability of threedimensional Navier–Stokes equations. Comm Math Phys 266(1):123–151 Smart DR (1974) Fixed point theorems, Cambridge Tracts in Mathematics, No 66. Cambridge University Press, New York Tucsnak M, Weiss G (2009) Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts. Birkhäuser, Basel Zhang X (2001) Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J Cont Optim 39:812–834 Zuazua E (1990) Exact controllability for the semilinear wave equation. J Math Pures Appl 69(1):1–31 Zuazua E (1993) Exact controllability for semilinear wave equations in one space dimension. Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129 Zuazua E (1998) Some problems and results on the controllability of Partial Differential Equations. Proceedings of the Second European Conference of Mathematics, Budapest, July 1996. Progress in Mathematics, vol 169. Birkhäuser, Basel, pp 276–311 Zuazua E (2002) Controllability of Partial Differential Equations and its Semi-Discrete Approximations. Discret Continuous Dyn Syst 8(2):469–513 Zuazua E (2005) Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev 47(2):197–243 Zuazua E (2006) Controllability and observability of partial differential equations: some results and open problems. In: Dafermos C, Feiresl E (eds) Handbook of differential equations: evolutionary differential equations, vol 3. Elsevier/North-Holland, Amsterdam
Inverse Scattering Transform and the Theory of Solitons
Inverse Scattering Transform and the Theory of Solitons TUNCAY AKTOSUN Department of Mathematics, University of Texas at Arlington, Arlington, USA Article Outline Glossary Definition of the Subject Introduction Inverse Scattering Transform The Lax Method The AKNS Method Direct Scattering Problem Time Evolution of the Scattering Data Inverse Scattering Problem Solitons Future Directions Bibliography Glossary AKNS method A method introduced by Ablowitz, Kaup, Newell, and Segur in 1973 that identifies the nonlinear partial differential equation (NPDE) associated with a given first-order system of linear ordinary differential equations (LODEs) so that the initial value problem (IVP) for that NPDE can be solved by the inverse scattering transform (IST) method. Direct scattering problem The problem of determining the scattering data corresponding to a given potential in a differential equation. Integrability A NPDE is said to be integrable if its IVP can be solved via an IST. Inverse scattering problem The problem of determining the potential that corresponds to a given set of scattering data in a differential equation. Inverse scattering transform A method introduced in 1967 by Gardner, Greene, Kruskal, and Miura that yields a solution to the IVP for a NPDE with the help of the solutions to the direct and inverse scattering problems for an associated LODE. Lax method A method introduced by Lax in 1968 that determines the integrable NPDE associated with a given LODE so that the IVP for that NPDE can be solved with the help of an IST. Scattering data The scattering data associated with a LODE usually consists of a reflection coefficient which is a function of the spectral parameter ; a finite number of constants j that correspond to the poles
of the transmission coefficient in the upper half complex plane, and the bound-state norming constants whose number for each bound-state pole j is the same as the order of that pole. It is desirable that the potential in the LODE is uniquely determined by the corresponding scattering data and vice versa. Soliton The part of a solution to an integrable NPDE due to a pole of the transmission coefficient in the upper half complex plane. The term soliton was introduced by Zabusky and Kruskal in 1965 to denote a solitary wave pulse with a particle-like behavior in the solution to the Korteweg-de Vries (KdV) equation. Time evolution of the scattering data The evolvement of the scattering data from its initial value S(; 0) at t D 0 to its value S(; t) at a later time t: Definition of the Subject A general theory to solve NPDEs does not seem to exist. However, there are certain NPDEs, usually first order in time, for which the corresponding IVPs can be solved by the IST method. Such NPDEs are sometimes referred to as integrable evolution equations. Some exact solutions to such equations may be available in terms of elementary functions, and such solutions are important to understand nonlinearity better and they may also be useful in testing accuracy of numerical methods to solve such NPDEs. Certain special solutions to some of such NPDEs exhibit particle-like behaviors. A single-soliton solution is usually a localized disturbance that retains its shape but only changes its location in time. A multi-soliton solution consists of several solitons that interact nonlinearly when they are close to each other but come out of such interactions unchanged in shape except for a phase shift. Integrable NPDEs have important physical applications. For example, the KdV equation is used to describe [14,23] surface water waves in long, narrow, shallow canals; it also arises [23] in the description of hydromagnetic waves in a cold plasma, and ion-acoustic waves in anharmonic crystals. The nonlinear Schrödinger (NLS) equation arises in modeling [24] electromagnetic waves in optical fibers as well as surface waves in deep waters. The sine-Gordon equation is helpful [1] in analyzing the magnetic field in a Josephson junction (gap between two superconductors). Introduction The first observation of a soliton was made in 1834 by the Scottish engineer John Scott Russell at the Union Canal between Edinburgh and Glasgow. Russell reported [21] his observation to the British Association of the Advance-
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Inverse Scattering Transform and the Theory of Solitons
ment of Science in September 1844, but he did not seem to be successful in convincing the scientific community. For example, his contemporary George Airy, the influential mathematician of the time, did not believe in the existence of solitary water waves [1]. The Dutch mathematician Korteweg and his doctoral student de Vries published [14] a paper in 1895 based on de Vries’ Ph.D. dissertation, in which surface waves in shallow, narrow canals were modeled by what is now known as the KdV equation. The importance of this paper was not understood until 1965 even though it contained as a special solution what is now known as the one-soliton solution. Enrico Fermi in his summer visits to the Los Alamos National Laboratory, together with J. Pasta and S. Ulam, used the computer named Maniac I to computationally analyze a one-dimensional dynamical system of 64 particles in which adjacent particles were joined by springs where the forces also included some nonlinear terms. Their main goal was to determine the rate of approach to the equipartition of energy among different modes of the system. Contrary to their expectations there was little tendency towards the equipartition of energy but instead the almost ongoing recurrence to the initial state, which was puzzling. After Fermi died in November 1954, Pasta and Ulam completed their last few computational examples and finished writing a preprint [11], which was never published as a journal article. This preprint appears in Fermi’s Collected Papers [10] and is also available on the internet [25]. In 1965 Zabusky and Kruskal explained [23] the Fermi-Pasta-Ulam puzzle in terms of solitary wave solutions to the KdV equation. In their numerical analysis they observed “solitary-wave pulses”, named such pulses “solitons” because of their particle-like behavior, and noted that such pulses interact with each other nonlinearly but come out of interactions unaffected in size or shape except for some phase shifts. Such unusual interactions among solitons generated a lot of excitement, but at that time no one knew how to solve the IVP for the KdV equation, except numerically. In 1967 Gardner, Greene, Kruskal, and Miura presented [12] a method, now known as the IST, to solve that IVP, assuming that the initial profile u(x; 0) decays to zero sufficiently rapidly as x ! ˙1: They showed that the integrable NPDE, i. e. the KdV equation, u t 6uu x C u x x x D 0 ;
(1)
is associated with a LODE, i. e. the 1-D Schrödinger equation
d2 C u(x; t) dx 2
D k2
;
(2)
and that the solution u(x, t) to (1) can be recovered from the initial profile u(x; 0) as explained in the diagram given in Sect. “Inverse Scattering Transform”. They also explained that soliton solutions to the KdV equation correspond to a zero reflection coefficient in the associated scattering data. Note that the subscripts x and t in (1) and throughout denote the partial derivatives with respect to those variables. In 1972 Zakharov and Shabat showed [24] that the IST method is applicable also to the IVP for the NLS equation iu t C u x x C 2ujuj2 D 0 ;
(3)
p where i denotes the imaginary number 1. They proved that the associated LODE is the first-order linear system 8 d ˆ ˆ D i C u(x; t) ; < dx ˆ ˆ : d D i u(x; t) ; dx
(4)
where is the spectral parameter and an overline denotes complex conjugation. The system in (4) is now known as the Zakharov–Shabat system. Soon afterwards, again in 1972 Wadati showed in a one-page publication [22] that the IVP for the modified Korteweg-de Vries (mKdV) equation u t C 6u2 u x C u x x x D 0 ;
(5)
can be solved with the help of the inverse scattering problem for the linear system 8 d ˆ ˆ D i C u(x; t) ; < dx ˆ ˆ : d D i u(x; t) : dx
(6)
Next, in 1973 Ablowitz, Kaup, Newell, and Segur showed [2,3] that the IVP for the sine-Gordon equation u x t D sin u ; can be solved in the same way by exploiting the inverse scattering problem associated with the linear system 8 d 1 ˆ ˆ D i u x (x; t) ; < dx 2 ˆ d 1 ˆ : D i C u x (x; t) : dx 2 Since then, many other NPDEs have been discovered to be solvable by the IST method.
Inverse Scattering Transform and the Theory of Solitons
direct scattering for LODE at tD0
u(x; 0) ! S(; 0) ? ? ? ?time evolution of scattering data solution to NPDEy y u(x; t)
S(; t)
inverse scattering for LODE at time t
Inverse Scattering Transform and the Theory of Solitons, Diagram 1 The method of inverse scattering transform
Our review is organized as follows. In the next section we explain the idea behind the IST. Given a LODE known to be associated with an integrable NPDE, there are two primary methods enabling us to determine the corresponding NPDE. We review those two methods, the Lax method and the AKNS method, in Sect. “The Lax Method” and in Sect. “The AKNS Method”, respectively. In Sect. “Direct Scattering” we introduce the scattering data associated with a LODE containing a spectral parameter and a potential, and we illustrate it for the Schrödinger equation and for the Zakharov–Shabat system. In Sect. “Time Evolution of the Scattering Data” we explain the time evolution of the scattering data and indicate how the scattering data sets evolve for those two LODEs. In Sect. “Inverse Scatering Problem” we summarize the Marchenko method to solve the inverse scattering problem for the Schrödinger equation and that for the Zakharov–Shabat system, and we outline how the solutions to the IVPs for the KdV equation and the NLS equation are obtained with the help of the IST. In Sect. “Solitons” we present soliton solutions to the KdV and NLS equations. A brief conclusion is provided in Sect. “Future Directions”.
the direct scattering problem for the LODE. On the other hand, the problem of determining u(x, t) from S(; t) is known as the inverse scattering problem for that LODE. The IST method for an integrable NPDE can be explained with the help of Diagram 1. In order to solve the IVP for the NPDE, i. e. in order to determine u(x, t) from u(x; 0); one needs to perform the following three steps: (i)
Solve the corresponding direct scattering problem for the associated LODE at t D 0; i. e. determine the initial scattering data S(; 0) from the initial potential u(x; 0): (ii) Time evolve the scattering data from its initial value S(; 0) to its value S(; t) at time t: Such an evolution is usually a simple one and is particular to each integrable NPDE. (iii) Solve the corresponding inverse scattering problem for the associated LODE at fixed t; i. e. determine the potential u(x, t) from the scattering data S(; t): It is amazing that the resulting u(x, t) satisfies the integrable NPDE and that the limiting value of u(x, t) as t ! 0 agrees with the initial profile u(x; 0):
Inverse Scattering Transform Certain NPDEs are classified as integrable in the sense that their corresponding IVPs can be solved with the help of an IST. The idea behind the IST method is as follows: Each integrable NPDE is associated with a LODE (or a system of LODEs) containing a parameter (usually known as the spectral parameter), and the solution u(x, t) to the NPDE appears as a coefficient (usually known as the potential) in the corresponding LODE. In the NPDE the quantities x and t appear as independent variables (usually known as the spatial and temporal coordinates, respectively), and in the LODE x is an independent variable and and t appear as parameters. It is usually the case that u(x, t) vanishes at each fixed t as x becomes infinite so that a scattering scenario can be created for the related LODE, in which the potential u(x, t) can uniquely be associated with some scattering data S(; t): The problem of determining S(; t) for all values from u(x, t) given for all x values is known as
The Lax Method In 1968 Peter Lax introduced [15] a method yielding an integrable NPDE corresponding to a given LODE. The basic idea behind the Lax method is the following. Given a linear differential operator L appearing in the spectral problem L D ; find an operator A (the operators A and L are said to form a Lax pair) such that: The spectral parameter does not change in time, i. e. t D 0: (ii) The quantity t A remains a solution to the same linear problem L D : (iii) The quantity L t C LA AL is a multiplication operator, i. e. it is not a differential operator.
(i)
From condition (ii) we get L( t A ) D ( t A ) ;
(7)
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Inverse Scattering Transform and the Theory of Solitons
and with the help of L obtain L t LA
D
D
t
and t D 0; from (7) we
A ( ) D @ t ( ) AL
D @ t (L ) AL
D Lt
CL
t
AL
;
A D 4@3x C 6u@x C 3u x :
(8)
where @ t denotes the partial differential operator with respect to t: After canceling the term L t on the left and right hand sides of (8), we get (L t C LA AL)
D0;
which, because of (iii), yields
L :D @2x C u(x; t) ;
(10)
where the notation :D is used to indicate a definition so that the quantity on the left should be understood as the quantity on the right hand side. Given the linear differential operator L defined as in (10), let us try to determine the associated operator A by assuming that it has the form A D ˛3 @3x C ˛2 @2x C ˛1 @x C ˛0 ;
(11)
where the coefficients ˛ j with j D 0; 1; 2; 3 may depend on x and t; but not on the spectral parameter : Note that L t D u t : Using (10) and (11) in (9), we obtain (
C(
C(
)@3x
C(
)@2x
C ( )@x C ( ) D 0 ;
(12)
where, because of (iii), each coefficient denoted by () must vanish. The coefficient of @5x vanishes automatically. Setj ting the coefficients of @x to zero for j D 4; 3; 2; 1; we obtain ˛3 D c1 ;
˛2 D c2 ;
For the Zakharov–Shabat system in (4), we proceed in a similar way. Let us write it as L D ; where the linear differential operator L is defined via # " # " 0 u(x; t) 1 0 : L :D i @x i 0 1 u(x; t) 0
"
(9)
Note that (9) is an evolution equation containing a firstorder time derivative, and it is the desired integrable NPDE. The equation in (9) is often called a compatibility condition. Having outlined the Lax method, let us now illustrate it to derive the KdV equation in (1) from the Schrödinger equation in (2). For this purpose, we write the Schrödinger equation as L D with :D k 2 and
)@4x
(13)
Then, the operator A is obtained as
L t C LA AL D 0 :
)@5x
in (1). Moreover, by letting c2 D c4 D 0; we obtain the operator A as
3 ˛1 D c3 c1 u; 2
3 ˛0 D c4 c1 u x c2 u ; 4 with c1 ; c2 ; c3 ; and c4 denoting arbitrary constants. Choosing c1 D 4 and c3 D 0 in the last coefficient in (12) and setting that coefficient to zero, we get the KdV equation
A D 2i
1
0
0
1
#
" @2x 2i
0
u
u
0
#
"
juj2
ux
ux
juj2
@x i
# ;
(14) and the compatibility condition (9) gives us the NLS equation in (3). For the first-order system (6), by writing it as L D ; where the linear operator L is defined by " " # # 1 0 0 u(x; t) L :D i @x i ; 0 1 u(x; t) 0 we obtain the corresponding operator A as " " " # # A D 4
1
0
0
1
@3x 6
u2
u x
ux
u2
@x
6uu x
3u x x
3u x x
6uu x
# ;
and the compatibility condition (9) yields the mKdV equation in (5). The AKNS Method In 1973 Ablowitz, Kaup, Newell, and Segur introduced [2,3] another method to determine an integrable NPDE corresponding to a LODE. This method is now known as the AKNS method, and the basic idea behind it is the following. Given a linear operator X associated with the first-order system v x D X v; we are interested in finding an operator T (the operators T and X are said to form an AKNS pair) such that: The spectral parameter does not change in time, i. e. t D 0: (ii) The quantity v t T v is also a solution to v x D X v; i. e. we have (v t T v)x D X (v t T v): (iii) The quantity X t Tx C XT T X is a (matrix) multiplication operator, i. e. it is not a differential operator. (i)
Inverse Scattering Transform and the Theory of Solitons
Then from the (1; 2)-entry in (17) we obtain
From condition (ii) we get
u t C 12 x x x u x 2 x (u ) D 0 :
v tx Tx v T v x D X v t XT v D (X v) t X t v XT v
Assuming a linear dependence of on the spectral parameter and hence letting D C in (19), we get
D (v x ) t X t v XT v D v x t X t v XT v :
(15)
Using v tx D v x t and replacing T v x by T X v on the left side and equating the left and right hand sides in (15), we obtain (X t Tx C XT T X )v D 0 ;
2 x 2 C
(16)
We can view (16) as an integrable NPDE solvable with the help of the solutions to the direct and inverse scattering problems for the linear system v x D X v: Like (9), the compatibility condition (16) yields a nonlinear evolution equation containing a first-order time derivative. Note that X contains the spectral parameter ; and hence T also depends on as well. This is in contrast with the Lax method in the sense that the operator A does not contain : Let us illustrate the AKNS method by deriving the KdV equation in (1) from the Schrödinger equation in (2). For this purpose we write the Schrödinger equation, by replacing the spectral parameter k2 with ; as a first-order linear system v x D X v; where we have defined " # " # 0 u(x; t) x v :D ; X :D : 1 0
1
2 x u C 2x u x C u t C 12 x x x 2x u u x D 0 :
2 x x x
Equating the coefficients of each power of to zero, we have D c1 ;
which in turn, because of (iii), implies X t T x C XT T X D 0 :
(19)
D 12 c1 u C c2 ; u t 32 c1 uu x c2 u x C 14 c1 u x x x D 0 ;
(20)
with c1 and c2 denoting arbitrary constants. Choosing c1 D 4 and c2 D 0; from (20) we obtain the KdV equation given in (1). Moreover, with the help of (18) we get ˛ D u x C c3 ; ˇ D 42 C 2u C 2u2 u x x ;
D 4 C 2u; D c3 u x ; where c3 is an arbitrary constant. Choosing c3 D 0; we find " # 42 C 2u C 2u2 u x x ux T D : 4 C 2u u x As for the Zakharov–Shabat system in (4), writing it as v x D X v; where we have defined # " i u(x; t) ; X :D i u(x; t)
Let us look for T in the form " # ˛ ˇ T D ;
we obtain the matrix operator T as " # 2i2 C ijuj2 2u C iu x T D ; 2u C iu x 2i2 ijuj2
where the entries ˛; ˇ; ; and may depend on x; t; and : The compatibility condition (16) yields
and the compatibility condition (16) yields the NLS equation in (3). As for the first-order linear system (6), by writing it as v x D X v; where " # i u(x; t) X :D ; u(x; t) i
˛x ˇ C (u ) u t ˇx C (u ) ˛(u ) x C ˇ (u ) x C ˛ " # 0 0 D : 0 0
(17) The (1; 1); (2; 1); and (2; 2)-entries in the matrix equation in (17) imply ˇ D ˛x C (u ) ;
D ˛ x ;
x D ˛x : (18)
we obtain the matrix operator T as T D
4i3 C 2iu 2 42 u C 2iu x u x x 2u 3 42 u C 2iu x C u x x C 2u 3 4i3 2iu 2
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Inverse Scattering Transform and the Theory of Solitons
and the compatibility condition (16) yields the mKdV equation in (5). As for the first-order system v x D X v; where 2 3 1 i (x; t) u x 6 7 2 7; X :D 6 41 5 i u x (x; t) 2 we obtain the matrix operator T as " # sin u i cos u T D : 4 sin u cos u Then, the compatibility condition (16) gives us the sineGordon equation u x t D sin u : Direct Scattering Problem The direct scattering problem consists of determining the scattering data when the potential is known. This problem is usually solved by obtaining certain specific solutions, known as the Jost solutions, to the relevant LODE. The appropriate scattering data can be constructed with the help of spatial asymptotics of the Jost solutions at infinity or from certain Wronskian relations among the Jost solutions. In this section we review the scattering data corresponding to the Schrödinger equation in (2) and to the Zakharov–Shabat system in (4). The scattering data sets for other LODEs can similarly be obtained. Consider (2) at fixed t by assuming that the potential u(x, t) belongs R 1to the Faddeev class, i. e. u(x, t) is real valued and 1 dx (1 C jxj) ju(x; t)j is finite. The Schrödinger equation has two types of solutions; namely, scattering solutions and bound-state solutions. The scattering solutions are those that consist of linear combinations of eikx and eikx as x ! ˙1; and they occur for k 2 R n f0g; i. e. for real nonzero values of k: Two linearly independent scattering solutions fl and fr ; known as the Jost solution from the left and from the right, respectively, are those solutions to (2) satisfying the respective asymptotic conditions fl (k; x; t) D e i kx C o(1) ; fl0 (k; x; t) D ikei kx C o(1) ;
x ! C1 ;
(21)
fr (k; x; t) D ei kx C o(1) ; fr0 (k; x; t) D ikei kx C o(1) ;
x ! 1 ;
where the notation o(1) indicates the quantities that vanish. Writing their remaining spatial asymptotics in the
form fl (k; x; t) D
fr (k; x; t) D
e i kx L(k; t) ei kx C C o(1) ; T(k; t) T(k; t) x ! 1 ;
(22)
ei kx R(k; t) e i kx C C o(1) ; T(k; t) T(k; t) x ! C1 ;
we obtain the scattering coefficients; namely, the transmission coefficient T and the reflection coefficients L and R; from the left and right, respectively. Let CC denote the upper half complex plane. A boundstate solution to (2) is a solution that belongs to L2 (R) in the x variable. Note that L2 (R) denotes the set of complexvalued functions whose absolute squares are integrable on the real line R: When u(x, t) is in the Faddeev class, it is known [5,7,8,9,16,17,18,19] that the number of bound states is finite, the multiplicity of each bound state is one, and the bound-state solutions can occur only at certain k-values on the imaginary axis in CC : Let us use N to denote the number of bound states, and suppose that the bound states occur at k D i j with the ordering 0 < 1 < < N : Each bound state corresponds to a pole of T in CC : Any bound-state solution at k D i j is a constant multiple of fl (i j ; x; t): The left and right boundstate norming constants cl j (t) and cr j (t); respectively, can be defined as
Z 1 1/2 dx fl (i j ; x; t)2 ; cl j (t) :D
Z cr j (t) :D
1 1 1
dx fr (i j ; x; t)2
1/2 ;
and they are related to each other through the residues of T via Res (T; i j ) D i cl j (t)2 j (t) D i
cr j (t)2 ;
j (t)
(23)
where the j (t) are the dependency constants defined as
j (t) :D
fl (i j ; x; t) : fr (i j ; x; t)
(24)
The sign of j (t) is the same as that of (1) N j ; and hence cr j (t) D (1) N j j (t) cl j (t) : The scattering matrix associated with (2) consists of the transmission coefficient T and the two reflection coefficients R and L; and it can be constructed from f j g NjD1 and one of the reflection coefficients. For example, if we start with the right reflection coefficient R(k, t) for k 2 R; we get
Inverse Scattering Transform and the Theory of Solitons
0
1 N Y k C i j A T(k; t) D @ k i j jD1 Z 1 1 log(1 jR(s; t)j2 ) exp ds ; 2 i 1 s k i0C k 2 CC [ R ; where the quantity i0+ indicates that the value for k 2 R must be obtained as a limit from CC : Then, the left reflection coefficient L(k, t) can be constructed via L(k; t) D
R(k; t) T(k; t) T(k; t)
;
k 2R:
We will see in the next section that T(k; t) D T(k; 0); jR(k; t)j D jR(k; 0)j, and jL(k; t)j D jL(k; 0)j. For a detailed study of the direct scattering problem for the 1-D Schrödinger equation, we refer the reader to [5,7, 8,9,16,17,18,19]. It is important to remember that u(x, t) for x 2 R at each fixed t is uniquely determined [5,7,8,9, 16,17,18] by the scattering data fR; f j g; fcl j (t)gg or one of its equivalents. Letting c j (t) :D cl j (t)2 ; we will work with one such data set, namely fR; f j g; fc j (t)gg, in Sect. “Time Evolution of the Scattering Data” and Sect. “Inverse Scattering Problem”. Having described the scattering data associated with the Schrödinger equation, let us briefly describe the scattering data associated with the Zakharov–Shabat system in (4). Assuming that u(x, t) for each t is integrable in x on R; the two Jost solutions (; x; t) and (; x; t); from the left and from the right, respectively, are those unique solutions to (4) satisfying the respective asymptotic conditions " # 0 (; x; t) D ix C o(1); x ! C1 ; e (25)
ix e C o(1) ; x ! 1 : (; x; t) D 0 The transmission coefficient T; the left reflection coefficient L; and the right reflection coefficient R are obtained via the asymptotics 2 3 L(; t) eix 6 T(; t) 7 6 7 (; x; t) D 6 7 C o(1) ; x ! 1 ; ix 4 5 e 2
T(; t) eix T(; t)
3
7 6 7 6 (; x; t) D 6 7 C o(1) ; 4 R(; t) e ix 5
x ! C1 :
T(; t) (26)
The bound-state solutions to (4) occur at those values corresponding to the poles of T in CC : Let us use f j g NjD1 to denote the set of such poles. It should be noted that such poles are not necessarily located on the positive imaginary axis. Furthermore, unlike the Schrödinger equation, the multiplicities of such poles may be greater than one. Let us assume that the pole j has multiplicity n j : Corresponding to the pole j ; one associates [4,20] nj boundstate norming constants c js (t) for s D 0; 1; : : : ; n j 1: We assume that, for each fixed t; the potential u(x, t) in the Zakharov–Shabat system is uniquely determined by the scattering data fR; f j g; fc js (t)gg and vice versa. Time Evolution of the Scattering Data As the initial profile u(x; 0) evolves to u(x, t) while satisfying the NPDE, the corresponding initial scattering data S(; 0) evolves to S(; t): Since the scattering data can be obtained from the Jost solutions to the associated LODE, in order to determine the time evolution of the scattering data, we can analyze the time evolution of the Jost solutions with the help of the Lax method or the AKNS method. Let us illustrate how to determine the time evolution of the scattering data in the Schrödinger equation with the help of the Lax method. As indicated in Sect. “The Lax Method”, the spectral parameter k and hence also the values j related to the bound states remain unchanged in time. Let us obtain the time evolution of fl (k; x; t); the Jost solution from the left. From condition (ii) in Sect. “The Lax Method”, we see that the quantity @ t fl A fl remains a solution to (2) and hence we can write it as a linear combination of the two linearly independent Jost solutions fl and fr as @ t fl (k; x; t) 4@3x C 6u@x C 3u x fl (k; x; t) D p(k; t) fl (k; x; t) C q(k; t) fr (k; x; t) ; (27) where the coefficients p(k, t) and q(k, t) are yet to be determined and A is the operator in (13). For each fixed t; assuming u(x; t) D o(1) and u x (x; t) D o(1) as x ! C1 and using (21) and (22) in (27) as x ! C1; we get @ t e i kx C 4@3x e i kx D p(k; t) e i kx
1 R(k; t) i kx i kx C q(k; t) C e e C o(1) : T(k; t) T(k; t) (28) Comparing the coefficients of eikx and eikx on the two sides of (28), we obtain q(k; t) D 0 ;
p(k; t) D 4ik 3 :
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Inverse Scattering Transform and the Theory of Solitons
Thus, fl (k; x; t) evolves in time by obeying the linear thirdorder PDE @ t fl A fl D 4ik 3 fl :
(29)
Proceeding in a similar manner, we find that fr (k; x; t) evolves in time according to @ t fr A fr D 4ik 3 fr :
(30)
Notice that the time evolution of each Jost solution is fairly complicated. We will see, however, that the time evolution of the scattering data is very simple. Letting x ! 1 in (29), using (22) and u(x; t) D o(1) and u x (x; t) D o(1) as x ! 1; and comparing the coefficients of eikx and eikx on both sides, we obtain @ t T(k; t) D 0 ;
The norming constants cj (t) appearing in the Marchenko kernel (38) are related to cl j (t) as c j (t) :D cl j (t)2 ; and hence their time evolution is described as
@ t L(k; t) D 8ik 3 L(k; t);
c j (t) D c j (0) e
8 3j t
:
(35)
As for the NLS equation and other integrable NPDEs, the time evolution of the related scattering data sets can be obtained in a similar way. For the former, in terms of the operator A in (14), the Jost solutions (; x; t) and (; x; t) appearing in (25) evolve according to the respective linear PDEs t
A
D 2i2
;
t A D 2i2 :
The scattering coefficients appearing in (26) evolve according to T(; t) D T(; 0) ;
yielding
2
R(; t) D R(; 0) e4i t ;
T(k; t) D T(k; 0) ;
L(k; t) D L(k; 0) e
8i k 3 t
:
L(; t) D L(; 0) e
In a similar way, from (30) as x ! C1; we get 3
R(k; t) D R(k; 0) e8i k t :
(31)
Thus, the transmission coefficient remains unchanged and only the phases of the reflection coefficients change as time progresses. Let us also evaluate the time evolution of the dependency constants j (t) defined in (24). Evaluating (29) at k D i j and replacing fl (i j ; x; t) by j (t) fr (i j ; x; t); we get fr (i j ; x; t) @ t j (t) C j (t) @ t fr (i j ; x; t) j (t)A fr (i j ; x; t) D 4 3j j (t) fr (i j ; x; t) :
(32)
On the other hand, evaluating (30) at k D i j ; we obtain
j (t) @ t fr (i j ; x; t) j (t) A fr (i j ; x; t) D 4 3j j (t) fr (i j ; x; t) :
(33)
Comparing (32) and (33) we see that @ t j (t) D 8 3j j (t); or equivalently
j (t) D j (0) e
8 3j t
:
(34)
Then, with the help of (23) and (34), we determine the time evolutions of the norming constants as cl j (t) D cl j (0) e
4 3j t
;
cr j (t) D cr j (0) e
4 3j t
:
4i2 t
(36) :
Associated with the bound-state pole j of T; we have the bound-state norming constants c js (t) appearing in the Marchenko kernel ˝(y; t) given in (41). Their time evolution is governed [4] by
c j(n j 1) (t) c j(n j 2) (t) : : : c j0 (t) 4i A2 t j ; D c j(n j 1) (0) c j(n j 2) (0) : : : c j0 (0) e (37)
where the n j n j matrix Aj appearing in the exponent is defined as 2 3 1 0 ::: 0 0 i j 6 0 1 : : : 0 0 7 i j 6 7 6 0 7 0 i : : : 0 0 j 6 7 A j :D 6 : 7: : : : : : :: 7 :: :: :: :: 6 :: 6 7 4 0 0 0 : : : i j 1 5 0 0 0 ::: 0 i j Inverse Scattering Problem In Sect. “Direct Scattering Problem” we have seen how the initial scattering data S(; 0) can be constructed from the initial profile u(x; 0) of the potential by solving the direct scattering problem for the relevant LODE. Then, in Sect. “Time Evolution of the Scattering Data” we have seen how to obtain the time-evolved scattering data S(; t) from the initial scattering data S(; 0): As the final step in the IST, in this section we outline how to obtain
Inverse Scattering Transform and the Theory of Solitons
u(x, t) from S(; t) by solving the relevant inverse scattering problem. Such an inverse scattering problem may be solved by the Marchenko method [5,7,8,9,16,17,18, 19]. Unfortunately, in the literature many researchers refer to this method as the Gel’fand–Levitan method or the Gel’fand–Levitan–Marchenko method, both of which are misnomers. The Gel’fand–Levitan method [5,7,16,17, 19] is a different method to solve the inverse scattering problem, and the corresponding Gel’fand–Levitan integral equation involves an integration on the finite interval (0; x) and its kernel is related to the Fourier transform of the spectral measure associated with the LODE. On the other hand, the Marchenko integral equation involves an integration on the semi-infinite interval (x; C1); and its kernel is related to the Fourier transform of the scattering data. In this section we first outline the recovery of the solution u(x, t) to the KdV equation from the corresponding time-evolved scattering data fR; f j g; fc j (t)gg appearing in (31) and (35). Later, we will also outline the recovery of the solution u(x, t) to the NLS equation from the corresponding time-evolved scattering data fR; f j g; fc js (t)gg appearing in (36) and (37). The solution u(x, t) to the KdV equation in (1) can be obtained from the time-evolved scattering data by using the Marchenko method as follows: (a) From the scattering data fR(k; t); f j g; fc j (t)gg appearing in (31) and (35), form the Marchenko kernel ˝ defined via Z 1 N X 1 ˝(y; t) :D dk R(k; t) e i k y C c j (t) e j y : 2 1 jD1
(38) (b) Solve the corresponding Marchenko integral equation K(x; y; t) C ˝(x C y; t) Z 1 dz K(x; z; t) ˝(z C y; t) D 0 ; C x
x < y < C1 ;
(39)
and obtain its solution K(x; y; t): (c) Recover u(x, t) by using u(x; t) D 2
@K(x; x; t) : @x
(40)
The solution u(x, t) to the NLS equation in (3) can be obtained from the time-evolved scattering data by using the Marchenko method as follows: (i)
From the scattering data fR(; t); f j g; fc js (t)gg appearing in (36) and (37), form the Marchenko kernel
˝ as 1 ˝(y; t) :D 2
Z
1
d R(; t) e iy
1
C
j 1 N nX X
jD1 sD0
c js (t)
y s i j y : e s!
(41)
(ii) Solve the Marchenko integral equation Z 1 K(x; y; t) ˝(x C y; t) C dz x Z 1 ds K(x; s; t) ˝(s C z; t) ˝(z C y; t) D 0 ; x
x < y < C1 ; and obtain its solution K(x; y; t): (iii) Recover u(x, t) from the solution K(x; y; t) to the Marchenko equation via u(x; t) D 2K(x; x; t) : (iv) Having determined K(x; y; t); one can alternatively get ju(x; t)j2 from ju(x; t)j2 D 2
@G(x; x; t) ; @x
where we have defined Z 1 dz K(x; z; t) ˝(z C y; t) : G(x; y; t) :D x
Solitons A soliton solution to an integrable NPDE is a solution u(x, t) for which the reflection coefficient in the corresponding scattering data is zero. In other words, a soliton solution u(x, t) to an integrable NPDE is nothing but a reflectionless potential in the associated LODE. When the reflection coefficient is zero, the kernel of the relevant Marchenko integral equation becomes separable. An integral equation with a separable kernel can be solved explicitly by transforming that linear equation into a system of linear algebraic equations. In that case, we get exact solutions to the integrable NPDE, which are known as soliton solutions. For the KdV equation the N-soliton solution is obtained by using R(k; t) D 0 in (38). In that case, letting X(x) :D e1 x e2 x : : : eN x ; 2 3 c1 (t) e1 y 6 c2 (t) e2 y 7 6 7 Y(y; t) :D 6 7; :: 4 5 : y N c N (t) e
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Inverse Scattering Transform and the Theory of Solitons
we get ˝(x C y; t) D X(x) Y(y; t). As a result of this separability the Marchenko integral equation can be solved algebraically and the solution has the form K(x; y; t) D H(x; t) Y(y; t), where H(x, t) is a row vector with N entries that are functions of x and t. A substitution in (39) yields K(x; y; t) D X(x) (x; t)1 Y(y; t);
(42)
where the N N matrix (x; t) is given by Z 1 (x; t) :D I C dz Y(z; t) X(z);
(43)
x
with I denoting the or N N identity matrix. Equivalently, the (j, l)-entry of is given by 2 xC8 3j t
c j (0) e j jl D ı jl C j C l
;
with ı jl denoting the Kronecker delta. Using (42) in (40) we obtain
x
As for the NLS equation, the well-known N-soliton solution (with simple bound-state poles) is obtained by choosing R(; t) D 0 and n j D 1 in (41). Proceeding as in the KdV case, we obtain the N-soliton solution in terms of the triplet A; B; C with A :D diagfi1 ; i2 ; : : : ; i N g;
(46)
where the complex constants j are the distinct poles of the transmission coefficient in CC ; B and C are as in (45) except for the fact that the constants c j (0) are now allowed to be nonzero complex numbers. In terms of the matrices P(x; t); M; and Q defined as P(x; t) :D 2
2
2
diagfe2i1 xC4i1 t ; e2i2 xC4i2 t ; : : : ; e2iN xC4iN t g;
@ X(x) (x; t)1 Y(x; t) u(x; t) D 2 @x @ Y(x; t) X(x) (x; t)1 ; D 2 tr @x
M jl :D
where tr denotes the matrix trace (the sum of diagonal entries in a square matrix). From (43) we see that Y(x; t) X(x) is equal to the x-derivative of (x; t) and hence the N-soliton solution can also be written as
@ @ (x; t) u(x; t) D 2 tr (x; t)1 @x @x " @ # det (x; t) @ @x D 2 ; (44) @x det (x; t) where det denotes the matrix determinant. When N D 1; we can express the one-soliton solution u(x, t) to the KdV equation in the equivalent form u(x; t) D 2 12 sech2 1 x 413 t C ; p with :D log 21 /c1 (0) : Let us mention that, using matrix exponentials, we can express [6] the N-soliton solution appearing in (44) in various other equivalent forms such as 3
u(x; t) D 4CeAxC8A t (x; t)1 A (x; t)1 eAx B ; where A :D diagf1 ; 2 ; : : : ; N g ; B :D 1 1 : : : 1 ; C :D c1 (0) c2 (0) : : : c N (0) :
Note that a dagger is used for the matrix adjoint (transpose and complex conjugate), and B has N entries. In this notation we can express (43) as Z 1 3 (x; t) D I C dz ezA BCezA e8tA :
(45)
i j l
;
Q jl :D
ic j c l j l
:
we construct the N-soliton solution u(x, t) to the NLS equation as u(x; t) D 2B [I C P(x; t) QP(x; t) M]1 P(x; t) C ; (47) or equivalently as
xC4i(A )2 t
u(x; t) D 2B eA x (x; t)1 eA
C ; (48)
where we have defined
Z 1 As4i A2 t As4i A2 t (x; t) :D I C ds (Ce ) (Ce ) x
Z 1 dz (eAz B)(eAz B) : (49) x
Using (45) and (46) in (49), we get the (j, l)-entry of (x; t) as 2
jl D ı jl
2
N X c j c l e i(2m j l )xC4i(m j )t mD1
(m j )(m l )
Note that the absolute square of u(x, t) is given by
@ @ (x; t) (x; t)1 ju(x; t)j2 D tr @x @x " @ # @ @x det (x; t) D : @x det (x; t)
:
Inverse Scattering Transform and the Theory of Solitons
For the NLS equation, when N D 1; from (47) or (48) we obtain the single-soliton solution 2t
u(x; t) D
8c 1 (Im[1 ])2 e2i1 x4i(1 )
2
4(Im[1 ])2 C jc1 j2 e4x(Im[1 ])8t(Im[1 ])
;
where Im denotes the imaginary part. Future Directions There are many issues related to the IST and solitons that cannot be discussed in such a short review. We will briefly mention only a few. Can we characterize integrable NPDEs? In other words, can we find a set of necessary and sufficient conditions that guarantee that an IVP for a NPDE is solvable via an IST? Integrable NPDEs seem to have some common characteristic features [1] such as possessing Lax pairs, AKNS pairs, soliton solutions, infinite number of conserved quantities, a Hamiltonian formalism, the Painlevé property, and the Bäcklund transformation. Yet, there does not seem to be a satisfactory solution to their characterization problem. Another interesting question is the determination of the LODE associated with an IST. In other words, given an integrable NPDE, can we determine the corresponding LODE? There does not yet seem to be a completely satisfactory answer to this question. When the initial scattering coefficients are rational functions of the spectral parameter, representing the timeevolved scattering data in terms of matrix exponentials results in the separability of the kernel of the Marchenko integral equation. In that case, one obtains explicit formulas [4,6] for exact solutions to some integrable NPDEs and such solutions are constructed in terms of a triplet of constant matrices A; B; C whose sizes are pp; p1; and 1p; respectively, for any positive integer p: Some special cases of such solutions have been mentioned in Sect. “Solitons”, and it would be interesting to determine if such exact solutions can be constructed also when p becomes infinite. Bibliography Primary Literature 1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge 2. Ablowitz MJ, Kaup DJ, Newell AC, Segur H (1973) Method for solving the sine–Gordon equation. Phys Rev Lett 30:1262– 1264 3. Ablowitz MJ, Kaup DJ, Newell AC, Segur H (1974) The inverse scattering transform-Fourier analysis for nonlinear problems. Stud Appl Math 53:249–315
4. Aktosun T, Demontis F, van der Mee C (2007) Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Problems 23:2171–2195 5. Aktosun T, Klaus M (2001) Chapter 2.2.4, Inverse theory: problem on the line, In: Pike ER, Sabatier PC (eds) Scattering. Academic Press, London, pp 770–785 6. Aktosun T, van der Mee C (2006) Explicit solutions to the Korteweg-de Vries equation on the half-line. Inverse Problems 22:2165–2174 7. Chadan K, Sabatier PC (1989) Inverse problems in quantum scattering theory. 2nd edn. Springer, New York 8. Deift P, Trubowitz E (1979) Inverse scattering on the line. Commun Pure Appl Math 32:121–251 9. Faddeev LD (1967) Properties of the S–matrix of the one– dimensional Schrödinger equation. Amer Math Soc Transl (Ser. 2) 65:139–166 10. Fermi E (1965) Collected papers, vol II: United States, 1939– 1954. University of Chicago Press, Chicago 11. Fermi E, Pasta J, Ulam S (1955) Studies of non linear problems, I. Document LA-1940, Los Alamos National Laboratory 12. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19:1095–1097 13. Gel’fand IM, Levitan BM (1955) On the determination of a differential equation from its spectral function. Amer Math Soc Transl (Ser. 2) 1:253–304 14. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves. Phil Mag 39:422–443 15. Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Commun Pure Appl Math 21:467–490 16. Levitan BM (1987) Inverse Sturm–Liouville problems. Science VNU Press, Utrecht 17. Marchenko VA (1986) Sturm–Liouville operators and applications. Birkhäuser, Basel 18. Melin A (1985) Operator methods for inverse scattering on the real line. Commun Partial Differ Equ 10:677–766 19. Newton RG (1983) The Marchenko and Gel’fand–Levitan methods in the inverse scattering problem in one and three dimensions. In: Bednar JB, Redner R, Robinson E, Weglein A (eds) Conference on inverse scattering: theory and application. SIAM, Philadelphia, pp 1–74 20. Olmedilla E (1987) Multiple pole solutions of the nonlinear Schrödinger equation. Phys D 25:330–346 21. Russell JS (1845) Report on waves, Report of the 14th meeting of the British Association for the Advancement of Science. John Murray, London, pp 311–390 22. Wadati M (1972) The exact solution of the modified Korteweg– de Vries equation. J Phys Soc Jpn 32:1681 23. Zabusky NJ, Kruskal MD (1965) Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243 24. Zakharov VE, Shabat AB (1972) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Phys JETP 34:62–69 25. http://www.osti.gov/accomplishments/pdf/A80037041/ A80037041.pdf
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Books and Reviews Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Philadelphia Aktosun T (2004) Inverse scattering transform, KdV, and solitons, In: Ball JA, Helton JW, Klaus M, Rodman L (ed), Current trends in operator theory and its applications. Birkhäuser, Basel, pp 1–22 Aktosun T (2005) Solitons and inverse scattering transform, In: Clemence DP, Tang G (eds) Mathematical studies in nonlinear wave propagation, Contemporary Mathematics, vol 379. Amer Math Soc, Providence, pp 47–62
Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1982) Solitons and nonlinear wave equations. Academic Press, London Drazin PG, Johnson RS (1988) Solitons: an introduction. Cambridge University Press, Cambridge Lamb GL Jr (1980) Elements of soliton theory. Wiley, New York Novikov S, Manakov SV, Pitaevskii LP, Zakharov VE (1984) Theory of solitons. Consultants Bureau, New York Scott AC, Chu FYF, McLaughlin D (1973) The soliton: a new concept in applied science. Proc IEEE 61:1443–1483
Isomorphism Theory in Ergodic Theory
Isomorphism Theory in Ergodic Theory CHRISTOPHER HOFFMAN Department of Mathematics, University of Washington, Seattle, USA Article Outline Glossary Definition of the Subject Introduction Basic Transformations Basic Isomorphism Invariants Basic Tools Isomorphism of Bernoulli Shifts Transformations Isomorphic to Bernoulli Shifts Transformations not Isomorphic to Bernoulli Shifts Classifying the Invariant Measures of Algebraic Actions Finitary Isomorphisms Flows Other Equivalence Relations Non-invertible Transformations Factors of a Transformation Actions of Amenable Groups Future Directions Bibliography Glossary Almost everywhere A property is said to hold almost everywhere (a.e.) if the set on which the property does not hold has measure 0. Bernoulli shift A Bernoulli shift is a stochastic process such that all outputs of the process are independent. Conditional measure For any measure space (X; B; ) is a C and -algebra C B the conditional measure R measurable function g such that (C) D C g d for all C 2 C . Coupling of two measure spaces A coupling of two measure spaces (X; ; B) and (Y; ; C ) is a measure on X Y such that (B Y) D (B) for all B 2 B and
(X C) D (C) for all C 2 C . Ergodic measure preserving transformation A measure preserving transformation is ergodic if the only invariant sets ((A4T 1 (A)) D 0) have measure 0 or 1. Ergodic theorem The pointwise ergodic theorem says that for any measure preserving transformation (X; B; ) and T and any L1 function f the time average n 1 X lim f (T i (x)) n!1 n iD1
converges a.e. If the transformation is ergodic then the R limit is the space average, f d a.e. Geodesic A geodesic on a Riemannian manifold is a distance minimizing path between points. Horocycle A horocycle is a circle in the hyperbolic disk which intersects the boundary of the disk in exactly one point. Invariant measure Likewise a measure is said to be invariant with respect to (X; T) provided that (T 1 (A)) D (A) for all measurable A 2 B. Joining of two measure preserving transformations A joining of two measure preserving transformations (X; T) and (Y; S) is a coupling of X and Y which is invariant under T S. Markov shift A Markov shift is a stochastic process such that the conditional distribution of the future outputs (fx n gn>0 ) of the process conditioned on the last output (x0 ) is the same as the distribution conditioned on all of the past outputs of the process (fx n gn0 ). Measure preserving transformation A measure preserving transformation consists of a probability space (X; T) and a measurable function T : X ! X such that (T 1 (A)) D (A) for all A 2 B. Measure theoretic entropy A numerical invariant of measure preserving transformations that measures the growth in complexity of measurable partitions refined under the iteration of the transformation. Probability space A probability space X D (X; ; B) is a measure space such that (B) D 1. Rational function, rational map A rational function f (z) D g(z)/h(z) is the quotient of two polynomials. The degree of f (z) is the maximum of the degrees of g(z) and h(z). The corresponding rational maps T f : z ! f (z) on the Riemann sphere C are a main object of study in complex dynamics. Stochastic process A stochastic process is a sequence of measurable functions fx n gn2Z (or outputs) defined on the same measure space, X. We refer to the value of the functions as outputs. Definition of the Subject Our main goal in this article is to consider when two measure preserving transformations are in some sense different presentations of the same underlying object. To make this precise we say two measure preserving maps (X; T) and (Y; S) are isomorphic if there exists a measurable map : X ! Y such that (1) is measure preserving, (2) is invertible almost everywhere and (3) (T(x)) D S((x)) for almost every x.
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The main goal of the subject is to construct a collection of invariants of a transformation such that a necessary condition for two transformations to be isomorphic is that the invariant be the same for both transformations. Another goal of the subject is to solve the much more difficult problem of constructing invariants such that the invariants being the same for two transformations is a sufficient condition for the transformation to be isomorphic. Finally we apply these invariants to many natural classes of transformation to see which of them are (or are not) isomorphic. Introduction In this article we look at the problem of determining of which measure preserving transformations are isomorphic. We look at a number of isomorphism invariants, the most important of which is the Kolmogorov–Sinai entropy. The central theorem in this field is Ornstein’s proof that any two Bernoulli shifts of the same entropy are isomorphic. We also discuss some of the consequences of this theorem, which transformations are isomorphic to Bernoulli shifts as well as generalizations of Ornstein’s theory. Basic Transformations In this section we list some of the basic classes of measure preserving transformations that we study in this article. Bernoulli shifts Some of the most fundamental transformations are the Bernoulli shifts. A probability vecP tor is a vector fp i gniD1 such that niD1 p i D 1 and p i 0 for all i. Let p D fp i gniD1 be a probability vector. The Bernoulli shift corresponding to p has state space f1; 2; : : : ; ngZ , the shift operator T(x) i D x iC1 . To specify the measure we only need to specify it on cylinder sets A D fx 2 X : x i D a i 8i 2 fm; : : : ; kgg for some m k 2 Z and a sequence a m ; : : : ; a k 2 f1; : : : ; ng. The measure on cylinder sets is defined by ˚ x 2 X : xi D ai
k Y for all i such that m i k D pai : iDm
For any d 2 N if p D (1/d; : : : ; 1/d) we refer to Bernoullip as the Bernoulli d shift. Markov shifts A Markov shift on state n symbols is defined by an n n matrix, M, such that fM(i; j)gnjD1 is a probability vector for each i. The Markov shift is
a measure preserving transformation with state space f1; 2; : : : ; ngZ , transformation T(x) i D x iC1 fx1 D a1 j x0 D a0 g D fx1 D a1 j x0 D a0 ; x1 D a1 ; x2 D a2 ; : : : g for all choices of a i , i 1. Let m D fm(i)gniD1 be a vector such that Mm D m. Then an invariant measure is defined by setting the measure on cylinder sets to be A D fx : x0 D a0 ; x1 D a1 ; : : : ; x n D a n g is Q given by (A) D m(a0 ) niD1 M(a i1; a i ). Shift maps More generally the shift map is the map : N Z ! N Z where (x) i D x iC1 for all x 2 N Z and i 2 Z. We also let designate the shift map on N N . For each measure that is invariant under the shift map there is a corresponding measure defined on N N that is invariant under the shift map. Let be an invariant measure under the shift map. For any measurable set of A N N we define A˜ on N Z by A˜ D f: : : ; x1 ; x0 ; x1 ; : : : : x0 ; x1 ; : : : 2 Ag : ˜ D ˜ defined by ( ˜ A) Then it is easy to check that (A) is invariant. If the original transformation was a Markov or Bernoulli shift then refer to the resulting transformations as a one sided Markov shifts or one sided Bernoulli shift respectively. Rational maps of the Riemann sphere We say that f (z) D g(z)/h(z) is a rational function of degree d 2 if both g(z) and h(z) are polynomials with max(deg(g(z)); deg(h(z))) D d. Then f induces a natural action on the Riemann sphere T f : z ! f (z) which is a d to one map (counting with multiplicity). In Subsect. “Rational Maps” we shall see that for every rational function f there is a canonical measure f such that T f is a measure preserving transformation. Horocycle flows The horocycle flow acts on SL(2; R)/ where is a discrete subgroup of SL(2; R) such that SL(2; R)/ has finite Haar measure. For any g 2 SL(2; R) and t 2 R we define the horocycle flow by 1 0 : h t ( g) D g t 1 Matrix actions Another natural class of actions is given by the action of matrices on tori. Let M be an invertible n n integer valued matrix. We define TM : [0; 1)n ! [0; 1)n by TM (x) i D M(x) i mod 1 for all i, 1 i n. It is easy to check that if M is surjective then Lebesgue measure is invariant under T M . T M is a jdet(M)j to one map. If n D 1 then M is an integer and we refer to the map as times M.
Isomorphism Theory in Ergodic Theory
The [T; T 1 ] transformations Let (X; T) be any invertible measure preserving transformation. Let be the shift operator on Y D f1; 1gZ . The space Y comes equipped with the Bernoulli (1/2; 1/2) product measure . The [T; T 1 ] transformation is a map on Y X which preserves . It is defined by T; T 1 (y; x) D (S(y); T y0 (x)) : Induced transformations Let (X; T) be a measure preserving transformation and let A X with 0 < (A) < 1. The transformation induced by A, (A; TA ; A ), is defined as follows. For any x 2 A TA (x) D T
n(x)
(x)
where n(x) D inffm > 0 : T m (x) 2 Ag. For any B A we have that A (B) D (B)/(A). Basic Isomorphism Invariants The main purpose of isomorphism theory is to classify which pairs of measure preserving transformation are isomorphic and which are not isomorphic. One of the main ways that we can show that two measure preserving transformation are not isomorphic is using isomorphism invariants. An isomorphism invariant is a function f defined on measure preserving transformations such that if (X; T) is isomorphic to (Y; S) then f ((X; T)) D f ((Y; S)). A measure preserving action is said to be ergodic if (A) D 0 or 1 for every A with A4T 1 (A) D 0 : A measure preserving action is said to be weak mixing if for every measurable A and B 1 ˇ 1 X ˇˇ A \ T n (B) (A)(B)ˇ D 0 : n nD1
A measure preserving action is said to be mixing if for every measurable A and B lim A \ T n (B) D (A)(B) : n!1
It is easy to show that all three of these properties are isomorphism invariants and that (X; T) is mixing ) (X; T) is weak mixing ) (X; T) is ergodic. We include one more definition before introducing an even stronger isomorphism invariant. The action
of a group (or a semigroup) G on a probability space (X; ; B) is a family of measure preserving transformations f f g g g2G such that for any g; h 2 G we have that f g ( f h (x)) D f gCh (x) for almost every x 2 X. Thus any invertible measure preserving transformation (X; T) induces a Z action by f n (x) D T n (x). We say that a group action (X; Tg ) is mixing of all orders if for every n 2 N and every collection of set A1 ; : : : ; A n 2 B then A1 \ Tg 2 (A2 ) \ Tg 2 Cg 3 (A3 ) : : : n Y \ Tg 2 Cg 3 C:::g n (A n ) ! (A i ) iD1
as the g i go to infinity. Basic Tools Birkhoff’s ergodic theorem states that for every measure preserving action the limits fˆ(x) D lim
n!1
n1 1X k f T x n kD0
exists for almost every x and if (X; T) is ergodic then the R limit is f¯ D f d [2]. A partition P of X is a measurable function defined on X. (For simplicity we often assume that a partition is a function to Z or some subset of Z.) We write Pi for P1 (i). For any partition P of (X; T) define the partition T i P by P ı T i . Thus for invertible T this is given by W i T i P(x) D P(T i (x)). Then define (P)T D i2Z T P. i Thus (P)T is the smallest -algebra which contains T (Pj ) for all i 2 Z and j 2 N. We say that (P)T is the -algebra generated by P. A partition P is a generator of (X; T) if (P)T D B. Many measure preserving transformations come equipped with a natural partition. Rokhlin’s theorem For any measure preserving transformation (X; T), any > 0 and any n 2 N there exists A X such that T i (A) \ T j (A) for all 0 i < j n and ([niD0 T i (A) > 1 . Moreover for any finite partition P of X we can choose A such that (Pi \ A) D (A)(Pi ) for all i 2 N [63]. Shannon–McMillan–Breiman theorem [3,57] For any measure preserving system (X; T) and any > 0 there exists n 2 N and a set G with (G) > 1 with the following property. For any sequence g1 ; : : : g n 2 N T let g D niD1 P(T i (x)) D g i . Then if (g \ G) > 0 then (g) 2 2 h((X;T)) ; 2 h((X;T))C :
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Krieger generator theorem If H((X; T)) < 1 then there exists a finite partition P such that (P)T D B. Thus every measure preserving transformation with finite entropy is isomorphic to a shift map on finitely many symbols [33]. Measure-theoretic entropy Entropy was introduced in physics by Rudolph Clausius in 1854. In 1948 Claude Shannon introduced the concept to information theory. Consider a process that generates a string of data of length n. The entropy of the process is the smallest number h such that you can condense the data to a string of zeroes and one of length hn and with high probability you can reconstruct the original data from the string of zeroes and ones. Thus the entropy of a process is the average amount of information transmitted per symbol of the process. Kolmogorov and Sinai introduced the concept of entropy to ergodic theory in the following way [31,32]. They defined the entropy of a partition Q is defined to be H(Q) D
k X
(Q i ) log (Q i ) :
(Y; S) is trivial if Y consists of only one point. We say that a transformation (X; T) has completely positive entropy if every non-trivial factor of (X; T) has positive entropy. Isomorphism of Bernoulli Shifts Kolmogorov–Sinai A long standing open question was for which p and q are Bernoullip and Bernoulliq isomorphic. In particular are the Bernoulli 2 shift and the Bernoulli 3 shift isomorphic. Both of these transformations have completely positive entropy and all other isomorphism invariants which were known at the time are the same for the two transformations. The first application of the Kolmogorov–Sinai entropy was to show that the answer to this question is no. Fix a probability vector p. The transformation Bernoullip has Qp : x ! x0 as a generating partition. By Sinai’s theorem H(Bernoulli p ) D H(Qp ) D
Finally, the measure-theoretic entropy of a dynamical system (X; T) is defined as h((X; T)) D sup h(X; T; Q) Q
where the supremum is taken over all finite measurable partitions. A theorem of Sinai showed that if Q is a generator of (X; T) then h(T) D h(T; Q) [73]. This shows that for every measure preserving function (X; T) there is an associated entropy h(T) 2 [0; 1]. It is easy to show from the definition that entropy is an isomorphism invariant. We say that (Y; S) is a factor of (X; T) if there exists a map : X ! Y such that (1) is measure preserving and (2) (T(x)) D S((x)) for almost every x. Each factor (Y; S) can be associated with 1 (C ), which is an invariant sub -algebra of B. We say that
p i log2 (p i ) :
iD1
iD1
The measure-theoretic entropy of a dynamical system (X; T) with respect to a partition Q : X ! f1; : : : ; kg is then defined as ! N _ 1 n T Q : H h(X; T; Q) D lim N!1 N nD1
n X
Thus the Bernoulli 2 shift (with entropy 1) is not isomorphic to the Bernoulli 3 shift (with entropy log2 (3)). Sinai also made significant progress toward showing that Bernoulli shifts with the same entropy are isomorphic by proving the following theorem. Theorem 1 [72] If (X; T) is a measure preserving system of entropy h and (Y; S) is a Bernoulli shift of entropy h0 h then (Y; S) is a factor of (X; T). This theorem implies that if p and q are probability vectors and H(p) D H(q) then Bernoullip is a factor in Bernoulliq and Bernoulliq is a factor in Bernoullip. Thus we say that Bernoullip and Bernoulliq are weakly isomorphic. Explicit Isomorphisms The other early progress on proving that Bernoulli shifts with the same entropy are isomorphic came from Meshalkin. He considered pairs of probability vectors p and q with H(p) D H(q) and all of the pi and qi are related by some algebraic relations. For many such pairs he was able to prove that the two Bernoulli shifts are isomorphic. In particular he proved the following theorem. Theorem 2 [40] Let p D (1/4; 1/4; 1/4; 1/4) and q D (1/2; 1/8; 1/8; 1/8; 1/8). Then Bernoullip and Bernoulliq are isomorphic.
Isomorphism Theory in Ergodic Theory
Ornstein The central theorem in the study of isomorphisms of measure preserving transformations is Ornstein’s isomorphism theorem. Theorem 3 [46] If p and q are probability vectors and H(p) D H(q) then the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic. To see how central this is to the field most of the rest of this article is a summary of: (1) (2) (3) (4)
The proof of Ornstein’s theorem, The consequences of Ornstein’s theorem, The generalizations of Ornstein’s theorem, and How the properties that Ornstein’s theorem implies that Bernoulli shifts must have differ from the properties of every other class of transformations.
The key to Ornstein’s proof was the introduction of the finitely determined property. To explain the finitely determined property we first define the Hamming distance of length n between sequences x; y 2 N Z by jfk 2 f1; : : : ; ng : x k D y k gj d¯n (x; y) D 1 : n Let (X; T) and (Y; S) be a measure preserving transformation and let P and Q be finite partitions of X and Y respectively. We say that (X; T) and P and (Y; S) and Q are within ı in n distributions if ˇ n o X ˇ ˇ x : P T i (x) D a i 8i D 1; : : : ; n ˇ (a 1 ;:::;a n )2Z n
o ˇˇ n y : Q T i (y) D a i 8i D 1; : : : ; n ˇˇ < ı :
A process (X; T) and P are finitely determined if for every " there exist n and ı such that if (Y; S) and Q are such that (1) (X; T) and P and Yand Q are within ı in n distributions and (2) jH(X; P) H(Y; Q)j < ı then there exists a joining of X and Y such that for all m Z d¯m (x; y) d (x; y) < :
the Rokhlin lemma and the Shannon–McMillan–Breiman theorem to prove a more robust version of Theorem 1. To describe Ornstein’s proof we use the a description due to Rothstein. We say that for a joining of (X; ) and (Y; ) that P ; C if there exists a partition P˜ of C such that X
Pi 4P˜i < : i
If P ; C for all > 0 then it is possible to show that there exists a partition P˜ of C such that X
Pi 4P˜i D 0 i
and we write P C . If P C then (X; T) is a factor of (Y; S) by the map that sends y ! x where P(T i x) D ˜ i y) for all i. P(S In this language Ornstein proved that if (X; T) is finitely determined, P is a generating partition of X and h((Y; S)) h((X; T)) then for every > 0 the set of joinings such that P ; C is an open and dense set. Thus by the Baire category theorem there exists such that P C . This reproves Theorem 1. Moreover if (X; T) and (Y; S) are finitely determined, h((X; T)) D h((Y; S)) and P and Q are generating partition of X and Y then by the Baire category theorem there exists such that P C and Q B. Then the map ˜ i y) for all i is an that sends y ! x where P(T i x) D P(S isomorphism. Properties of Bernoullis Now we define the very weak Bernoulli property which is the most effective property for showing that a measure preserving transformation is isomorphic to a Bernoulli shift. Given X and a partition P define the past of x by n o Ppast (x) D x 0 : T i P x 0 D T i P(x) 8i 2 N and denote the measure conditioned on Ppast (x) by j Ppast (x) . Define d¯n;x; D inf
Z
d¯n x; x 0 d x; x 0 ;
x;y
A transformation (X; T) is finitely determined if it is finitely determined for every finite partition P. It is fairly straightforward to show that Bernoulli shifts are finitely determined. Ornstein used this fact along with
where the inf is taken over all which are couplings of j Ppast (x) and . Also define d¯n;Ppast ;(X;T) D
Z
d¯n;x; d :
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We say that (X; T) and P are very weak Bernoulli if for every > 0 there exists n such that d¯n;Ppast ;(X;T) < . We say that (X; T) is very weakly Bernoulli if there exists a generating partition P such that (X; T) and P are very weak Bernoulli. Ornstein and Weiss were able to show that the very weak Bernoulli property is both necessary and sufficient to be isomorphic to a Bernoulli shift. Theorem 4 [45,53] For transformations (X; T) the following conditions are equivalent: (1) (X; T) is finitely determined, (2) (X; T) is very weak Bernoulli and (3) (X; T) is isomorphic to a Bernoulli shift.
to a Bernoulli shift then the very weak Bernoulli property is more useful. There have been many classes of transformations that have been proven to be isomorphic to a Bernoulli shift. Here we mention two. The first class are the Markov chains. Friedman and Ornstein proved that if a Markov chain is mixing then it is isomorphic to a Bernoulli shift [9]. The second are automorphisms of [0; 1)n . Let M be any n n matrix with integer coefficients and jdet(M)j D 1. If none of the eigenvalues f i gniD1 of M are roots of unity then Katznelson proved that T M is isomorphic to the Bernoulli shift with P entropy niD1 max(0; log( i )) [29].
Transformations not Isomorphic to Bernoulli Shifts Using the fact that a transformation is finitely determined or very weak Bernoulli is equivalent to it being isomorphic to a Bernoulli shift we can prove the following theorem. Theorem 5 [44] (1) If (X; T n ) is isomorphic to a Bernoulli shift then (X; T) is isomorphic to a Bernoulli shift. (2) If (X; T) is a factor of a Bernoulli shift then (X; T) is isomorphic to a Bernoulli shift. (3) If (X; T) is isomorphic to a Bernoulli shift then there exists a measure preserving transformation (Y; S) such that (Y; S n ) is isomorphic to (X; T).
Rudolph Structure Theorem An important application of the very weak Bernoulli condition is the following theorem of Rudolph. Theorem 6 [67] Let (X; T) be isomorphic to a Bernoulli shift, G be a compact Abelian group with Haar measure G and : X ! G be a measurable map. Then let S : X G ! X G be defined by S(x; g) D (T(x); g C (x)) : Then (X G; S; G ) is isomorphic to a Bernoulli shift. Transformations Isomorphic to Bernoulli Shifts One of the most important features of Ornstein’s isomorphism theory is that it can be used to check whether specific transformations (or families of transformations) are isomorphic to Bernoulli shifts. The finitely determined property is the key to the proof of Ornstein’s theorem and the proof of many of the consequences listed in Subsect. “Properties of Bernoullis”. However if one wants to show a particular transformation is isomorphic
Recall that a measure preserving transformation (X; T) has completely positive entropy if for every nontrivial (Y; S) which is a factor of (X; T) we have that H((Y; S)) > 0. It is easy to check that Bernoulli shifts have completely positive entropy. It is natural to ask if the converse true? We shall see that the answer is an emphatic no. While the isomorphism class of Bernoulli shifts is given by just one number, the situation for transformations of completely positive entropy is infinitely more complicated. Ornstein constructed the first example of a transformation with completely positive entropy which is not isomorphic to a Bernoulli shift [47]. Ornstein and Shields built upon this construction to prove the following theorem. Theorem 7 [50] For every h > 0 there is an uncountable family of completely positive entropy transformations which all have entropy h but no two distinct members of the family are isomorphic. Now that we see there are many isomorphic transformations that have completely positive entropy it is natural to ask if (X; T) is not isomorphic to a Bernoulli shift then is there any reasonable condition we can put on (Y; S) that implies the two transformations are isomorphic. For example if (X; T 2 ) and (Y; S 2 ) are completely positive entropy transformations which are isomorphic does that necessarily imply that (X; T) and (Y; S) are isomorphic? The answer turns out to be no [66]. We could also ask if (X; T) and (Y; S) are completely positive entropy transformations which are weakly isomorphic does that imply that (X; T) and (Y; S) are isomorphic? Again the answer is no [17]. The key insight to answering questions like this is due to Rudolph who showed that such questions about the isomorphism of transformations can be reduced to questions about conjugacy of permutations.
Isomorphism Theory in Ergodic Theory
Rudolph’s Counterexample Machine
Theorem 10 [17]
Given any transformation (X; T) and any permutation in Sn (or SN ) we can define the transformation (X n ; T ; n ; B) by
(1) There exist measure preserving transformations (X; T) and (Y; S) which both have completely positive entropy and are weakly isomorphic but not isomorphic. (2) There exist measure preserving transformations (X; T) and (Y; S) which both have completely positive entropy and are not isomorphic but (X; T k ; ) is isomorphic to (Y; S k ; ) for every k > 1.
T (x1 ; x2 ; : : : ; x n ) D T x(1) ; T x(2) ; : : : ; T x(n) where n is the direct product of n copies of . Rudolph introduced the concept of a transformation having minimal self joinings. If a transformation has minimal self joinings then for every it is possible to list all of the measures on X n which are invariant under T . If there exists an isomorphism between T and (Y; S) then there is a corresponding measure on X Y which is supported on points of the form (x; (x)) and has marginals and . Thus if we know all of the measures on X Y which are invariant under T S then we know all of the isomorphisms between (X; T) and (Y; S). Using this we get the following theorem. Theorem 8 [65] There exists a nontrivial transformation with minimal self joinings. For any transformation (X; T) with minimal self joinings the corresponding transformation T1 is isomorphic to T2 if and only if the permutations 1 and 2 are conjugate. There are two permutations on two elements, the flip 1 D (12) and the identity 2 D (1)(2). For both permutations, the square of the permutation is the identity. Thus there are two distinct permutations whose square is the same. Rudolph showed that this fact can be used to generate two transformations which are mixing that are not isomorphic but their squares are isomorphic. The following theorem gives more examples of the power of this technique. Theorem 9 [65] (1) There exists measure preserving transformations (X; T) and (Y; S) which are weakly isomorphic but not isomorphic. (2) There exists measure preserving transformations (X; T) and (Y; S) which are not isomorphic but (X; T k ; ) is isomorphic to (Y; S k ; ) for every k > 1, and (3) There exists a mixing transformation with no non trivial factors. If (X; T) has minimal self joinings then it has zero entropy. However Hoffman constructed a transformation with completely positive entropy that shares many of the properties of transformations with minimal self joinings listed above.
T, T Inverse All of the transformations that have completely positive entropy but are not isomorphic to a Bernoulli shift described above are constructed by a process called cutting and stacking. These transformations have little inherent interest outside of their ergodic theory properties. This led many people to search for a “natural” example of such a transformation. The most natural examples are the [T; T 1 ] transformation and many other transformations derived from it. It is easy to show that the [T; T 1 ] transformation has completely positive entropy [39]. Kalikow proved that for many T the corresponding [T; T 1 ] transformation is not isomorphic to a Bernoulli shift. Theorem 11 [24] If h(T) > 0 then the [T; T 1 ] transformation is not isomorphic to a Bernoulli shift. The basic idea of Kalikow’s proof has been used by many others. Katok and Rudolph used the proof to construct smooth measure preserving transformations on infinite differentiable manifolds which have completely positive entropy but are not isomorphic to Bernoulli shifts [27,68]. Den Hollander and Steif did a thorough study of the ergodic theory properties of [T; T 1 ] transformations where T is simple random walk on a wide family of graphs [4]. Classifying the Invariant Measures of Algebraic Actions This problem of classifying all of the invariant measures like Rudolph did with his transformation with minimal self joinings comes up in a number of other settings. Ratner characterized the invariant measures for the horocycle flow and thus characterized the possible isomorphisms between a large family of transformations generated by the horocycle flow [59,60,61]. This work has powerful applications to number theory. There has also been much interest in classifying the measures on [0; 1) that are invariant under both times 2 and times 3. Furstenberg proved that the only closed infinite set on the circle which is invariant under times 2 and
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times 3 is [0; 1) itself and made the following measure theoretic conjecture.
Schmidt proved that this happens only in the trivial case that p and q are rearrangements of each other.
Conjecture 1 [10] The only nonatomic measure on [0; 1) which is invariant under times 2 and times 3 is Lebesgue measure.
Theorem 14 [70] If p and q are probability vectors and the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic and there exists an isomorphism such that and 1 are both finitary and have finite expected coding time then p is a rearrangement of q.
Rudolph improved on the work of Lyons [36] to provide the following partial answer to this conjecture. Theorem 12 [69] The only measure on [0; 1) which is invariant under multiplication by 2 and by 3 and has positive entropy under multiplication by 2 is Lebesgue measure. Johnson then proved that for all relatively prime p and q that p and q can be substituted for 2 and 3 in the theorem above. This problem can be generalized to higher dimensions by studying the actions of commuting integer matrices of determinant greater than one on tori. Katok and Spatzier [28] and Einsiedler and Lindenstrauss [5] obtained results similar to Rudolph’s for actions of commuting matrices. Finitary Isomorphisms By Ornstein’s theorem we know that there exists an isomorphism between any two Bernoulli shifts (or mixing Markov shifts) of the same entropy. There has been much interest in studying how “nice” the isomorphism can be. By this we mean can be chosen so that the map x ! ((x))0 is continuous and if so what is its best possible modulus of continuity? We say that a map from N Z to N Z is finitary if in order to determine ((x))0 we only need to know finitely many coordinates of x. More precisely if for almost every x there exists m(x) such that ˚ x 0 : x 0i D x i for all jij m(x) and x 0 0 ¤ (x)0 D 0 : R We say that m(x) has finite tth moment if m(x) t d< 1 and that has finite expected coding length if the first moment of m(x) is finite. Keane and Smorodinsky proved the following strengthening of Ornstein’s isomorphism theorem. Theorem 13 [30] If p and q are probability vectors with H(p) D H(q) then the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic and there exists an isomorphism such that and 1 are both finitary. The nicest that we could hope to be is if both and 1 are finitary and have finite expected coding length.
The best known result about this problem is the following theorem of Harvey and Peres. Theorem 15 [14] If p and q are probability vectors with P P H(p) D H(q) then i (p i )2 log(p i ) D i (q i )2 log(q i ) if and only if the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic and there exists an isomorphism such that and 1 are both finitary and have finite one half moment. Flows A flow is a measure preserving action of the group R on a measure space (X; T). A cross section is any measurable set in C X such that for almost every x 0 < inf ft : Tt (x) 2 Cg < 1 : For any flow (X; T) and fTt g t2R and any cross section C we define the return time map for C R : C ! C as follows. For any x 2 C define t(x) D inf ft : Tt (x) 2 Cg then set R(x) D Tt(x) (x). There is a standard method to project the probability measure on X to an invariant probability measure C on C as well as the -algebra B on X to a -algebra BC on C such that (C; C ; BC ) and R is a measure preserving transformation. First we show that there is a natural analog of Bernoulli shifts for flows. Theorem 16 [45] There exists a flow (X; T) and fTt g t2R such that for every t > 0 the map (X; T) and T t is isomorphic to a Bernoulli shift. Moreover for any h 2 (0; 1] there exists (X; T) and fTt g t2R such that h(T1 ) D h. We say that such a flow (X; f f t g t2R ; ; B) is a Bernoulli flow. This next version of Ornstein’s isomorphism theorem shows that up to isomorphism and a change in time (considering the flow X and fTc t g instead of X and fTt g) there are only two Bernoulli flows, one with positive but finite entropy and one with infinite entropy. Theorem 17 [45] If (X; T) and fTt g t2R and (Y; S) fS t g t2R are Bernoulli flows and h(T1 ) D h(S1 ) then they are isomorphic.
Isomorphism Theory in Ergodic Theory
As in the case of actions of Z there are many natural examples of flows that are isomorphic to the Bernoulli flow. The first is for geodesic flows. In the 1930s Hopf proved that geodesic flows on compact surfaces of constant negative curvature are ergodic [22]. Ornstein and Weiss extended Hopf’s proof to show that the geodesic flow is also Bernoulli [52]. The second class of flows comes from billiards on a square table with one circular bumper. The state space X consists of all positions and velocities for a fixed speed. The flow T t is frictionless movement for time t with elastic collisions. This flow is also isomorphic to the Bernoulli flow [11]. Other Equivalence Relations In this section we will discuss a number of equivalence relations between transformations that are weaker than isomorphism. All of these equivalence relations have a theory that is parallel to Ornstein’s theory. Kakutani Equivalence We say that two transformations (X; T) and (Y; S) are Kakutani equivalent if there exist subsets A X and B Y such that (TA ; A; A ) and (S B ; B; B ) are isomorphic. This is equivalent to the existence of a flow (X; T) and fTt g t2R with cross sections C and C 0 such that the return time maps of C and C 0 are isomorphic to (X; T) and (Y; S) respectively. Using the properties of entropy of the induced map we have that if (X; T) and (Y; S) are Kakutani equivalent then either h((X; T)) D h((Y; S)) D 0, 0 < h((X; T)); h((Y; S)) < 1 or h((X; T)) D h((Y; S)) D 1. In general the answer to the question of which pairs of measure preserving transformations are isomorphic is quite poorly understood. But if one of the transformations is a Bernoulli shift then Ornstein’s theory gives a fairly complete answer to the question. A similar situation exists for Kakutani equivalence. In general the answer to the question of which pairs of measure preserving transformation are Kakutani equivalent is also quite poorly understood. But the more specialized question of which transformations are isomorphic to a Bernoulli shift has a more satisfactory answer. Feldman constructed a transformation (X; T) which has completely positive entropy but (X; T) is not Kakutani equivalent to a Bernoulli shift. Ornstein, Rudolph and Weiss extended Feldman’s work to construct a complete theory of the transformations that are Kakutani equivalent to a Bernoulli shift [49] for positive entropy transformations and a theory of the transformations that are
Kakutani equivalent to an irrational rotation [49] for zero entropy transformations. (The zero entropy version of this theorem had been developed independently (and earlier) by Katok [26].) They defined two class of transformations called loosely Bernoulli and finitely fixed. The definitions of these properties are the same as the definitions of very weak Bernoulli and finitely determined except that the d¯ metric is replaced by the f¯ metric. For x; y 2 N Z we define k f¯n (x; y) D 1 n where k is the largest number such that sequences 1 i1 < i2 < < i k n and 1 j1 < j2 < < j k n such that x i l D y j l for all j, 1 j k. (In computer science this metric is commonly referred to as the edit distance.) Note that d¯n (x; y) f¯n (x; y). They proved the following analog of Theorem 5. Theorem 18 For transformations (X; T) with h((X; T)) > 0 the following conditions are equivalent: (1) (2) (3) (4)
(X; T) is finitely fixed, (X; T) is loosely Bernoulli, (X; T) is Kakutani equivalent to a Bernoulli shift and There exists a Bernoulli flow Y and fF t g t2R and a cross section C such that the return time map for C is isomorphic to (X; T).
Restricted Orbit Equivalence Using the d¯ metric we got a theory of which transformations are isomorphic to a Bernoulli shift. Using the f¯ metric we got a strikingly similar theory of which transformations are Kakutani equivalent to a Bernoulli shift. Rudolph showed that it is possible to replace the d¯ metric (or the f¯ metric) with a wide number of other metrics and produce parallel theories for other equivalence relations. For instance, for each of these theories we get a version of Theorem 5. This collection of theories is called restricted orbit equivalence [64]. Non-invertible Transformations The question of which noninvertible measure preserving transformations are isomorphic turns out to be quite different from the same question for invertible transformations. In one sense it is easier because of an additional isomorphism invariant. For any measure preserving transformation (X; T) the probability measure j T 1 (x) on T 1 (x) is defined for almost every x 2 X. (If (X; T) is invertible then this measure
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is trivial as jfT 1 (x)gj D 1 and j T 1 (x) (T 1 (x)) D 1 for almost every x.) It is easy to check that if is an isomorphism from (X; T) to (Y; S) then for almost every x and x 0 2 T 1 (x) we have ˇ ˇ ˇ T 1 (x) x 0 D ˇS 1 ((x)) x 0 : From this we can easily see that if p D fp i gniD1 and q D fq i gm iD1 are probability vectors then the corresponding one sided Bernoulli shifts are isomorphic only if m D n and there is a permutation 2 S n such that p(i) D q i for all i. (In this case we say p is a rearrangement of q.) If p is a rearrangement of q then it is easy to construct an isomorphism between the corresponding Bernoulli shifts. Thus the analog of Ornstein’s theorem for Bernoulli endomorphisms is trivial. However we will see that there still is an analogous theory classifying the class of endomorphism that are isomorphic to Bernoulli endomorphisms. We say that an endomorphism is uniformly d to 1 if for almost every x we have that jfT 1 (x)gj D d and j T 1 (x) (y) D 1/d for all y 2 T 1 (x). Hoffman and Rudolph defined two classes of noninvertible transformations called tree very week Bernoulli and tree finitely determined and proved the following theorem. Theorem 19 The following three conditions are equivalent for uniformly d to 1 endomorphisms. (1) (X; T) is tree very weak Bernoulli, (2) (X; T) is tree finitely determined, and (3) (X; T) is isomorphic to the one sided Bernoulli d shift. Jong extended this theorem to say that if there exists a probability vector p such that for almost every x the distribution of j T 1 (x) is the same as the distribution of p then (X; T) is isomorphic to the one sided Bernoulli d shift if and only if it is tree finitely determined and if and only if it is tree very weak Bernoulli [23]. Markov Shifts We saw that mixing Markov chains are isomorphic if they have the same entropy. As we have seen there are additional isomorphism invariants for noninvertible transformations. Ashley, Marcus and Tuncel managed to classify all one sided mixing Markov chains up to isomorphism [1]. Rational Maps Rational maps are the main object of study in complex dynamics. For every rational function f (z) D p(z)/q(z) there is a nonempty compact set J f which is called the Julia set.
Roughly speaking this is the set of points for which every neighborhood acts “chaotically” under repeated iterations of f . In order to consider rational maps as measure preserving transformations we need to specify an invariant measure. The following theorem of Gromov shows that for every rational map there is one canonical measure to consider. Theorem 20 [12] For every f rational function of degree d there exists a unique invariant measure f of maximal entropy. We have that h( f ) D log2 d and f (J f ) D 1. The properties of this measure were studied by Freire, Lopes and Mañé [8]. Mañé that analysis to prove the following theorem. Theorem 21 [38] For every rational function f of degree d there exists n such that (C; f n ; f ) (where f n (z) D f ( f ( f : : : (z))) is composition) is isomorphic to the one sided Bernoulli dn shift. Heicklen and Hoffman used the tree very weak Bernoulli condition to show that we can always take n to be one. Theorem 22 [16] For every rational function f of degree d 2 the corresponding map ((C; f ; B); T f ) is isomorphic to the one sided Bernoulli d shift. Differences with Ornstein’s Theory Unlike Kakutani equivalence and restricted orbit equivalence which are very close parallels to Ornstein’s theory, the theory of which endomorphisms are isomorphic to a Bernoulli endomorphism contains some significant differences. One of the principal results of Ornstein’s isomorphism theory is that if (X; T) is an invertible transformation and (X; T 2 ) is isomorphic to a Bernoulli shift then (X; T) is also isomorphic to a Bernoulli shift. There is no corresponding result for noninvertible transformations. Theorem 23 [19] There is a uniformly two to one endomorphism (X; T) which is not isomorphic to the one sided Bernoulli 2 shift but (X; T 2 ) is isomorphic to the one sided Bernoulli 4 shift. Factors of a Transformation In this section we study the relationship between a transformation and its factors. There is a natural way to associate a factor of (X; T) with a sub -algebra of B. Let (Y; S) be a factor of (X; T) with factor map : X ! Y. Then the -algebra associated with (Y; S) is BY D 1 (C ). Thus the study of factors of a transformation is the study of its sub -algebras.
Isomorphism Theory in Ergodic Theory
Almost every property that we have discussed above has an analog in the study of factors of a transformation. We give three such examples. We say that two factors C and D of (X; T) are relatively isomorphic if there exists an isomorphism : X ! X of (X; T) with itself such that (C ) D D. We say that (X; T) has relatively completely positive entropy with respect to C if every factor D which contains D has h(D) > h(C ). We say that C is relatively Bernoulli if there exists a second factor D which is independent of D and B D C _ D. Thouvenot defined properties of factors called relatively very weak Bernoulli and relatively finitely determined. Then he proved an analog of Theorem 3. This says that a factor being relatively Bernoulli is equivalent to it being relatively finitely determined (and also equivalent to it being relatively very weak Bernoulli). The Pinsker algebra is the maximal -algebra P such that h(P ) D 0. The Pinsker conjecture was that for every measure preserving transformation (X; T) there exists a factor C such that (1) C is independent of the Pinsker algebra P (2) B D C _ P and (3) (X; C ) has completely positive entropy. Ornstein found a counterexample to the Pinsker conjecture [48]. After Thouvenot developed the relative isomorphism theory he came up with the following question which is referred to as the weak Pinsker conjecture. Conjecture 2 For every measure preserving transformation (X; T) and every > 0 there exist invariant -algebras C ; D B such that (1) (2) (3) (4)
C is independent of D BDC_D (X; T; ; D) is isomorphic to a Bernoulli shift h((X; T; ; C )) < :
There is a wide class of transformations which have been proven to satisfy the weak Pinsker conjecture. This class includes almost all measure preserving transformations which have been extensively studied. Actions of Amenable Groups All of the discussion above has been about the action of a single invertible measure preserving transformation (actions of N and Z) or flows (actions of R). We now consider more general group actions. If we have two actions S and T on a measure space (X; ) which commute (S(T(x)) D T(S(x)) for almost every x) then there is an action of Z2 on (X; ) given by f(n;m) (x) D S n (T m (x)).
A natural question to ask is do there exist a version of entropy theory and Ornstein’s isomorphism theory for actions of two commuting automorphisms. More generally for each of the results discussed above we can ask what is the largest class of groups such that an analogous result is true. It turns out that for most of the properties described above the right class of groups is discrete amenable groups. A Følner sequence F n in a group G is a sequence of subsets F n of G such that for all g 2 G we have that lim n!1 jg(Fn )j/jFn j. A countable group is amenable if and only if it has a Følner sequence. For nonamenable groups it is much more difficult to generalize Birkhoff’s ergodic theorem [41,42]. Lindenstrauss proved that for every discrete amenable group there is an analog of the ergodic theorem [35]. For every amenable group G and every probability vector p we can define a Bernoulli action of G. There are also analogs of Rokhlin’s lemma and the Shannon–McMillan– Breiman theorem for actions of all discrete amenable groups [33,51,54]. Thus we have all of the ingredients to prove a version of Ornstein’s isomorphism theorem. Theorem 24 If p and q are probability vectors and H(p) D H(q) then the Bernoulli actions of G corresponding to p and q are isomorphic. Also all of the aspects of Rudolph’s theory of restricted orbit equivalence can be carried out for actions of amenable groups [25]. Differences Between Actions of Z and Actions of Other Groups Although generalizations of Ornstein theory and restricted orbit equivalence carry over well to the actions of discrete amenable groups there do turn out to be some significant differences between the possible actions of Z and those of other discrete amenable groups. Many of these have to do with the generalization of Markov shifts. For actions of Z2 these are called Markov random fields. By the result of Friedman and Ornstein if a Markov chain is mixing then it has completely positive entropy and it is isomorphic to a Bernoulli shift. Mixing Markov random fields can have very different properties. Ledrappier constructed a Z2 action which is a Markov random field and is mixing but has zero entropy [34]. Even more surprising even though it is mixing it is not mixing of all orders. The existence of a Z action which is mixing but not mixing of all orders is one of the longest standing open questions in ergodic theory [13]. Even if we try to strengthen the hypothesis of Friedman and Ornstein’s theorem to assume that the Markov
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random field has completely positive entropy we will not succeed as there exists a Markov random field which has completely positive entropy but is not isomorphic to a Bernoulli shift [18]. Future Directions In the future we can expect to see progress of isomorphism theory in a variety of different directions. One possible direction for future research is better understand the properties of finitary isomorphisms between various transformations and Bernoulli shifts described in Sect. “Finitary Isomorphisms”. Another possible direction would be to find a theory of equivalence relations for Bernoulli endomorphisms analogous to the one for invertible Bernoulli transformations described in Sect. “Other Equivalence Relations”. As the subject matures the focus of research in isomorphism theory will likely shift to connections to other fields. Already there are deep connections between isomorphism theory and both number theory and statistical physics. Finally one hopes to see progress made on the two dominant outstanding conjectures in the field: Thouvenot weak Pinsker conjecture (Conjecture 2) and Furstenberg’s conjecture (Conjecture 1) about measures on the circle invariant under both the times 2 and times 3 maps. Progress on either of these conjectures would invariably lead the field in exciting new directions. Bibliography 1. Ashley J, Marcus B, Tuncel S (1997) The classification of onesided Markov chains. Ergod Theory Dynam Syst 17(2):269–295 2. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Natl Acad Sci USA 17:656–660 3. Breiman L (1957) The individual ergodic theorem of information theory. Ann Math Statist 28:809–811 4. Den Hollander F, Steif J (1997) Mixing E properties of the generalized T; T 1 -process. J Anal Math 72:165–202 5. Einsiedler M, Lindenstrauss E (2003) Rigidity properties of Zd actions on tori and solenoids. Electron Res Announc Amer Math Soc 9:99–110 6. Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann Math (2) 164(2):513–560 7. Feldman J (1976) New K-automorphisms and a problem of Kakutani. Isr J Math 24(1):16–38 8. Freire A, Lopes A, Mañé R (1983) An invariant measure for rational maps. Bol Soc Brasil Mat 14(1):45–62 9. Friedman NA, Ornstein DS (1970) On isomorphism of weak Bernoulli transformations. Adv Math 5:365–394 10. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theory 1:1–49 11. Gallavotti G, Ornstein DS (1974) Billiards and Bernoulli schemes. Comm Math Phys 38:83–101
12. Gromov M (2003) On the entropy of holomorphic maps. Enseign Math (2) 49(3–4):217–235 13. Halmos PR (1950) Measure theory. D Van Nostrand, New York 14. Harvey N, Peres Y. An invariant of finitary codes with finite expected square root coding length. Ergod Theory Dynam Syst, to appear 15. Heicklen D (1998) Bernoullis are standard when entropy is not an obstruction. Isr J Math 107:141–155 16. Heicklen D, Hoffman C (2002) Rational maps are d-adic Bernoulli. Ann Math (2) 156(1):103–114 17. Hoffman C (1999) A K counterexample machine. Trans Amer Math Soc 351(10):4263–4280 18. Hoffman C (1999) A Markov random field which is K but not Bernoulli. Isr J Math 112:249–269 19. Hoffman C (2004) An endomorphism whose square is Bernoulli. Ergod Theory Dynam Syst 24(2):477–494 20. Hoffman C (2003) The scenery factor of the [T; T 1 ] transformation is not loosely Bernoulli. Proc Amer Math Soc 131(12):3731–3735 21. Hoffman C, Rudolph D (2002) Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann Math (2) 156(1):79–101 22. Hopf E (1971) Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull Amer Math Soc 77: 863–877 23. Jong P (2003) On the isomorphism problem of p-endomorphisms. Ph D thesis, University of Toronto 24. Kalikow SA (1982) T; T 1 transformation is not loosely Bernoulli. Ann Math (2) 115(2):393–409 25. Kammeyer JW, Rudolph DJ (2002) Restricted orbit equivalence for actions of discrete amenable groups. Cambridge tracts in mathematics, vol 146. Cambridge University Press, Cambridge 26. Katok AB (1975) Time change, monotone equivalence, and standard dynamical systems. Dokl Akad Nauk SSSR 223(4):789–792; in Russian 27. Katok A (1980) Smooth non-Bernoulli K-automorphisms. Invent Math 61(3):291–299 28. Katok A, Spatzier RJ (1996) Invariant measures for higherrank hyperbolic abelian actions. Ergod Theory Dynam Syst 16(4):751–778 29. Katznelson Y (1971) Ergodic automorphisms of T n are Bernoulli shifts. Isr J Math 10:186–195 30. Keane M, Smorodinsky M (1979) Bernoulli schemes of the same entropy are finitarily isomorphic. Ann Math (2) 109(2):397–406 31. Kolmogorov AN (1958) A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl Akad Nauk SSSR (NS) 119:861–864; in Russian 32. Kolmogorov AN (1959) Entropy per unit time as a metric invariant of automorphisms. Dokl Akad Nauk SSSR 124:754–755; in Russian 33. Krieger W (1970) On entropy and generators of measure-preserving transformations. Trans Amer Math Soc 149:453–464 34. Ledrappier F (1978) Un champ markovien peut être d’entropie nulle et mélangeant. CR Acad Sci Paris Sér A–B 287(7):A561– A563; in French 35. Lindenstrauss E (2001) Pointwise theorems for amenable groups. Invent Math 146(2):259–295 36. Lyons R (1988) On measures simultaneously 2- and 3-invariant. Isr J Math 61(2):219–224 37. Mañé R (1983) On the uniqueness of the maximizing measure for rational maps. Bol Soc Brasil Mat 14(1):27–43
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38. Mañé R (1985) On the Bernoulli property for rational maps. Ergod Theory Dynam Syst 5(1):71–88 39. Meilijson I (1974) Mixing properties of a class of skew-products. Isr J Math 19:266–270 40. Meshalkin LD (1959) A case of isomorphism of Bernoulli schemes. Dokl Akad Nauk SSSR 128:41–44; in Russian 41. Nevo A (1994) Pointwise ergodic theorems for radial averages on simple Lie groups. I. Duke Math J 76(1):113–140 42. Nevo A, Stein EM (1994) A generalization of Birkhoff’s pointwise ergodic theorem. Acta Math 173(1):135–154 43. Ornstein DS (1973) A K automorphism with no square root and Pinsker’s conjecture. Adv Math 10:89–102 44. Ornstein D (1970) Factors of Bernoulli shifts are Bernoulli shifts. Adv Math 5:349–364 45. Ornstein D (1970) Two Bernoulli shifts with infinite entropy are isomorphic. Adv Math 5:339–348 46. Ornstein D (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352 47. Ornstein DS (1973) An example of a Kolmogorov automorphism that is not a Bernoulli shift. Adv Math 10:49–62 48. Ornstein DS (1973) A mixing transformation for which Pinsker’s conjecture fails. Adv Math 10:103–123 49. Ornstein DS, Rudolph DJ, Weiss B (1982) Equivalence of measure preserving transformations. Mem Amer Math Soc 37(262). American Mathematical Society 50. Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88 51. Ornstein D, Weiss B (1983) The Shannon–McMillan–Breiman theorem for a class of amenable groups. Isr J Math 44(1): 53–60 52. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Isr J Math 14:184–198 53. Ornstein DS, Weiss B (1974) Finitely determined implies very weak Bernoulli. Isr J Math 17:94–104 54. Ornstein DS, Weiss B (1987) Entropy and isomorphism theorems for actions of amenable groups. J Anal Math 48:1–141 55. Parry W (1981) Topics in ergodic theory. Cambridge tracts in mathematics, vol 75. Cambridge University Press, Cambridge 56. Parry W (1969) Entropy and generators in ergodic theory. WA Benjamin, New York 57. Petersen K (1989) Ergodic theory. Cambridge studies in advanced mathematics, vol 2. Cambridge University Press, Cambridge 58. Pinsker MS (1960) Dynamical systems with completely pos-
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itive or zero entropy. Dokl Akad Nauk SSSR 133:1025–1026; in Russian Ratner M (1978) Horocycle flows are loosely Bernoulli. Isr J Math 31(2):122–132 Ratner M (1982) Rigidity of horocycle flows. Ann Math (2) 115(3):597–614 Ratner M (1983) Horocycle flows, joinings and rigidity of products. Ann Math (2) 118(2):277–313 Ratner M (1991) On Raghunathan’s measure conjecture. Ann Math (2) 134(3):545–607 Halmos PR (1960) Lectures on ergodic theory. Chelsea Publishing, New York Rudolph DJ (1985) Restricted orbit equivalence. Mem Amer Math Soc 54(323). American Mathematical Society Rudolph DJ (1979) An example of a measure preserving map with minimal self-joinings, and applications. J Anal Math 35:97–122 Rudolph DJ (1976) Two nonisomorphic K-automorphisms with isomorphic squares. Isr J Math 23(3–4):274–287 Rudolph DJ (1983) An isomorphism theory for Bernoulli free Zskew-compact group actions. Adv Math 47(3):241–257 Rudolph DJ (1988) Asymptotically Brownian skew products give non-loosely Bernoulli K-automorphisms. Invent Math 91(1):105–128 Rudolph DJ (1990) 2 and 3 invariant measures and entropy. Ergod Theory Dynam Syst 10(2):395–406 Schmidt K (1984) Invariants for finitary isomorphisms with finite expected code lengths. Invent Math 76(1):33–40 Shields P (1973) The theory of Bernoulli shifts. Chicago lectures in mathematics. University of Chicago Press, Chicago Sina˘ı JG (1962) A weak isomorphism of transformations with invariant measure. Dokl Akad Nauk SSSR 147:797–800; in Russian Sina˘ı J (1959) On the concept of entropy for a dynamic system. Dokl Akad Nauk SSSR 124:768–771; in Russian Thouvenot J-P (1975) Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli. Conference on ergodic theory and topological dynamics, Kibbutz, Lavi, 1974. Isr J Math 21(2–3):177–207; in French Thouvenot J-P (1975) Une classe de systèmes pour lesquels la conjecture de Pinsker est vraie. Conference on ergodic theory and topological dynamics, Kibbutz Lavi, 1974. Isr J Math 21(2–3):208–214; in French
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Minimal self-joinings Let k 2 be an integer. The ergodic measure-preserving dynamical system T has k-fold minimal self-joinings if, for any ergodic joining of k copies of T, we can partition the set f1; : : : ; kg of coordinates into subsets J1 ; : : : ; J` such that
Joinings in Ergodic Theory THIERRY DE LA RUE Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen, Saint Étienne du Rouvray, France
1. For j1 and j2 belonging to the same J i , the marginal of on the coordinates j1 and j2 is supported on the graph of T n for some integer n (depending on j1 and j2 ); 2. For j1 2 J1 ; : : : ; j` 2 J` , the coordinates j1 ; : : : ; j` are independent.
Article Outline Glossary Definition of the Subject Introduction Joinings of Two or More Dynamical Systems Self-Joinings Some Applications and Future Directions Bibliography Glossary Disjoint measure-preserving systems The two measurepreserving dynamical systems (X; A; ; T) and (Y; B; ; S) are said to be disjoint if their only joining is the product measure ˝ . Joining Let I be a finite or countable set, and for each i 2 I, let (X i ; A i ; i ; Ti ) be a measure-preserving dynamical system. A joining of these systems is a probability Q measure on the Cartesian product i2I X i , which has the i ’s as marginals, and which is invariant under the N product transformation i2I Ti . Marginal of a probability measure on a product space Let be a probability measure on the Cartesian product of aQ finite or countable collection of measurable N spaces X ; A i , and let J D f j1 ; : : : ; j k g i2I i i2I be a finite subset of I. The k-fold marginal of on X j 1 ; : : : ; X j k is the probability measure defined by: 8A1 2 A j 1 ; : : : ; A k 2 A j k ; 0 (A1 A k ) :D @ A1 A k
Y
1 Xi A :
i2InJ
Markov intertwining Let (X; A; ; T) and (Y; B; ; S) be two measure-preserving dynamical systems. We call Markov intertwining of T and S any operator P : L2 (X; ) ! L2 (Y; ) enjoying the following properties: PU T D U S P, where U T and U S are the unitary operators on L2 (X; ) and L2 (Y; ) associated respectively to T and S (i. e. U T f (x) D f (Tx), and U S g(y) D g(Sy)). P1 X D 1Y , f 0 implies P f 0, and g 0 implies P g 0, where P is the adjoint operator of P.
We say that T has minimal self-joinings if T has k-fold minimal self-joinings for every k 2. Off-diagonal self-joinings Let (X; A; ; T) be a measure-preserving dynamical system, and S be an invertible measure-preserving transformation of (X; A; ) commuting with T. Then the probability measure S defined on X X by S (A B) :D (A \ S 1 B)
(1)
is a 2-fold self-joining of T supported on the graph of S. We call it an off-diagonal self-joining of T. Process in a measure-preserving dynamical systems Let (X; A; ; T) be a measure-preserving dynamical system, and let (E; B(E)) be a measurable space (which may be a finite or countable set, or Rd , or C d . . . ). For any E-valued random variable defined on the probability space (X; A; ), we can consider the stochastic process ( i ) i2Z defined by i :D ı T i : Since T preserves the probability measure , ( i ) i2Z is a stationary process: For any ` and n, the distribution of (0 ; : : : ; ` ) is the same as the probability distribution of ( n ; : : : ; nC` ). Self-joining Let T be a measure-preserving dynamical system. A self-joining of T is a joining of a family (X i ; A i ; i ; Ti ) i2I of systems where each T i is a copy of T. If I is finite and has cardinal k, we speak of a k-fold self-joining of T. Simplicity For k 2, we say that the ergodic measurepreserving dynamical system T is k-fold simple if, for any ergodic joining of k copies of T, we can partition the set f1; : : : ; kg of coordinates into subsets J1 ; : : : ; J` such that 1. for j1 and j2 belonging to the same J i , the marginal of on the coordinates j1 and j2 is supported on the graph of some S 2 C(T) (depending on j1 and j2 );
Joinings in Ergodic Theory
2. for j1 2 J1 ; : : : ; j` 2 J` , the coordinates j1 ; : : : ; j` are independent. We say that T is simple if T is k-fold simple for every k 2. Definition of the Subject The word joining can be considered as the counterpart in ergodic theory of the notion of coupling in probability theory (see e. g. [48]): Given two or more processes defined on different spaces, what are the possibilities of embedding them together in the same space? There always exists the solution of making them independent of each other, but interesting cases arise when we can do this in other ways. The notion of joining originates in ergodic theory from pioneering works of H. Furstenberg [14], who introduced the fundamental notion of disjointness, and D.J. Rudolph, who laid the basis of joining theory in his article on minimal self-joinings [41]. It has today become an essential tool in the classification of measure-preserving dynamical systems and in the study of their intrinsic properties. Introduction A central question in ergodic theory is to tell when two measure-preserving dynamical systems are essentially the same, i. e. when they are isomorphic. When this is not the case, a finer analysis consists of asking what these two systems could share in common: For example, do there exist stationary processes which can be observed in both systems? This latter question can also be asked in the following equivalent way: Do these two systems have a common factor? The arithmetical flavor of this question is not fortuitous: There are deep analogies between the arithmetic of integers and the classification of measure-preserving dynamical systems, and these analogies were at the starting point of the study of joinings in ergodic theory. In the seminal paper [14] which introduced the concept of joinings in ergodic theory, H. Furstenberg observed that two operations can be done with dynamical systems: We can consider the product of two dynamical systems, and we can also take a factor of a given system. Like the multiplication of integers, the product of dynamical systems is commutative, associative, it possesses a neutral element (the trivial single-point system), and the systems S and T are both factors of their product S T. It was then natural to introduce the property for two measure-preserving systems to be relatively prime. As far as integers are concerned, there are two equivalent ways of characterizing the relative primeness: First, the integers a and b are relatively prime if their unique positive common factor is 1.
Second, a and b are relatively prime if, each time both a and b are factors of an integer c, their product ab is also a factor of c. It is a well-known theorem in number theory that these two properties are equivalent, but this was not clear for their analog in ergodic theory. Furstenberg reckoned that the second way of defining relative primeness was the most interesting property in ergodic theory, and called it disjointness of measure-preserving systems (we will discuss precisely in Subsect. “From Disjointness to Isomorphy” what the correct analog is in the setting of ergodic theory). He also asked whether the non-existence of a non-trivial common factor between two systems was equivalent to their disjointness. He was able to prove that disjointness implies the impossibility of a non-trivial common factor, but not the converse. And in fact, the converse turns out to be false: In 1979, D.J. Rudolph exhibited a counterexample in his paper introducing the important notion of minimal self-joinings. The relationships between disjointness and the lack of common factor will be presented in details in Sect. “Joinings and Factors”. Given two measure-preserving dynamical systems S and T, the study of their disjointness naturally leads one to consider all the possible ways these two systems can be both seen as factors of a third system. As we shall see, this is precisely the study of their joinings. The concept of joining turns out to be related to many important questions in ergodic theory, and a large number of deep results can be stated and proved inside the theory of joinings. For example, the fact that the dynamical systems S and T are isomorphic is equivalent to the existence of a special joining between S and T, and this can be used to give a joining proof of Krieger’s finite generator theorem, as well as Ornstein’s isomorphism theorem (see Sect. “Joinings Proofs of Ornstein’s and Krieger’s Theorems”). As it already appears in Furstenberg’s article, joinings provide a powerful tool in the classification of measure-preserving dynamical systems: Many classes of systems can be characterized in terms of their disjointness with other systems. Joinings are also strongly connected with difficult questions arising in the study of the convergence almost everywhere of nonconventional averages (see Sect. “Joinings and Pointwise Convergence”). Amazingly, a situation in which the study of joinings leads to the most interesting results consists of considering two or more identical systems. We then speak of the self-joinings of the dynamical system T. Again, the study of self-joinings is closely related to many ergodic properties of the system: Its mixing properties, the structure of its factors, the transformations which commute with T, and so on. . . We already mentioned minimal self-joinings, and we will see in Sect. “Minimal Self-Joinings” how this
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property may be used to get many interesting examples, such as a transformation with no root, or a process with no non-trivial factor. In the same section we will also discuss a very interesting generalization of minimal self-joinings: the property of being simple. The range of applications of joinings in ergodic theory is very large; only some of them will be given in Sect. “Some Applications and Future Directions”: The use of joinings in proving Krieger’s and Ornstein’s theorems, the links between joinings and some questions of pointwise convergence, and the strong connections between the study of self-joinings and Rohlin’s famous question on multifold mixing, which was first posed in 1949 [39]. Joinings of Two or More Dynamical Systems In the following, we are given a finite or countable family (X i ; A i ; i ; Ti ) i2I of measure-preserving dynamical systems: T i is an invertible measure-preserving transformation of the standard Borel probability space (X i ; A i ; i ). When it is not ambiguous, we shall often use the symbol T i to denote both the transformation and the system. A joining of the T i ’s (see the definition in the Glossary) defines a new measure-preserving dynamical system: The product transformation O
Ti : (x i ) i2I 7! (Ti x i ) i2I
i2I
Q acting on the Cartesian product i2I X i , and preserving the N probability measure . We will denote this big system by T . Since all marginals of are given by the i i2I original probabilities i , observing only the coordinate i in the big system is the same as observing N only the system T i . Thus, each system T i is a factor of i2I Ti , via the homomorphism i which maps any point in the Cartesian product to its ith coordinate. Conversely, if we are given a measure-preserving dynamical system (Z; C ; ; R) admitting each T i as a factor via some homomorphism ' i : Z ! X i , then we can conQ struct the map ' : Z ! i2I X i sending z to (' i (z)) i2I . We can easily check that the image of the probability measure is then a joining of the T i ’s. Therefore, studying the joinings of a family of measure-preserving dynamical system amounts to studying all the possible ways these systems can be seen together as factors in another big system. The Set of Joinings The set of all joinings of the T i ’s will be denoted by J(Ti ; i 2 I). Before anything else, we have to observe
that this set is never empty. Indeed, whatever the systems N are, the product measure i2I i always belongs to this set. Note also that any convex combination of joinings is a joining: J(Ti ; i 2 I) is a convex set. The set of joinings is turned into a compact metrizable space, equipped with the topology defined by the following notion of convergence: n ! if and only if, for all n!1 Q families of measurable subsets (A i ) i2I 2 i2I A i , finitely many of them being different from X i , we have ! ! Y Y n A i ! Ai : (2) i2I
n!1
i2I
We can easily construct a distance defining this topology by observing that it is enough to check (2) when each of the Ai ’s is chosen in some countable algebra C i generating the -algebra A i . We can also point out that, when the X i ’s are themselves compact metric spaces, this topology on the set of joinings is nothing but the restriction to J(Ti ; i 2 I) of the usual weak* topology. It is particularly interesting to study ergodic joinings of the T i ’s, whose set will be denoted by Je (Ti ; i 2 I). Since any factor of an ergodic system is itself ergodic, a necessary condition for Je (Ti ; i 2 I) not to be empty is that all the T i ’s be themselves ergodic. Conversely, if all the T i ’s are ergodic, we can prove by considering the ergodic deN composition of the product measure i2I i that ergodic joinings do exist: Any ergodic measure appearing in the ergodic decomposition of some joining has to be itself a joining. This result can also be stated in the following way: Proposition 1 If all the T i ’s are ergodic, the set of their ergodic joinings is the set of extremal points in the compact convex set J(Ti ; i 2 I). From Disjointness to Isomorphy In this section, as in many others in this article, we are focusing on the case where our family of dynamical systems is reduced to two systems. We will then rather call them S and T, standing for (Y; B; ; S) and (X; A; ; T). We are interested here in two extremal cases for the set of joinings J(T; S). The first one occurs when the two systems are as far as possible from each other: They have nothing to share in common, and therefore their set of joinings is reduced to the singleton f ˝ g: This is called the disjointness of S and T. The second one arises when the two systems are isomorphic, and we will see how this property shows through J(T; S). Disjointness Many situations where disjointness arises were already given by Furstenberg in [14]. Particularly in-
Joinings in Ergodic Theory
teresting is the fact that classes of dynamical systems can be characterized through disjointness properties. We list here some of the main examples of disjoint classes of measure-preserving systems. Theorem 2 1. T is ergodic if and only if it is disjoint from every identity map. 2. T is weakly mixing if and only if it is disjoint from any rotation on the circle. 3. T has zero entropy if and only if it is disjoint from any Bernoulli shift. 4. T is a K-system if and only if it is disjoint from any zeroentropy system. The first result is the easiest, but is quite important, in particular when it is stated in the following form: If is a joining of T and S, with T ergodic, and if is invariant by T Id, then D ˝ . The second, third and fourth results were originally proved by Furstenberg. They can also be seen as corollaries of the theorems presented in Sect. “Joinings and Factors”, linking the non-disjointness property with the existence of a particular factor. Both the first and the second results can be derived from the next theorem, giving a general spectral condition in which disjointness arises. The proof of this theorem can be found in [49]. It is a direct consequence of the fact that, if f and g are square-integrable functions in a given dynamical system, and if their spectral measures are mutually singular, then f and g are orthogonal in L2 . Theorem 3 If the reduced maximum spectral types of T and S are mutually singular, then T and S are disjoint. As we already said in the introduction, disjointness was recognized by Furstenberg as the most pertinent way to define the analog of the arithmetic property “a and b are relatively prime” in the context of measure-preserving dynamical systems. We must however point out that the statement (i) S and T are disjoint is, in general, strictly stronger than the straightforward translation of the arithmetic property: (ii) Each time both S and T appear as factors in a third dynamical system, then their product S T also appears as a factor in this system. Indeed, contrary to the situation in ordinary arithmetic, there exist non-trivial dynamical systems T which are isomorphic to T T: For example, this is the case when T is the product of countably many copies of a single non-triv-
ial system. Now, if T is such a system and if we take S D T, then S and T do not satisfy statement (i): A non-trivial system is never disjoint from itself, as we will see in the next Section. However they obviously satisfy the statement (ii). A correct translation of the arithmetic property is the following: S and T are disjoint if and only if, each time T and S appear as factors in some dynamical system through the respective homomorphisms T and S , T S also appears as a factor through a homomorphism TS such that X ı TS D T and Y ı TS D S , where X and Y are the projections on the coordinates in the Cartesian product X Y (see the diagram below).
Joinings and Isomorphism We first introduce some notations: For any probability measure on a measurable
space, let ‘A D B’ stand for ‘(AB) D 0’. Similarly, if C
and D are -algebras of measurable sets, we write ‘C D’ if, for any C 2 C , we can find some D 2 D such that
C D D, and by ‘C D D’ we naturally mean that both
C D and D C hold.
Let us assume now that our two systems S and T are isomorphic: This means that we can find some measurable one-to-one map ' : X ! Y, with T() D , and ' ı T D S ı '. With such a ', we construct the measurable map : X ! X Y by setting (x) :D (x; '(x) : Let ' be the image measure of by . This measure is supported on the graph of ', and is also characterized by 8A 2 A ; 8B 2 B ; ' (A B) D (A \ ' 1 B) : (3) We can easily check that, ' being an isomorphism of T and S, ' is a joining of T and S. And this joining satisfies very special properties: '
For any measurable A X, A Y D X '(A); '
Conversely, for any measurable B Y, X B D ' 1 (B) Y.
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satisfies the weaker one:
fX; ;g ˝ B A ˝ fY; ;g :
(5)
The existence of a joining satisfying (5) is a criterion for S being a factor of T. For more details on the results stated in this section, we refer the reader to [6]. Joinings and Factors
Joinings in Ergodic Theory, Figure 1 The joining ' identifies the sets A Y and X '(A)
Thus, in the case where S and T are isomorphic, we can find a special joining of S and T, which is supported on the graph of an isomorphism, and which identifies the two -algebras generated by the two coordinates. What is remarkable is that the converse is true: The existence of an isomorphism between S and T is characterized by the existence of such a joining, and we have the following theorem: Theorem 4 The measure-preserving dynamical systems S and T are isomorphic if and only if there exists a joining of S and T such that
fX; ;g ˝ B D A ˝ fY; ;g :
(4)
When is such a joining, it is supported on the graph of an isomorphism of T and S, and both systems are isomorphic to the joint system (T ˝ S) . This theorem finds nice applications in the proof of classical isomorphism results. For example, it can be used to prove that two discrete-spectrum systems which are spectrally isomorphic are isomorphic (see [49] or [6]). We will also see in Sect.“ Joinings Proofs of Ornstein’s and Krieger’s Theorems” how it can be applied in the proofs of Krieger’s and Ornstein’s deep theorems. Consider now the case were T and S are no longer isomorphic, but where S is only a factor of T. Then we have a factor map : X ! Y which has the same properties as an isomorphism ', except that it is not one-to-one ( is only onto). The measure , constructed in the same way as ' , is still a joining supported on the graph of , but it does not identify the two -algebras generated by the two coordinates anymore: Instead of condition (4), only
The purpose of this section is to investigate the relationships between the disjointness of two systems S and T, and the lack of a common factor. The crucial fact which was pointed out by Furstenberg is that the existence of a common factor enables one to construct a very special joining of S and T: The relatively independent joining over this factor. Let us assume that our systems S and T share a common factor (Z; C ; ; R), which means that we have measurable onto maps X : X ! Z and Y : Y ! Z, respectively sending and to , and satisfying X ı T D R ı X and Y ı S D R ı Y . We can then consider the joinings supported on their graphs X 2 J(T; R) and Y 2 J(S; R), as defined in the preceding section. Next, we construct a joining of the three systems S, T and R. Heuristically, is the probability distribution of the triple (x; y; z) when we first pick z according to the probability distribution , then x and y according to their conditional distribution knowing z in the respective joinings X and Y , but independently of each other. More precisely, is defined by setting, for all A 2 A, B 2 B and C2C (A B C) :D Z E X [1x2A jz]EY [1 y2B jz] d (z) :
(6)
C
Observe that the two-fold marginals of on X Z and Y Z are respectively X and Y , which means that we have z D X (x) D Y (y) -almost surely. In other words, we have identified in the two systems T and S the projections on their common factor R. The two-fold marginal of on X Y is itself a joining of T and S, which we call the relatively independent joining over the common factor R. This joining will be denoted by ˝ R . (Be careful: The projections X and Y are hidden in this notation, but we have to know them to define this joining.) From (6), we immediately get the formula defining ˝R : 8A 2 A ; 8B 2 B ; ˝ R (A B) :D Z E X [1x2A jz]EY [1 y2B jz] d (z) : Z
(7)
Joinings in Ergodic Theory
rived from the proof of Theorem 6: For any joining of S and T, for any bounded measurable function f on X, the factor -algebra of S generated by the function E [ f (x)jy] is a T-factor -algebra of S. With the notion of T-factor, Theorem 6 has been extended in [31] in the following way, showing the existence of a special T-factor -algebra of S concentrating anything in S which could lead to a non-trivial joining between T and S. Theorem 7 Given two measure-preserving dynamical systems (X; A; ; T) and (Y; B; ; S), there always exists a maximum T-factor -algebra of S, denoted by FT . Under any joining of T and S, the -algebras A ˝ f;; Yg and f;; Xg ˝ B are independent conditionally to the -algebra f;; Xg ˝ FT .
Joinings in Ergodic Theory, Figure 2 The relatively independent joining ˝R and its disintegration over Z
This definition of the relatively independent joining over a common factor can easily be extended to a finite or countable family of systems sharing the same common factor. Note that ˝ R coincides with the product measure ˝ if and only if the common factor is the trivial onepoint system. We therefore get the following result: Theorem 5 If S and T have a non-trivial common factor, then these systems are not disjoint. As we already said in the introduction, Rudolph exhibited in [41] a counterexample showing that the converse is not true. There exists, however, an important result, which was published in [20,29] allowing us to derive some information on factors from the non-disjointness of two systems. Theorem 6 If T and S are not disjoint, then S has a nontrivial common factor with some joining of a countable family of copies of T. This result leads to the introduction of a special class of factors when some dynamical system T is given: For any other dynamical system S, call T-factor of S any common factor of S with a joining of countably many copies of T. If (Z; C ; ; R) is a T-factor of S and : Y ! Z is a factor map, we say that the -algebra 1 (C ) is a T-factor -algebra of S. Another way to state Theorem 6 is then the following: If S and T are not disjoint, then S has a non-trivial T-factor. In fact, an even more precise result can be de-
Theorem 6 gives a powerful tool to prove some important disjointness results, such as those stated in Theorem 2. These results involve properties of dynamical systems which are stable under the operations of taking joinings and factors. We will call these properties stable properties. This is, for example, the case of the zero-entropy property: We know that any factor of a zero-entropy system still has zero entropy, and that any joining of zero-entropy systems also has zero entropy. In other words, T has zero entropy implies that any T-factor has zero entropy. But the property of S being a K-system is precisely characterized by the fact that any non-trivial factor of S has positive entropy. Hence a K-system S cannot have a non-trivial T-factor if T has zero entropy, and is therefore disjoint from T. The converse is a consequence of Theorem 5: If S is not a K-system, then it possesses a non-trivial zero-entropy factor, and therefore there exists some zero-entropy system from which it is not disjoint. The same argument also applies to the disjointness of discrete-spectrum systems with weakly mixing systems, since a discrete spectrum is a stable property, and weakly mixing systems are characterized by the fact that they do not have any discrete-spectrum factor. Markov Intertwinings and Composition of Joinings There is another way of defining joinings of two measurepreserving dynamical systems involving operators on L2 spaces, mainly put to light by Ryzhikov (see [47]): Observe that for any joining 2 J(T; S), we can consider the operator P : L2 (X; ) ! L2 (Y; ) defined by P ( f ) :D E f (x)jy : It is easily checked that P is a Markov intertwining of T and S. Conversely, given any Markov intertwining P of T
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and S, it can be shown that the measure P defined on X Y by P (A B) :D hP1A ; 1B iL 2 (Y;) is a joining of T and S. This reformulation of the notion of joining is useful when joinings are studied in connection with spectral properties of the transformations (see e. g. [22]). It also provides us with a convenient setting to introduce the composition of joinings: If we are given three dynamical systems (X; A; ; T), (Y; B; ; S) and (Z; C ; ; R), a joining 2 J(T; S) and a joining 0 2 J(S; R), the composition of the Markov intertwinings P and P0 is easily seen to give a third Markov intertwining, which itself corresponds to a joining of T and R denoted by ı 0 . When R D S D T, i. e. when we are speaking of 2-fold self-joinings of a single system T (cf. next section), this operation turns J(T; T) D J 2 (T) into a semigroup. Ahn and Lema´nczyk [1] have shown that the subset Je2 (T) of ergodic two-fold self-joinings is a sub-semigroup if and only if T is semisimple (see Sect. “Simple Systems”). Self-Joinings We now turn to the case where the measure-preserving dynamical systems we want to join together are all copies of a single system T. For k 2, any joining of k copies of T is called a k-fold self-joining of T. We denote by J k (T) the set of all k-fold self-joinings of T, and by Jek (T) the subset of ergodic k-fold self-joinings. Self-Joinings and Commuting Transformations As soon as T is not the trivial single-point system, T is never disjoint from itself: Since T is obviously isomorphic to itself, we can always find a two-fold self-joining of T which is not the product measure by considering self-joinings supported on graphs of isomorphisms (see Sect. “Joinings and Isomorphism”). The simplest of them is obtained by taking the identity map as an isomorphism, and we get that J 2 (T) always contains the diagonal measure 0 :D Id . In general, an isomorphism of T with itself is an invertible measure-preserving transformation S of (X; A; ) which commutes with T. We call commutant of T the set of all such transformations (it is a subgroup of the group of automorphisms of (X; A; )), and denote it by C(T). It always contains, at least, all the powers T n , n 2 Z. Each element S of C(T) gives rise to a two-fold selfjoining S supported on the graph of S. Such self-joinings are called off-diagonal self-joinings. They also belong to Jek (T) if T is ergodic.
It follows that properties of the commutant of an ergodic T can be seen in its ergodic joinings. As an example of application, we can cite Ryzhikov’s proof of King’s weak closure theorem for rank-one transformations1 . Rank-one measure-preserving transformations form a very important class of zero-entropy, ergodic measure-preserving transformations. They have many remarkable properties, among which the fact that their commutant is reduced to the weak limits of powers of T. In other words, if T is rank one, for any S 2 C(T) there exists a subsequence of integers (n k ) such that, 8A 2 A ; T n k AS 1 A ! 0 : k!1
(8)
King proved this result in 1986 [26], using a very intricate coding argument. Observing that (8) was equivalent to the convergence, in J 2 (T), of T n k to S , Ryzhikov showed in [45] that King’s theorem could be seen as a consequence of the following general result concerning two-fold selfjoinings of rank-one systems: Theorem 8 Let T be a rank-one measure-preserving transformation, and 2 Je2 (T). Then there exist t 1/2, a subsequence of integers (n k ) and another two-fold self-joining 0 of T such that T n k ! t C (1 t)0 : k!1
Minimal Self-Joinings For any measure-preserving dynamical system T, the set of two-fold self-joinings of T contains at least the product measure ˝, the off-diagonal joinings T n for each n 2 Z, and any convex combination of these. Rudolph [41] discovered in 1979 that we can find systems for which there are no other two-fold self-joinings than these obvious ones. When this is the case, we say that T has 2-foldminimal self-joinings, or for short: T 2 MSJ(2). It can be shown (see e. g. [42]) that, as soon as the underlying probability space is not atomic (which we henceforth assume), two-fold minimal self-joinings implies that T is weakly mixing, and therefore that ˝ and T n , n 2 Z, are ergodic two-fold self-joinings of T. That is why two-fold minimal self-joinings are often defined by the following: T 2 MSJ(2) () Je2 (T) D f ˝ g [ fT n ; n 2 Zg :
(9)
1 An introduction to finite-rank transformations can be found e. g. in [32]; we also refer the reader to the quite complete survey [12].
Joinings in Ergodic Theory
Systems with two-fold minimal self-joinings have very interesting properties. First, since for any S in C(T), S belongs to Je2 (T), we immediately see that the commutant of T is reduced to the powers of T. In particular, it is impossible to find a square root of T, i. e. a measure-preserving S such that S ı S D T. Second, the existence of a non-trivial factor -algebra of T would lead, via the relatively independent self-joining over this factor, to some ergodic two-fold self-joining of T which is not in the list prescribed by (9). Therefore, any factor -algebra of a system with two-fold minimal self-joinings must be either the trivial -algebra f;; Xg or the whole -algebra A. This has the remarkable consequence that if is any random variable on the underlying probability space which is not almost-surely constant, then the process ( ı T n )n2Z always generates the whole -algebra A. This also implies that T has zero entropy, since positive-entropy systems have many non-trivial factors. The notion of two-fold minimal self-joinings extends for any integer k 2 to k-fold minimal self-joinings, which roughly means that there are no other k-fold ergodic selfjoinings than the “obvious” ones: Those for which the k coordinates are either independent or just translated by some power of T (see the Glossary for a more precise definition). We denote in this case: T 2 MSJ(k). If T has k-fold minimal self-joinings for all k 2, we simply say that T has minimal self joinings. Rudolph’s construction of a system with two-fold minimal self-joinings [41] was inspired by a famous work of Ornstein [35], giving the first example of a transformation with no roots. It turned out that Ornstein’s example is a mixing rank-one system, and all mixing rank-one systems were later proved by J. King [27] to have 2-fold minimal self-joinings. This can also be viewed as a consequence of Ryzhikov’s Theorem 8. Indeed, in the language of joinings, the mixing property of T translates as follows: T is mixing () T n ! ˝ : jnj!1
(10)
Therefore, if in Theorem 8 we further assume that T is mixing, then either the sequence (n k ) we get in the conclusion is bounded, and then is some T n , or it is unbounded and then D ˝ . T 2 MSJ(k) obviously implies T 2 MSJ(k 0 ) for any 2 k 0 k, but the converse is not known. The question whether two-fold minimal self-joinings implies k-fold minimal self-joinings for all k is related to the important open problem of pairwise independent joinings (see Sect. “Pairwise-Independent Joinings”). But the latter problem is solved for some special classes of systems,
in particular in the category of mixing rank-one transformations. It follows that, if T is mixing and rank one, then T has minimal self-joinings. In 1980, Del Junco, Rahe and Swanson proved that Chacon’s transformation also has minimal self-joinings [9]. This well-known transformation is also a rankone system, but it is not mixing (it had been introduced by R.V. Chacon in 1969 [4] as the first explicit example of a weakly mixing transformation which is not mixing). For another example of a transformation with two-fold minimal self-joinings, constructed as an exchange map on three intervals, we refer to [7]. The existence of a transformation with minimal selfjoinings has been used by Rudolph as a wonderful tool to construct a large variety of striking counterexamples, such as A transformation T which has no roots, while T 2 has roots of any order, A transformation with a cubic root but no square root, Two measure-preserving dynamical systems which are weakly isomorphic (each one is a factor of the other) but not isomorphic. . . Let us now sketch the argument showing that we can find two systems with no common factor but which are not disjoint: We start with a system T with minimal self-joinings. Consider the direct product of T with an independent copy T 0 of itself, and take the symmetric factor S of T ˝ T 0 , that is to say the factor we get if we only look at the non-ordered pair of coordinates fx; x 0 g in the Cartesian product. Then S is surely not disjoint from T, since the pair fx; x 0 g is not independent of x. However, if S and T had a non-trivial common factor, then this factor should be isomorphic to T itself (because T has minimal self-joinings). Therefore we could find in the direct product T ˝ T 0 a third copy T˜ of T, which is measurable with respect to the symmetric factor. In particular, T˜ is invariant by the flip map (x; x 0 ) 7! (x 0 ; x), and this prevents T˜ from being measurable with respect to only one coordinate. Then, since T 2 MSJ(3), the systems T, T 0 and T˜ have no choice but being independent. But this contradicts the fact that T˜ is measurable with respect to the -algebra generated by x and x 0 . Hence, T and S have no non-trivial common factor. We can also cite the example given by Glasner and Weiss [21] of a pair of horocycle transformations which have no nontrivial common factor, yet are not disjoint. Their construction relies on the deep work by Ratner [38], which describes the structure of joinings of horocycle flows.
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Simple Systems An important generalization of two-fold minimal selfjoinings has been proposed by William A. Veech in 1982 [51]. We say that the measure-preserving dynamical system T is two-fold simple if it has no other ergodic two-fold self-joinings than the product measure f ˝ g and joinings supported on the graph of a transformation S 2 C(T). (The difference with MSJ(2) lies in the fact that C(T) may contain other transformations than the powers of T.) It turns out that simple systems may have non-trivial factors, but the structure of these factors can be explicitly described: They are always associated with some compact subgroup of C(T). More precisely, if K is a compact subgroup of C(T), we can consider the factor -algebra F K :D fA 2 A : 8S 2 K ; A D S(A)g ;
and the corresponding factor transformation TjFK (called a group factor). Then Veech proved the following theorem concerning the structure of factors of a two-fold simple system. Theorem 9 If the dynamical system T is two-fold simple, and if F A is a non-trivial factor -algebra of T, then there exists a compact subgroup K of the group C(T) such that F D F K . There is a natural generalization of Veech’s property to the case of k-fold self-joinings, which has been introduced by A. Del Junco and D.J. Rudolph in 1987 [10] (see the precise definition of simple systems in the Glossary). In their work, important results concerning the structure of factors and joinings of simple systems are proved. In particular, they are able to completely describe the structure of the ergodic joinings between a given simple system and any ergodic system (see also [18,49]). Recall that, for any r 1, the symmetric factor of T ˝r is the system we get if we observe the r coordinates of the point in X r and forget their order. This is a special case of group factor, associated with the subgroup of C(T ˝r ) consisting of all permutations of the coordinates. We denote this symmetric factor by T hri . Theorem 10 Let T be a simple system and S an ergodic system. Assume that is an ergodic joining of T and S which is different from the product measure. Then there exists a compact subgroup K of C(T) and an integer r 1 such that (TjFK )hri is a factor of S, is the projection on X Y of the relatively independent joining of T ˝r and S over their common factor (TjFK )hri .
If we further assume that the second system is also simple, then in the conclusion we can take r D 1. In other words, ergodic joinings of simple systems S and T are either the product measure or relatively independent joinings over a common group factor. This leads to the following corollary: Theorem 11 Simple systems without non-trivial common factor are disjoint. As for minimal self-joining, it is not known in general whether two-fold simplicity implies k-fold simplicity for all k. This question is studied in [19], where sufficient spectral conditions are given for this implication to hold. It is also proved that any three-fold simple weakly mixing transformation is simple of all order. Relative Properties with Respect to a Factor In fact, Veech also introduced a weaker, “relativized”, version of the two-fold simplicity. If F A is a non-trivial factor -algebra of T, let us denote by J 2 (T; F ) the twofold self-joinings of T which are “constructed over F ”, which means that their restriction to the product -algebra F ˝ F coincides with the diagonal measure. (The relatively-independent joining over F is the canonical example of such a joining.) For the conclusion of Theorem 9 to hold, it is enough to assume only that the ergodic elements of J 2 (T; F ) be supported on the graph of a transformation S 2 C(T). This is an important situation where the study of J 2 (T; F ) gives strong information on the way F is embedded in the whole system T, or, in other words, on the relative properties of T with respect to the factor TjF . A simple example of such a relative property is the relative weak mixing with respect to F , which is characterized by the ergodicity of the relatively-independent joining over F (recall that weak-mixing is itself characterized by the ergodicity of the direct product T ˝ T). For more details on this subject, we refer the reader to [30]. We also wish to mention the generalization of simplicity called semisimplicity proposed by Del Junco, Lema´nczyk and Mentzen in [8], which is precisely characterized by the fact that, for any 2 Je2 (T), the system (T ˝ T) is a relatively weakly mixing extension of T. Some Applications and Future Directions Joinings Proofs of Ornstein’s and Krieger’s Theorems We have already seen that joinings could be used to prove isomorphisms between systems. This fact found a nice application in the proofs of two major theorems in ergodic theory: Ornstein’s isomorphism theorem [34], stating that
Joinings in Ergodic Theory
two Bernoulli shifts with the same entropy are isomorphic, and Krierger’s finite generator theorem [28], which says that any dynamical system with finite entropy is isomorphic to the shift transformation on a finite-valued stationary process. The idea of this joining approach to the proofs of Krieger’s and Ornstein’s theorems was originally due to Burton and Rothstein, who circulated a preliminary report on the subject which was never published [3]. The first published and fully detailed exposition of these proofs can be found in Rudolph’s book [42] (see also in Glasner’s book [18]). In fact, Ornstein’s theorem goes far more beyond the isomorphism of two given Bernoulli shifts: It also gives a powerful tool for showing that a specific dynamical system is isomorphic to a Bernoulli shift. In particular, Ornstein introduced the property for an ergodic stationary process to be finitely determined. We shall not give here the precise definition of this property (for a complete exposition of Ornstein’s theory, we refer the reader to [36]), but simply point out that Bernoulli shifts and mixing Markov chains are examples of finitely determined processes. Rudolph’s argument to show Ornstein’s theorem via joinings makes use of Theorem 4, and of the topology of J(T; S). Theorem 12 (Ornstein’s Isomorphism Theorem) Let T and S be two ergodic dynamical systems with the same entropy, and both generated by finitely determined stationary processes. Then the set of joinings of T and S which are supported on graphs of isomorphisms forms a dense Gı in Je (T; S). Krieger’s theorem is not as easily stated in terms of joinings, because it does not refer to the isomorphism of two specific systems, but rather to the isomorphism of one given system with some other system which has to be found. We have therefore to introduce a larger set of joinings: Given an integer n, we denote by Y n the set of doublesided sequences taking values in f1; : : : ; ng. We consider on Y n the shift transformation S, but we do not determine yet the invariant measure. Now, for a specific measurepreserving dynamical system T, consider the set J(n; T) of all possible joinings of T with some system (Yn ; ; S), when ranges over all possible shift-invariant probability measures on Y n . J(n; T) can also be equipped with a topology which turns it into a compact convex metric space, and as soon as T is ergodic, the set Je (n; T) of ergodic elements of J(n; T) is not empty. In this setting, Krieger’s theorem can be stated as follows: Theorem 13 (Krieger’s Finite Generator Theorem) Let T be an ergodic dynamical system with entropy h(T) < log2 n. Then the set of 2 J(n; T) which are supported
on graphs of isomorphisms between T and some system (Yn ; ; S) forms a dense Gı in Je (n; T). Since any system of the form (Yn ; ; S) obviously has an n-valued generating process, we obtain as a corollary that T itself is generated by an n-valued process. Joinings and Pointwise Convergence The study of joinings is also involved in questions concerning pointwise convergence of (non-conventional) ergodic averages. As an example, we present here the relationships between disjointness and the following wellknown open problem: Given two commuting measurepreserving transformations S and T acting on the same probability space (X; A; ), is it true that for any f and g in L2 (), the sequence ! n1 X 1 k k f (T x)g(S x) (11) n kD0
n>0
converges -almost surely? It turns out that disjointness of T and S is a sufficient condition for this almost-sure convergence to hold. Indeed, let us first consider the case where T and S are defined on a priori different spaces (X; A; ) and (Y; B; ) respectively, and consider the ergodic average in the product 1 n
n1 X
f (T k x)g(S k y) ;
(12)
kD0
which can be viewed as the integral of the function f ˝ g with respect to the empirical distribution ın (x; y) :D
1 n
n1 X
ı(T k x;S k y) :
kD0
We can always assume that T and S are continuous transformations of compact metric spaces (indeed, any measure-preserving dynamical system is isomorphic to such a transformation on a compact metric space; see e. g. [15]). Then the set of probability measures on X Y equipped with the topology of weak convergence is metric compact. Now, here is the crucial point where joinings appear: If T and S are ergodic, we can easily find subsets X 0 X and Y0 Y with (X0 ) D (Y0 ) D 1, such that for all (x; y) 2 X 0 Y0 , any cluster point of the sequence (ın (x; y))n>0 is automatically a joining of T and S. (We just have to pick x and y among the “good points” for the ergodic theorem in their respective spaces.) When T and S are disjoint, there is therefore only one possible cluster
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point to the sequence ın (x; y) which is ˝. This ensures that, for continuous f and g, (12) converges to the product of the integrals of f and g as soon as (x; y) is picked in X 0 Y0 . The subspace of continuous functions being dense in L2 , the classical ergodic maximal inequality (see [17]) ensures that, for any f and g in L2 (), (12) converges for any (x; y) in a rectangle of full measure X 0 Y0 . Coming back to the original question where the spaces on which T and S act are identified, we observe that with probability one, x belongs both to X 0 and Y 0 , and therefore the sequence (11) converges. The existence of a rectangle of full measure in which the sequence of empirical distributions (ın (x; y))n>0 always converges to some joining has been studied in [31] as a natural generalization of the notion of disjointness. This property was called weak disjointness of S and T, and it is indeed strictly weaker than disjointness, since there are examples of transformations which are weakly disjoint from themselves. There are other situations in which joinings can be used in the study of everywhere convergence, among which we can cite Rudolph’s joinings proof of Bourgain’s return time theorem [43]. Joinings and Rohlin’s Multifold Mixing Question We have already seen that the property of T being mixing could be expressed in terms of two-fold self-joinings of T (see (10)). Rohlin proposed in 1949 [39] a generalization of this property, called multifold mixing: The measure-preserving transformation T is said to be k-fold mixing if 8A1 ; A2 ; : : : ; A k 2 A, lim
n 2 ;n 3 ;:::;n k !1
(A1 \T n 2 A2 \ \T
(n 2 CCn k )
Ak ) D
k Y
(A i ) :
iD1
Again, this definition can easily be translated into the language of joinings: T is k-fold mixing when the sequence (T n2 ;:::;T n2 CCn k ) converges in J k (T) to ˝k as n2 ; : : : ; n k go to infinity, where (T n2 ;:::;T n2 CCn k ) is the obvious generalization of T n to the case of k-fold selfjoinings. The classical notion of mixing corresponds in this setting to two-fold mixing. (We must point out that Rohlin’s original definition of k-fold mixing involved k C1 sets, thus the classical mixing property was called 1-fold mixing. However it seems that the convention we adopt here is now used by most authors, and we find it more coherent when translated in the language of multifold selfjoinings.)
Obviously, 3-fold mixing is stronger than 2-fold mixing, and Rohlin asked in his article whether the converse is true. This question is still open today, even though many important works have dealt with it and supplied partial answers. Most of these works directly involve self-joinings via the argument exposed in the following section. Pairwise-Independent Joinings Let T be a two-fold mixing dynamical system. If T is not three-fold mixing, (T n ;T nCm ) does not converge to the product measure as n and m go to 1. By compactness of J 3 (T), we can find subsequences (n k ) and (m k ) such that (T n k ;T n k Cm k ) converges to a cluster point ¤ ˝ ˝ . However, by two-fold mixing, the three coordinates must be pairwise independent under . We therefore get a three-fold selfjoining with the unusual property that has pairwise independent coordinates, but is not the product measure. In fact, systems with this kind of pairwise-independent but non-independent three-fold self-joining are easy to find (see e. g. [6]), but the examples we know so far are either periodic transformations (which cannot be counterexamples to Rohlin’s question since they are not mixing!), or transformations with positive entropy. But using an argument provided by Thouvenot, we can prove that, if there exists a two-fold mixing T which is not threefold mixing, then we can find such a T in the category of zero-entropy dynamical systems (see e. g. [5]). Therefore, a negative answer to the following question would solve Rohlin’s multifold mixing problem: Question 14 Does there exist a zero-entropy, weakly mixing dynamical system T with a self-joining 2 J 3 (T) for which the coordinates are pairwise independent but which is different from ˝ ˝ ? (non) existence of such pairwise-independent joinings is also related to the question of whether MSJ(2) implies MSJ(3), or whether two-fold simplicity implies three-fold simplicity. Indeed, any counter-example to one of these implication would necessarily be of zero entropy, and would possess a pairwise-independent three-fold self-joining which is not the product measure. Question 14 has been answered by B. Host and V. Ryzhikov for some special classes of zero-entropy dynamical systems. Host’s and Ryzhikov’s Theorems The following theorem, proved in 1991 by B. Host [23] (see also [18,33]), establishes a spectacular connection between the spectral properties of a finite family of dynamical systems and the non-existence of a pairwise-independent, non-independent joining:
Joinings in Ergodic Theory
Theorem 15 (Host’s Theorem on Singular Spectrum) Let (X i ; A i ; i ; Ti )1ir be a finite family of measure-preserving dynamical systems with purely singular spectrum. Then any pairwise-independent joining 2 J(T1 ; : : : ; Tr ) is the product measure 1 ˝ ˝ r . Corollary 16 If a dynamical system with singular spectrum is two-fold mixing, then it is k-fold mixing for any k 2. The multifold-mixing problem for rank-one measure-preserving systems was solved in 1984 by S. Kalikow [25], using arguments which do not involve the theory of joinings. In 1993, V. Ryzhikov [46] extended Kalikow’s result to finite-rank systems, by giving a negative answer to Question 14 in the category of finite-rank mixing systems: Theorem 17 (Ryzhikov’s Theorem for Finite Rank Systems) Let T be a finite-rank mixing transformation, and k 2. Then the only pairwise-independent k-fold self-joining of T is the product measure. Corollary 18 If a finite-rank transformation is two-fold mixing, then it is k-fold mixing for any k 2. Future Directions A lot of important open questions in ergodic theory involve joinings, and we already have cited several of them: Joinings are a natural tool when we want to deal with some problems of pointwise convergence involving several transformations (see Sect. “Joinings and Pointwise Convergence”). Their use is also fundamental in the study of Rohlin’s question on multifold mixing. As far as this latter problem is concerned, we may mention a recent approach to Question 14: Start with a transformation for which some special pairwise-independent self-joining exists, and see what this assumption entails. In particular, we can ask under which conditions there exists a pairwise-independent three-fold self-joining of T under which the third coordinate is a function of the two others. It has already been proven in [24] that if this function is sufficiently regular (continuous for some topology), then T is periodic or has positive entropy. And there is strong evidence leading to the conjecture that, when T is weakly mixing, such a situation can only arise when T is a Bernoulli shift of entropy log n for some integer n 2. A question in the same spirit was raised by Ryzhikov, who asked in [44] under which conditions we can find a factor of the direct product T T which is independent of both coordinates. There is also a lot of work to do with joinings in order to understand the structure of factors of some dynamical systems, and how different classes of systems are related. An example of such a work is given in the class of
Gaussian dynamical systems, i. e. dynamical systems constructed from the shift on a stationary Gaussian process: For some of them (which are called GAG, from the French Gaussien à Autocouplages Gaussiens), it can be proven that any ergodic self-joining is itself a Gaussian system (see [29,50]), and this gives a complete description of the structure of their factors. This kind of analysis is expected to be applicable to other classes of dynamical systems. In particular, Gaussian joinings find a nice generalization in the notion of infinitely divisible joinings, studied by Roy in [40]). These ID joinings concern a wider class of dynamical systems of probabilistic origin, among which we can also find Poisson suspensions. The counterpart of Gaussian joinings in this latter class are Poisson joinings, which have been introduced by Derriennic, Fraczek, ˛ Lema´nczyk and Parreau in [11]. As far as Poisson suspensions are concerned, the analog of the GAG property in the Gaussian class can also be considered, and a family of Poisson suspension for which the only ergodic self-joinings are Poisson joinings has been given recently by Parreau and Roy [37]. In [11], a general joining property is described: T satisfies the ELF property (from the French: Ergodicité des Limites Faibles) if any joining which can be obtained as a limit of off-diagonal joinings T n k is automatically ergodic. It turns out that this property is satisfied by any system arising from an infinitely divisible stationary process (see [11,40]). It is proven in [11] that ELF property implies disjointness with any system which is two-fold simple and weakly mixing but not mixing. ELF property is expected to give a useful tool to prove disjointness between dynamical systems of probabilistic origin and other classes of systems (see e. g. [13] in the case of R-action for disjointness between ELF systems and a class of special flows over irrational rotations). Many other questions involving joinings have not been mentioned here. We should at least cite filtering problems, which were one of the motivations presented by Furstenberg for the introduction of the disjointness property in [14]. Suppose we are given two real-valued stationary processes (X n ) and (Yn ), with their joint distribution also stationary. We can interpret (X n ) as a signal, perturbed by a noise (Yn ), and the question posed by Furstenberg is: Under which condition can we recover the original signal (X n ) from the observation of (X n C Yn )? Furstenberg proved that it is always possible if the two processes (X n ) and (Yn ) are integrable, and if the two measure-preserving dynamical systems constructed as the shift of the two processes are disjoint. Furstenberg also observed that the integrability assumption can be removed if a stronger disjointness property is satisfied: A perfect filtering exists if the system T generated by (X n ) is dou-
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bly disjoint from the system S generated by (Yn ), in the sense that T is disjoint from any ergodic self-joining of S. Several generalizations have been studied (see [2,16]), but the question of whether the integrability assumption of the processes can be removed is still open.
Bibliography ´ 1. Ahn Y-H, Lemanczyk M (2003) An algebraic property of joinings. Proc Amer Math Soc 131(6):1711–1716 (electronic) ´ 2. Bułatek W, Lemanczyk M, Lesigne E (2005) On the filtering problem for stationary random Z2 -fields. IEEE Trans Inform Theory 51(10):3586–3593 3. Burton R, Rothstein A (1977) Isomorphism theorems in ergodic theory. Technical report, Oregon State University 4. Chacon RV (1969) Weakly mixing transformations which are not strongly mixing. Proc Amer Math Soc 22:559–562 5. de la Rue T (2006) 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bull Braz Math Soc (NS) 37(4):503–521 6. de la Rue T (2006) An introduction to joinings in ergodic theory. Discret Contin Dyn Syst 15(1):121–142 7. del Junco A (1983) A family of counterexamples in ergodic theory. Isr J Math 44(2):160–188 ´ 8. del Junco A, Lemanczyk M, Mentzen MK (1995) Semisimplicity, joinings and group extensions. Studia Math 112(2):141–164 9. del Junco A, Rahe M, Swanson L (1980) Chacon’s automorphism has minimal self-joinings. J Analyse Math 37:276–284 10. del Junco A, Rudolph DJ (1987) On ergodic actions whose selfjoinings are graphs. Ergod Theory Dynam Syst 7(4):531–557 ´ 11. Derriennic Y, Fraczek ˛ K, Lemanczyk M, Parreau F (2008) Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings. Colloq Math 110:81–115 12. Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1): 35–65 ´ 13. Fraczek ˛ K, Lemanczyk M (2004) A class of special flows over irrational rotations which is disjoint from mixing flows. Ergod Theory Dynam Syst 24(4):1083–1095 14. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theory 1:1–49 15. Furstenberg H (1981) Recurrence in ergodic theory and combinatorial number theory. In: M.B. Porter Lectures. Princeton University Press, Princeton 16. Furstenberg H, Peres Y, Weiss B (1995) Perfect filtering and double disjointness. Ann Inst H Poincaré Probab Stat 31(3):453–465 17. Garsia AM (1970) Topics in almost everywhere convergence. In: Lectures in Advanced Mathematics, vol 4. Markham Publishing Co, Chicago, IL 18. Glasner E (2003) Ergodic theory via joinings. In: Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence 19. Glasner E, Host B, Rudolph DJ (1992) Simple systems and their higher order self-joinings. Isr J Math 78(1):131–142 20. Glasner E, Thouvenot J-P, Weiss B (2000) Entropy theory without a past. Ergod Theory Dynam Syst 20(5):1355–1370 21. Glasner S, Weiss B (1983) Minimal transformations with no common factor need not be disjoint. Isr J Math 45(1):1–8
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Kolmogorov–Arnold–Moser (KAM) Theory
Kolmogorov–Arnold–Moser (KAM) Theory LUIGI CHIERCHIA Dipartimento di Matematica, Università “Roma Tre”, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Kolmogorov Theorem Arnold’s Scheme The Differentiable Case: Moser’s Theorem Future Directions A The Classical Implicit Function Theorem B Complementary Notes Bibliography Glossary Action-angle variables A particular set of variables (y; x) D ((y1 ; : : : ; y d ); (x1 ; : : : ; x d )), xi (“angles”) defined modulus 2, particularly suited to describe the general behavior of an integrable system. Fast convergent (Newton) method Super-exponential algorithms, mimicking Newton’s method of tangents, used to solve differential problems involving small divisors. Hamiltonian dynamics The dynamics generated by a Hamiltonian system on a symplectic manifold, i. e., on an even-dimensional manifold endowed with a symplectic structure. Hamiltonian system A time reversible, conservative (without dissipation or expansion) dynamical system, which generalizes classical mechanical systems (solutions of Newton’s equation m i x¨ i D f i (x), with 1 i d and f D ( f1 ; : : : ; f d ) a conservative force field); they are described by the flow of differential equations (i. e., the time t map associating to an initial condition, the solution of the initial value problem at time t) on a symplectic manifold and, locally, look like the flow associated with the system of differential equation p˙ D H q (p; q), q˙ D H p (p; q) where p D (p1 ; : : : ; p d ), q D (q1 ; : : : ; qd ). Integrable Hamiltonian systems A very special class of Hamiltonian systems, whose orbits are described by linear flows on the standard d-torus: (y; x) ! (y; x C ! t) where (y; x) are action-angle variables and t is time; the ! i ’s are called the “frequencies” of the orbit.
Invariant tori Manifolds diffeomorphic to tori invariant for the flow of a differential equation (especially of Hamiltonian differential equations); establishing the existence of tori invariant for Hamiltonian flows is the main object of KAM theory. KAM Acronym from the names of Kolmogorov (Andrey Nikolaevich Kolmogorov, 1903–1987), Arnold (Vladimir Igorevich Arnold, 1937) and Moser (Jürgen K. Moser, 1928–1999), whose results, in the 1950’s and 1960’s, in Hamiltonian dynamics, gave rise to the theory presented in this article. Nearly-integrable Hamiltonian systems Hamiltonian systems which are small perturbations of an integrable system and which, in general, exhibit a much richer dynamics than the integrable limit. Nevertheless, KAM theory asserts that, under suitable assumptions, the majority (in the measurable sense) of the initial data of a nearly-integrable system behaves as in the integrable limit. Quasi-periodic motions Trajectories (solutions of a system of differential equations), which are conjugate to linear flow on tori. Small divisors/denominators Arbitrary small combinaP tions of the form ! k :D djD1 ! i k i with ! D (!1 ; : : : ; !d ) 2 Rd a real vector and k 2 Zd an integer vector different from zero; these combinations arise in the denominators of certain expansions appearing in the perturbation theory of Hamiltonian systems, making (when d > 1) convergent arguments very delicate. Physically, small divisors are related to “resonances”, which are a typical feature of conservative systems. Stability The property of orbits of having certain properties similar to a reference limit; more specifically, in the context of KAM theory, stability is normally referred to as the property of action variables of staying close to their initial values. Symplectic structure A mathematical structure (a differentiable, non-degenerate, closed 2-form) apt to describe, in an abstract setting, the main geometrical features of conservative differential equations arising in mechanics. Definition of the Subject KAM theory is a mathematical, quantitative theory which has as its primary object the persistence, under small (Hamiltonian) perturbations, of typical trajectories of integrable Hamiltonian systems. In integrable systems with bounded motions, the typical trajectory is quasi-periodic, i. e., may be described through the linear flow x 2 T d ! x C ! t 2 T d where T d denotes the standard d-dimen-
Kolmogorov–Arnold–Moser (KAM) Theory
sional torus (see Sect. “Introduction” below), t is time, and ! D (!1 ; : : : ; !d ) 2 Rd is the set of frequencies of the trajectory (if d D 1, 2/! is the period of the motion). The main motivation for KAM theory is related to stability questions arising in celestial mechanics which were addressed by astronomers and mathematicians such as Kepler, Newton, Lagrange, Liouville, Delaunay, Weierstrass, and, from a more modern point of view, Poincaré, Birkhoff, Siegel, . . . The major breakthrough in this context, was due to Kolmogorov in 1954, followed by the fundamental work of Arnold and Moser in the early 1960s, who were able to overcome the formidable technical problem related to the appearance, in perturbative formulae, of arbitrarily small divisors1 . Small divisors make the use of classical analytical tools (such as the standard Implicit Function Theorem, fixed point theorems, etc.) impossible and could be controlled only through a “fast convergent method” of Newton-type2 , which allowed, in view of the super-exponential rate of convergence, counterbalancing the divergences introduced by small divisors. Actually, the main bulk of KAM theory is a set of techniques based, as mentioned, on fast convergent methods, and solving various questions in Hamiltonian (or generalizations of Hamiltonian) dynamics. By now, there are excellent reviews of KAM theory – especially Sect. 6.3 of [6] and [60] – which should complement the reading of this article, whose main objective is not to review but rather to explain the main fundamental ideas of KAM theory. To do this, we re-examine, in modern language, the main ideas introduced, respectively, by the founders of KAM theory, namely Kolmogorov (in Sect. “Kolmogorov Theorem”), Arnold (in Sect. “Arnold’s Scheme”) and Moser (Sect. “The Differentiable Case: Moser’s Theorem”). In Sect. “Future Directions” we briefly and informally describe a few developments and applications of KAM theory: this section is by no means exhaustive and is meant to give a non technical, short introduction to some of the most important (in our opinion) extensions of the original contributions; for more detailed and complete reviews we recommend the above mentioned articles Sect. 6.3 of [6] and [60]. Appendix A contains a quantitative version of the classical Implicit Function Theorem. A set of technical notes (such as notes 17, 18, 19, 21, 24, 26, 29, 30, 31, 34, 39), which the reader not particularly interested in technical mathematical arguments may skip, are collected in Appendix B and complete the mathematical expositions. Appendix B also includes several other complementary notes, which contain either standard material or further references or side comments.
Introduction In this article we will be concerned with Hamiltonian flows on a symplectic manifold (M; dy ^ dx); for general information, see, e. g., [5] or Sect. 1.3 of[6]. Notation, main definitions and a few important properties are listed in the following items. (a) As symplectic manifold (“phase space”) we shall consider M :D B T d with d 2 (the case d D 1 is trivial for the questions addressed in this article) where: B is an open, connected, bounded set in Rd ; T d :D Rd /(2Zd ) is the standard flat d-dimensional torus with periods3 2 P (b) dy ^ dx :D diD1 dy i ^ dx i , (y 2 B, x 2 T d ) is the standard symplectic form4 (c) Given a real-analytic (or smooth) function H : M ! R, the Hamiltonian flow governed by H is the oneparameter family of diffeomorphisms Ht : M ! M, which to z 2 M associates the solution at time t of the differential equation5 z˙ D J2d rH(z) ;
z(0) D z ;
(1)
where z˙ D dz/dt, J2d is the standard symplectic (2d 2d)-matrix J2d D
0 1d
1d 0
;
1d denotes the unit (d d)-matrix, 0 denotes a (d d) block of zeros, and r denotes gradient; in the symplectic coordinates (y; x) 2 B T d , equations (1) reads
y˙ D H x (y; x) x˙ D H y (y; x)
;
y(0) D y x(0) D x
(2)
Clearly, the flow Ht is defined until y(t) eventually reaches the border of B. Equations (1) and (2) are called the Hamilton’s equations with Hamiltonian H; usually, the symplectic (or “conjugate”) variables (y; x) are called action-angle variables6; the number d (= half of the dimension of the phase space) is also referred to as “the number of degrees of freedom7 ”. The Hamiltonian H is constant over trajectories Ht (z), as it follows immediately by differentiating t ! H( Ht (z)). The constant value E D H( Ht (z)) is called the energy of the trajectory Ht (z). Hamilton equations are left invariant by symplectic (or “canonical”) change of variables, i. e., by diffeomorphisms on M which preserve the 2-form dy ^ dx; i. e.,
811
812
Kolmogorov–Arnold–Moser (KAM) Theory
if : (y; x) 2 M ! (; ) D (y; x) 2 M is a diffeomorphism such that d ^ d D dy ^ dx, then t ı Ht ı 1 D Hı 1 :
(3)
An equivalent condition for a map to be symplectic is that its Jacobian 0 is a symplectic matrix, i. e., 0T
J2d 0 D J2d
(4)
where J2d is the standard symplectic matrix introduced above and the superscript T denotes matrix transposition. By a (generalization of a) theorem of Liouville, the Hamiltonian flow is symplectic, i. e., the map (y; x) ! (; ) D Ht (y; x) is symplectic for any H and any t; see Corollary 1.8, [6]. A classical way of producing symplectic transformations is by means of generating functions. For example, if g(; x) is a smooth function of 2d variables with det
@2 g ¤0; @@x
then, by the Implicit Function Theorem (IFT; see [36] or Sect. “A The Classical Implicit Function Theorem” below), the map : (y; x) ! (; ) defined implicitly by the relations yD
@g ; @x
D
@g ; @
: 2 T d ! ( ) :D (v( ); U( )) 2 M If U is a smooth diffeomorphism of T d (so that, in particular9 det U ¤ 0) then is an embedding of T d into M and the set T! D T!d :D (T d )
d X iD1
!i ni D 0 H) n D 0 ;
(5)
and if there exist smooth (periodic) functions v; u : T d ! Rd such that8 ( y(t) D v(! t) (6) x(t) D ! t C u(! t) :
(7)
is an embedded d-torus invariant for Ht and on which the motion is conjugated to the linear (Kronecker) flow ! C ! t, i. e., 1 ı Ht ı ( ) D C ! t ;
8 2 T d :
(8)
Furthermore, the invariant torus T! is a graph over T d and is Lagrangian, i. e., (T! has dimension d and) the restriction of the symplectic form dy ^ dx on T! vanishes10 . (f) In KAM theory a major role is played by the numerical properties of the frequencies !. A typical assumption is that ! is a (homogeneously) Diophantine vector: ! 2 Rd is called Diophantine or (; )-Diophantine if, for some constants 0 < min i j! i j and d 1, it verifies the following inequalities: j! nj
yields a local symplectic diffeomorphism; in such a case, g is called the generating function of the transformation . For example, the function x is the generating function of the identity map. For general information about symplectic changes of coordinates, generating functions and, in general, about symplectic structures we refer the reader to [5] or [6]. (d) A solution z(t) D (y(t); x(t)) of (2) is a maximal quasi-periodic solution with frequency vector ! D (!1 ; : : : ; !d ) 2 Rd if ! is a rationally-independent vector, i. e., 9n 2 Zd s:t: ! n :D
(e) Let !, u and v be as in the preceding item and let U and denote, respectively, the maps ( U : 2 T d ! U( ) :D C u( ) 2 T d
; jnj
8n 2 Zd n f0g ;
(9)
(normally, for integer vectors n, jnj denotes jn1 jC C jnd j, but other norms may well be used). We shall refer to and as the Diophantine constants of !. The set of Diophantine numbers in Rd with constants and d ; the union over all > 0 of will be denoted by D; d D; will be denoted by Dd and the union over all d 1 of Dd will be denoted by Dd . Basic facts about these sets are11 : if < d 1 then Dd D ;; if > d 1 then the Lebesgue measure of Rd n Dd is zero; if D d 1, the Lebesgue measure of Dd is zero but its intersection with any open set has the cardinality of R. (g) The tori T! defined in (e) with ! 2 Dd will be called maximal KAM tori for H. (h) A Hamiltonian function (; ) 2 M ! H(; ) having a maximal KAM torus (or, more generally, a maximal invariant torus as in (e) with ! rationally independent) T! , can be put into the form12 K(y; x) :D E C ! y C Q(y; x) ; with @˛y Q(0; x) D 0; 8˛ 2 N d ; j˛j 1 ;
(10)
Kolmogorov–Arnold–Moser (KAM) Theory
compare, e. g., Sect. 1 of [59]; in the variables (y; x), the torus T! is simply given by fy D 0g T d and E is its (constant) energy. A Hamiltonian in the form (10) is said to be in Kolmogorov normal form. If deth@2y Q(0; )i ¤ 0 ;
(11)
(where the brackets denote an average over T d and @2y the Hessian with respect to the y-variables) we shall say that the Kolmogorov normal form K in (10) is nondegenerate; similarly, we shall say that the KAM torus T! for H is non-degenerate if H can be put in a nondegenerate Kolmogorov normal form.
Thus, if the “frequency map” y 2 B ! !(y) is a diffeomorphism (which is guaranteed if det K y y (y0 ) ¤ 0, for some y0 2 B and B is small enough), in view of (f), for almost all initial data, the trajectories (13) belong to maximal KAM tori fygT d with !(y) 2 Dd . The main content of (classical) KAM theory, in our language, is that, if the frequency map ! D K y of a (real-analytic) integrable Hamiltonian K(y) is a diffeomorphism, KAM tori persist under small (smooth enough) perturbations of K; compare Remark 7–(iv) below. The study of the dynamics generated by the flow of a one-parameter family of Hamiltonians of the form K(y) C "P(y; x; ") ;
Remark 1 (i)
A classical theorem by H. Weyl says that the flow 2 Td ! C !t 2 Td ;
t2R
is dense (ergodic) in T d if and only if ! 2 Rd is rationally independent (compare [6], Theorem 5.4 or Sect. 1.4 of [33]). Thus, trajectories on KAM tori fill them densely (i. e., pass in any neighborhood of any point). (ii) In view of the preceding remark, it is easy to see that if ! is rationally independent, (y(t); x(t)) in (6) is a solution of (2) if and only if the functions v and u satisfy the following quasi-linear system of PDEs on Td: ( D! v D H x (v( ); C u( )) (12) ! C D! u D H y (v( ); C u( )) where D! denotes the directional derivative ! @ D Pd @ iD1 ! i @ i . (iii) Probably, the main motivation for studying quasi-periodic solutions of Hamiltonian systems on Rd T d comes from perturbation theory of nearly-integrable Hamiltonian systems: a completely integrable system may be described by a Hamiltonian system on M :D B(y0 ; r) T d Rd T d with Hamiltonian H D K(y) (compare Theorem 5.8, [6]); here B(y0 ; r) denotes the open ball fy 2 Rd : jy y0 j < rg centered at y0 2 Rd . In such a case the Hamiltonian flow is simply
Kt (y; x) D y; x C !(y)t ; !(y) :D K y (y) :D
@K (y) : @y
(13)
0 0 and for any j"j "0 a real-analytic symplectic transformation : M :D B(0; r ) T d ! M, for some 0 < r < r, putting H in non-degenerate Kolmogorov normal form, H ı D K , with K :D E C ! y 0 C Q (y 0 ; x 0 ). Furthermore13 , k idkC 1 (M ) , jE Ej, and kQ QkC 1 (M ) are small with ". Remark 2 (i)
From Theorem 1 it follows that the torus T!;" :D (0; T d )
is a maximal non-degenerate KAM torus for H and the H-flow on T!;" is analytically conjugated (by ) to the translation x 0 ! x 0 C ! t with the same frequency vector of T!;0 :D f0g T d , while the energy of T!;" , namely E , is in general different from the energy E of T!;0 . The idea of keeping the frequency fixed is a key idea introduced by Kolmogorov and its importance will be made clear in the analysis of the proof. (ii) In fact, the dependence upon " is analytic and therefore the torus T!;" is an analytic deformation of the unperturbed torus T!;0 (which is invariant for K); see Remark 7–(iii) below. (iii) Actually, Kolmogorov not only stated the above result but gave also a precise outline of its proof, which is based on a fast convergent “Newton” scheme, as we shall see below; compare also [17]. The map is obtained as D lim 1 ı ı j ; j!1
where the j ’s are ("-dependent) symplectic transformations of M closer and closer to the identity. It is enough to describe the construction of 1 ; 2 is then obtained by replacing H0 :D H with H1 D H ı 1 and so on. We proceed to analyze the scheme of Kolmogorov’s proof, which will be divided into three main steps. Step 1: Kolmogorov Transformation The map 1 is close to the identity and is generated by g(y 0 ; x) :D y 0 x C " b x C s(x) C y 0 a(x)
where s and a are (respectively, scalar and vector-valued) real-analytic functions on T d with zero average and b 2 Rd : setting ˇ0 D ˇ0 (x) :D b C s x ; A D A(x) :D a x
and
0
(18) 0
ˇ D ˇ(y ; x) :D ˇ0 C Ay ; (s x D @x s D (s x 1 ; : : : ; s x d ) and ax denotes the matrix i (a x ) i j :D @a ) 1 is implicitly defined by @x j (
y D y 0 C "ˇ(y 0 ; x) :D y 0 C "(ˇ0 (x) C A(x)y 0 ) x 0 D x C "a(x) :
(19)
Thus, for " small, x 2 T d ! x C "a(x) 2 T d defines a diffeomorphism of T d with inverse x D '(x 0 ) :D x 0 C "˛(x 0 ; ") ;
(20)
for a suitable real-analytic function ˛, and 1 is explicitly given by ( 0
0
1 : (y ; x ) !
y D y 0 C "ˇ y 0 ; '(x 0 ) x D '(x 0 ) :
(21)
Remark 3 (i) Kolmogorov transformation 1 is actually the composition of two “elementary” symplectic transformations: 1 D 1(1) ı 1(2) where 1(2) : (y 0 ; x 0 ) ! (; ) is the symplectic lift of the T d -diffeomorphism given by x 0 D C "a() (i. e., 1(2) is the symplectic map generated by y 0 C "y 0 a()), while 1(1) : (; ) ! (y; x) is the angle-dependent action translation generated by x C "(b x C s(x)); 1(2) acts in the “angle direction” and will be needed to straighten out the flow up to order O("2 ), while 1(1) acts in the “action direction” and will be needed to keep the frequency of the torus fixed. (ii) The inverse of 1 has the form ( (y; x) !
y 0 D M(x)y C c(x) x 0 D (x)
(22)
with M a (d d)-invertible matrix and a diffeomorphism of T d (in the present case M D (1d C "A(x))1 D 1d C O(") and D id C "a) and it is easy to see that the symplectic diffeomorphisms of the form (22) form a subgroup of the symplectic diffeomorphisms, which we shall call the group of Kolmogorov transformations.
Kolmogorov–Arnold–Moser (KAM) Theory
Determination of Kolmogorov transformation Following Kolmogorov, we now try to determine b, s and a so that the “new Hamiltonian” (better: “the Hamiltonian in the new symplectic variables”) takes the form H1 :D H ı 1 D K1 C "2 P1 ;
(23)
with K 1 in the Kolmogorov normal form K1 D E1 C ! y 0 C Q1 (y 0 ; x 0 ) ;
Q1 D O(jy 0 j2 ) : (24)
To proceed we insert y D y 0 C "ˇ(y 0 ; x) into H and, after some elementary algebra and using Taylor formula, we find14
where, defining 8 (1) Q :D Q y (y 0 ; x) (a x y 0 ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ Q (2) :D [Q y (y 0 ; x) Q y y (0; x)y 0 ] ˇ0 ˆ ˆ ˆ Z 1 ˆ ˆ ˆ ˆ D (1 t)Q y y y (ty 0 ; x)y 0 y 0 ˇ0 dt ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ ˆ Q (3) :D P(y 0 ; x) P(0; x) Py (0; x)y 0 ˆ ˆ ˆ Z 1 ˆ ˆ ˆ ˆ D (1 t)Py y (ty 0 ; x)y 0 y 0 dt ˆ ˆ ˆ 0 < 1 P(1) :D 2 [Q(y 0 C "ˇ; x) Q(y 0 ; x) ˆ ˆ ˆ " ˆ ˆ ˆ ˆ "Q y (y 0 ; x) ˇ] ˆ ˆ ˆ Z ˆ 1 ˆ ˆ ˆ ˆ (1 t)Q y y (y 0 C t"ˇ; x)ˇ ˇdt D ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ 1 ˆ ˆ P(2) :D [P(y 0 C "ˇ; x) P(y 0 ; x)] ˆ ˆ " ˆ ˆ Z 1 ˆ ˆ ˆ ˆ : Py (y 0 C t"ˇ; x) ˇdt ; D
(25)
(26)
!j
jD1
Q0
F0
F 0 (y 0 ; x)
(i) F 0 is a first degree polynomial in y 0 so that (28) is equivalent to ( ! b C D! s C P(0; x) D const ; (29) D! a C Q y y (0; x)ˇ0 C Py (0; x) D 0 :
D! u D f ;
(30)
for some given function f real-analytic on the average over T d shows that h f i D 0, and we see that (30) can be solved only if f has vanishing mean value h f i D f0 D 0 ;
@ @x j Q 0 (y 0 ; x),
(28)
T d . Taking
(recall that Q y (0; x) D 0) and denoting the !-directional derivative d X
F 0 (y 0 ; x) D const :
Indeed, the second equation is necessary to keep the torus frequency fixed and equal to ! (which, as we shall see in more detail later, is a key ingredient introduced by Kolmogorov). (ii) In solving (28) or (29), we shall encounter differential equations of the form
0
D! :D
recall, also, that Q D O(jyj2 ) so that Q y D O(y) and Q 0 D O(jy 0 j2 ). Notice that, as an intermediate step, we are considering H as a function of mixed variables y 0 and x (and this causes no problem, as it will be clear along the proof). Thus, recalling that x is related to x 0 by the (y 0 -independent) diffeomorphism x D x 0 C "˛(x 0 ; ") in (21), we see that in order to achieve relations (23)–(24), we have to determine b, s and a so that
Remark 4
H(y 0 C "ˇ; x) D E C ! y 0 C Q(y 0 ; x) C "Q 0 (y 0 ; x) C "F 0 (y 0 ; x) C "2 P0 (y 0 ; x)
where D! a is the vector function with kth entry Pd Pd @a k @a k 0 0 0 jD1 ! j @x j ; D! a y D ! (a x y ) D j;kD1 ! j @x j y k ;
P0
one sees that D D and D P0 (y 0 ; x) are given by, respectively 8 ˆ Q 0 (y 0 ; x) :D Q (1) C Q (2) C Q (3) D O(jy 0 j2 ) ˆ ˆ ˆ ˆ < 0 0 F (y ; x) :D ! b C D! s C P(0; x) (27) ˚ ˆ ˆ C D! a C Q y y (0; x)ˇ0 C Py (0; x) y 0 ˆ ˆ ˆ : 0 P :D P(1) C P(2) ;
in such a case, expanding in Fourier series15 , one sees that (30) is equivalent to X X i! nu n e i nx D f n e i nx ; (31) n2Z d n¤0
n2Z d n¤0
so that the solutions of (30) are given by X fn e i nx ; u D u0 C i! n d
(32)
n2Z n¤0
for an arbitrary u0 . Recall that for a continuous functionf over T d to be analytic it is necessary and sufficient that its Fourier coefficients f n decay exponentially fast in n, i. e., that there exist positive constants M and such that j f n j Mejnj ;
8n :
(33)
815
816
Kolmogorov–Arnold–Moser (KAM) Theory
d one has that (for n ¤ 0) Now, since ! 2 D;
jnj 1 j! nj
(34)
and one sees that if f is analytic so is u in (32) (although the decay constants of u will be different from those of f ; see below) Summarizing, if f is real-analytic on T d and has vanishing mean value f 0 , then there exists a unique realanalytic solution of (30) with vanishing mean value, which is given by 1 D! f :D
X n2Z d n¤0
f n i nx ; e i! n
(35)
all other solutions of (30) are obtained by adding an 1 f as in (32) with u arbitrary. arbitrary constant to D! 0 Taking the average of the first relation in (29), we may the determine the value of the constant denoted const, namely, const D ! b C P0 (0) :D ! b C hP(0; )i :
(36)
Thus, by (ii) of Remark 4, we see see that 1 s D D! (P(0; x) P0 (0)) D
X Pn (0) e i nx ; (37) i! n d
n2Z n¤0
Finally, if '(x 0 ) D x 0 C "˛(x 0 ; ") is the inverse diffeomorphism of x ! x C "a(x) (compare (20)), then, by Taylor’s formula, Z 1 Q x (y 0 ; x 0 C "˛t) ˛dt : Q(y 0 ; '(x 0 )) D Q(y 0 ; x 0 ) C " 0
In conclusion, we have Proposition 1 If 1 is defined in (19)–(18) with s, b and a given in (37), (39) and (40) respectively, then (23) holds with 8 E1 :D E C "e E ˆ ˆ ˆ ˆ ˆ e ˆ E :D ! b C P0 (0) ˆ ˆ ˆ < Q (y 0 ; x 0 ) :D Q(y 0 ; x 0 ) C "e Q(y 0 ; x 0 ) 1 Z 1 ˆ ˆ ˆ e ˆ Q x (y 0 ; x 0 C t"˛) ˛dt C Q 0 (y 0 ; '(x 0 )) Q :D ˆ ˆ ˆ 0 ˆ ˆ : P1 (y 0 ; x 0 ) :D P0 (y 0 ; '(x 0 )) (41) with Q 0 and P0 defined in (26), (27) and ' in (20). Remark 5 The main technical problem is now transparent: because of the appearance of the small divisors ! n 1 f (which may become arbitrarily small), the solution D! is less regular than f so that the approximation scheme cannot work on a fixed function space. To overcome this fundamental problem – which even Poincaré was unable to solve notwithstanding his enormous efforts (see, e. g., [49]) – three ingredients are necessary: (i)
where Pn (0) denote the Fourier coefficients of x ! P(0; x); indeed s is determined only up to a constant by the relation in (29) but we select the zero-average solution. Thus, s has been completely determined. To solve the second (vector) equation in (29) we first have to require that the left hand side (l.h.s.) has vanishing mean value, i. e., recalling that ˇ0 D b C s x (see (18)), we must have hQ y y (0; )ib C hQ y y (0; )s x i C hPy (0; )i D 0 :
(38)
In view of (11) this relation is equivalent to b D hQ y y (0; )i1 hQ y y (0; )s x i C hPy (0; )i ; (39) which uniquely determines b. Thus ˇ 0 is completely determined and the l.h.s. of the second equation in (29) has average zero; thus its unique zero-average solution (again zero-average of a is required as a normalization condition) is given by 1 Q y y (0; x)ˇ0 C Py (0; x) : a D D!
(40)
To set up a Newton scheme: this step has just been performed and it has been summarized in the above Proposition 1; such schemes have the following features: they are “quadratic” and, furthermore, after one step one has reproduced the initial situation (i. e., the form of H 1 in (23) has the same properties of H 0 ). It is important to notice that the new perturbation "2 P1 is proportional to the square "; thus, if one could iterate j times, at the jth step, would find j
H j D H j1 ı j D K j C "2 Pj :
(42)
The appearance of the exponential of the exponential of " justifies the term “super-convergence” used, sometimes, in connection with Newton schemes. (ii) One needs to introduce a scale of Banach function spaces fB : > 0g with the property that B 0 B when < 0 : the generating functions j will belong to B j for a suitable decreasing sequence j ; (iii) One needs to control the small divisors at each step and this is granted by Kolmogorov’s idea of keeping the frequency fixed in the normal form so that one can systematically use the Diophantine estimate (9).
Kolmogorov–Arnold–Moser (KAM) Theory
Kolmogorov in his paper very neatly explained steps (i) and (iii) but did not provide the details for step (ii); in this regard he added: “Only the use of condition (9) for proving the convergence of the recursions, j , to the analytic limit for the recursion is somewhat more subtle”. In the next paragraph we shall introduce classical Banach spaces and discuss the needed straightforward estimates.
(P5) Assume that x ! f (x) 2 B has a zero average (all above definitions may be easily adapted to functions d (recall depending only on x); assume that ! 2 D; Sect. “Introduction”, point (f)), and let p 2 N. Then, there exist constants B¯ p D B¯ p (d; ) 1 and k p D k p (d; ) 1 such that, for any multi-index ˇ 2 N d with jˇj p and for any 0 0 < one has 1 f k 0 B¯ p k@ˇx D!
Step 2: Estimates For 1, we denote by B the space of function f : B(0; ) T d ! R analytic on W :D D(0; ) Td ;
(43)
where D(0; ) :D fy 2 C d : jyj < g and Td :D fx 2 C d : jImx j j < g/(2Zd )
sup
(i)
A proof of (48) is easily obtained observing that by (35) and (46), calling ı :D 0 , one has 1 k@ˇx D! f k 0
X jnjjˇ j j f n j 0 e jnj j! nj d
n2Z n¤0
(44)
k f k jfj ;
D(0;)T d
D
(in other words, Td denotes the complex points x with real parts Rex j defined modulus 2 and imaginary part Imx j with absolute value less than ). The following properties are elementary:
j f n (y)j k f k ejnj ;
8n 2 Zd ; 8y 2 D(0; ) : (46)
Another elementary property, which together with (P3) may found in any book of complex variables (e. g., [1]), is the following “Cauchy estimate” (which is based on Cauchy’s integral formula): (P4) let f 2 B and let p 2 N then there exists a constant B p D B p (d) 1 such that, for any multi-index (˛; ˇ) 2 N d N d with j˛j C jˇj p (as above P for integer vectors ˛, j˛j D j j˛ j j) and for any 0 0 < one has k@˛y @ˇx f k 0 B p k f k ( 0 )(j˛jCjˇ j) :
(47)
1 f , i. e., on solutions Finally, we shall need estimates on D! of (30):
X jnjjˇ jC eıjnj d
n2Z n¤0
(45)
(P1) B equipped with the k k norm is a Banach space; (P2) B 0 B when < 0 and k f k k f k 0 for any f 2 B 0 ; (P3) if f 2 B , and f n (y) denotes the n-Fourier coefficient of the periodic function x ! f (y; x), then
(48)
Remark 6
with finite sup-norm k f k :D
k f k ( 0 )k p :
k f k (jˇ jCCd) ı X [ıjnj]jˇ jC eıjnj ı d n2Z d n¤0
const
k f k ( 0 )(jˇ jCCd) ;
where the last estimate comes from approximating the sum with the Riemann integral Z jyjjˇ jC ejyj dy : Rd
More surprising (and much more subtle) is that (48) holds with k p D jˇj C ; such an estimate has been obtained by Rüssmann [54,55]. For other explicit estimates see, e. g., [11] or [12]. (ii) If jˇj > 0 it is not necessary to assume that h f i D 0. (iii) Other norms may be used (and, sometimes, are more useful); for example, rather popular are Fourier norms X j f n jejnj ; (49) k f k0 :D n2Z d
see, e. g., [13] and references therein. By the hypotheses of Theorem 1 it follows that there exist 0 < 1, > 0 and d 1 such that H 2 B and d . Denote ! 2 D; T :D hQ y y (0; )i1 ;
M :D kPk :
(50)
817
818
Kolmogorov–Arnold–Moser (KAM) Theory
and let C > 1 be a constant such that16 jEj; j!j; kQk ; kTk < C
(51)
(i. e., each term on the l.h.s. is bounded by the r.h.s.); finally, fix 0 1, that17 8 ks x k¯ ; jbj; je Ej; kak¯ ; ka x k¯ ; kˇ0 k¯ ; kˇk¯ ; ˆ ˆ < kQ 0 k¯ ; k@2y 0 Q 0 (0; )k0 c¯C ¯ ı ¯ M D: L¯ ; (53) ˆ ˆ : 0 ¯ : kP k¯ c¯C ¯ ı ¯ M 2 D LM The estimate in the first line of (53) allows us to construct, for " small enough, the symplectic transformation 1 , whose main properties are collected in the following Lemma 1 If j"j "0 and "0 satisfies "0 L¯
ı ; 3
(54)
then the map " (x) :D x C "a(x) has an analytic inverse '(x 0 ) D x 0 C "˛(x 0 ; ") such that, for all j"j < "0 , k˛k 0 L¯
and
' D id C "˛ : Td0 ! T¯d :
(55)
Furthermore, for any (y 0 ; x) 2 W¯ , jy 0 C "ˇ(y 0 ; x)j < , so that 1 D y 0 C "ˇ(y 0 ; '(x 0 )); '(x 0 ) : W 0 ! W ; and k1 idk 0 j"j L¯ ; (56) finally, the matrix 1d C "a x is, for any x 2 T¯d , invertible with inverse 1d C "S(x; ") satisfying kSk¯
ka x k¯ 1 j"jka x k¯
1 and positive integers D (d; ), such that if j"j " :D
c kPk C
(68)
then one can construct a near-to-identity Kolmogorov transformation (Remark 3–(ii)) : W ! W such that the thesis of Theorem 1 holds together with the estimates k idk ; jE E j; kQ Q k ; kT T k
j"j D j"jc kPk C : "
(69)
det K y y (y0 ) ¤ 0 :
(70)
In fact, without loss of generality we may assume that ! :D H00 is a diffeomorphism on B(y0 ; 2r) and det K y y (y) ¤ 0 for all y 2 B(y0 ; 2r). Furthermore, letting B D B(y0 ; r), fixing > d 1 and denoting by `d the Lebesgue measure on Rd , from the remark in note 11 and from the fact that ! is a diffeomorphism, there follows that there exists a constant c# depending only on d, and r such that d d `d (!(B)nD; ); `d (fy 2 B : !(y) … D; g) < c# :
(71) d g (which, Now, let B; :D fy 2 B : !(y) 2 D; by (71) has Lebesgue measure `d (B; ) `d (B) c# ), then for any y¯ 2 B; we can make the trivial symplectic change of variables y ! y¯ C y, x ! x so that K can be written as in (10) with
E :D K( y¯) ;
! :D K y ( y¯ ) ;
Q(y; x) D Q(y) :D K(y) K( y¯) K y ( y¯) y ; (where, for ease of notation, we did not change names to the new symplectic variables) and P( y¯ C y; x) replacing (with a slight abuse of notation) P(y; x). By Taylor’s formula, Q D O(jyj2 ) and, furthermore (since Q(y; x) D Q(y), h@2y Q(0; x)i D Q y y (0) D
819
820
Kolmogorov–Arnold–Moser (KAM) Theory
K y y ( y¯), which is invertible according to out hypotheses. Thus K is Kolmogorov non-degenerate and Theorem 1 can be applied yielding, for j"j < "0 , a KAM torus T!;" , with ! D K y ( y¯), for each y¯ 2 B; . Notice that the measure of initial phase points, which, perturbed, give rise to KAM tori, has a small complementary bounded by c# (see (71)). (v) In the nearly-integrable setting described in the preceding point, the union of KAM tori is usually called the Kolmogorov set. It is not difficult to check that the dependence upon y¯ of the Kolmogorov transformation is Lipschitz23 , implying that the measure of the complementary of Kolmogorov set itself is also bounded by cˆ# with a constant cˆ# depending only on d, and r. Indeed, the estimate on the measure of the Kolmogorov set can be made more quantitative (i. e., one can see how such an estimate depends upon " as " ! 0). In fact, revisiting the estimates discussed in Step 2 above one sees easily that the constant c defined in (53) has the form24 c D cˆ 4 :
(72)
where cˆ D cˆ(d; ) depends only on d and (here the Diophantine constant is assumed, without loss of generality, to be smaller than one). Thus the small¯ 1 with some ness condition (65) reads "0 4 D ¯ independent of : such condition is satconstant D ¯ 0 )1/4 and since cˆ# was an isfied by choosing D ( D" upper bound on the complementary of Kolmogorov set, we see that the set of phase points which do not lie on KAM tori may be bounded by a constant times p 4 " . Actually, it turns that this bound is not optimal, 0 as we shall see in the next section: see Remark 10. (vi) The proof of claim C follows easily by induction on the number j of the iterative steps25 . Arnold’s Scheme The first detailed proof of Kolmogorov Theorem, in the context of nearly-integrable Hamiltonian systems (compare Remark 1–(iii)), was given by V.I. Arnold in 1963. Theorem 2 (Arnold [2]) Consider a one-parameter family of nearly-integrable Hamiltonians H(y; x; ") :D K(y) C "P(y; x) (" 2 R)
(73)
with K and P real-analytic on M :D B(y0 ; r) T d (endowed with the standard symplectic form dy^dx) satisfying K y (y0 ) D ! 2
d D;
;
det K y y (y0 ) ¤ 0 :
(74)
Then, if " is small enough, there exists a real-analytic embedding : 2 Td ! M
(75)
close to the trivial embedding (y0 ; id), such that the d-torus T!;" :D T d (76) is invariant for H and Ht ı ( ) D ( C ! t) ;
(77)
showing that such a torus is a non-degenerate KAM torus for H. Remark 8 (i)
The above Theorem is a corollary of Kolmogorov Theorem 1 as discussed in Remark 7–(iv). (ii) Arnold’s proof of the above Theorem is not based upon Kolmogorov’s scheme and is rather different in spirit – although still based on a Newton method – and introduces several interesting technical ideas. (iii) Indeed, the iteration scheme of Arnold is more classical and, from the algebraic point of view, easier than Kolmogorov’s, but the estimates involved are somewhat more delicate and introduce a logarithmic correction, so that, in fact, the smallness parameter will be :D j"j(log j"j1 )
(78)
(for some constant D (d; ) 1) rather than j"j as in Kolmogorov’s scheme; see, also, Remark 9–(iii) and (iv) below. Arnold’s Scheme Without loss of generality, one may assume that K and P have analytic and bounded extension to Wr; (y0 ) :D D(y0 ; r) Td for some > 0, where, as above, D(y0 ; r) denotes the complex ball of center y0 and radius r. We remark that, in what follows, the analyticity domains of actions and angles play a different role The Hamiltonian H in (73) admits, for " D 0 the (KAM) invariant torus T!;0 D fy0 g T d on which the K-flow is given by x ! x C ! t. Arnold’s basic idea is to find a symplectic transformation 1 : W1 :D D(y1 ; r1 )Td1 ! W0 :D D(y0 ; r)Td ; (79) so that W1 W0 and ( H1 :D H ı 1 D K1 C "2 P1 ; @ y K1 (y1 ) D ! ;
det @2y K1 (y1 )
K1 D K1 (y) ; ¤0
(80)
Kolmogorov–Arnold–Moser (KAM) Theory
(with abuse of notation we denote here the new symplectic variables with the same name of the original variables; as above, dependence on " will, often, not be explicitly indicated). In this way the initial set up is reconstructed and, for " small enough, one can iterate the scheme so as to build a sequence of symplectic transformations j : Wj :D D(y j ; r j ) Tdj ! Wj1 so that ( j H j :D H j1 ı j D K j C "2 Pj ; @ y K j (y j ) D ! ;
det @2y K j (y j )
determined by a generating function of the form ( y D y 0 C "g x (y 0 ; x) 0 0 y x C "g(y ; x) ; x 0 D x C "g y 0 (y 0 ; x) : Inserting y D y 0 C "g x (y 0 ; x) into H, one finds
(81)
K j D K j (y) ;
¤0:
(82)
H(y 0 C "g x ; x) D K(y 0 ) C " K y (y 0 ) g x C P(y 0 ; x) C "2 P(1) C P(2)
1 [K(y 0 C "g x ) K(y 0 ) "K y (y 0 ) g x ] "2 Z 1 (1 t)K y y (y 0 C t"g x ; x) g x g x dt D
P(1) :D
0
( ) :D lim ˚ j (y j ; ) ;
P
(83)
˚ j :D 1 ı ı j : Wj ! W0 ; defines a real-analytic embedding of T d into the phase space B(y0 ; r) T d , which is close to the trivial embedding (y0 ; id); furthermore, the torus T!;" :D (T d ) D lim ˚ j (y j ; T d )
(87)
with (compare (26))
Arnold’s transformations, as in Kolmogorov’s case, are closer and closer to the identity, and the limit j!1
(86)
(2)
1 :D [P(y 0 C "g x ; x) P(y 0 ; x)] " Z 1 Py (y 0 C t"g x ; x) g x dt : D
(88)
0
Remark 9 (i)
The (naive) idea is to try determine g so that
(84)
K y (y 0 ) g x C P(y 0 ; x) D function of y 0 only; (89)
is invariant for H and (77) holds as announced in Theorem 2. Relation (77) follows from the following argument. The radius rj will turn out to tend to 0 but in a much slower j way than "2 Pj . This fact, together with the rapid convergence of the symplectic transformation ˚ j in (83) implies
however, such a relation is impossible to achieve. First of all, by taking the x-average of both sides of (89) one sees that the “function of y 0 only” has to be the mean of P(y 0 ; ), i. e., the zero-Fourier coefficient P0 (y 0 ), so that the formal solution of (89), is (by Fourier expansion) 8 X Pn (y 0 ) ˆ ˆ 1, 2 ZC , and are positive integers depending on d and . Now, by32 Lemma 1 and (118), one has that map x ! x C "g y (y 0 ; x) has, for any y 0 2 Dr¯ (y0 ), an analytic inverse ' D x 0 C "˛(x 0 ; y 0 ; ") D: '(y 0 ; x 0 ) on T ¯d ı provided (54), with L¯ replaced by L 3
in (118), holds, in which case (55) holds (for any j"j "0 and any y 0 2 Dr (y0 )). Furthermore, under the above hypothesis, it follows that33 0 8 0 0 0 0 0 ˆ < 1 :D y C "g x (y ; '(y ; x )); '(y ; x ) : Wr¯/2;ı (y1 ) ! Wr; (y0 ) ˆ : k1 idkr¯/2;ı j"jL :
(119)
Finally, letting P1 (y 0 ; x 0 ) :D P0 (y 0 ; '(y 0 ; x 0 )) one sees that P1 is real-analytic on Wr¯/2;ı ¯ (y 1 ) and bounded on that domain by kP1 kr¯/2;ı LM :
(120)
In order to iterate the above construction, we fix 0 < < and set C :D 2 maxf1; kK y kr ; kK y y kr ; kTkg ;
:D 3C ; ı0 :D
(121)
( 1)( ) ;
j and ı j as in (63) but with ı 0 as in (121); we also define, for any j 0, j
; j :D 2 j D log "2 0 ; r j :D C1 4 5C1 C(ı 1 j j)
(122)
(this part is adapted from Step 3 in Sect. “Kolmogorov Theorem”; see, in particular, (103)). With such choices it is not difficult to check that the iterative construction may be carried out infinitely many times yielding, as a byproduct, Theorem 2 with real-analytic on Td , provided j"j "0 with "0 satisfying34 8 ˆ ˇ ˆ ˆ < "0 e "0 DBkPk 1 ˆ ˆ ˆ :
with with
1 ı0 C1 5 Cr D :D 3c 2 ı0(C1) C ;
ˇ :D
B :D C1 (log "1 0 ) :
(123) Remark 10 Notice that the power of 1 (the inverse of the Diophantine constant) in the second smallness condition in (123) is two, which implies (compare Remark 7–
(v)) that the measure of the complement of the Kolp mogorov set may be bounded by a constant times "0 . This bound is optimal as the trivial example (y12 C y22 )/2 C " cos(x1 ) shows: the Hamiltonian is integrable and the phase portrait shows that the separatrices ofpthe pendulum y12 /2 C " cos x1 bound a region of area j"j with no KAM tori (as the librational curves within such region are not graphs over the angles).
The Differentiable Case: Moser’s Theorem J.K. Moser, in 1962, proved a perturbation (KAM) Theorem, in the framework of area-preserving twist mappings of an annulus35 [0; 1] S1 , for integrable analytic systems perturbed by a Ck perturbation, [42] and [43]. Moser’s original setup corresponded to the Hamiltonian case with d D 2 and the required smoothness was Ck with k D 333. Later, this number was brought down to 5 by H. Rüssmann, [53]. Moser’s original approach, similarly to the approach that led J. Nash to prove its theorem on the smooth embedding problem of compact Riemannian manifolds [48], is based on a smoothing technique (via convolutions), which re-introduces at each step of the Newton iteration a certain number of derivatives which one loses in the inversion of the small divisor operator. The technique, which we shall describe here, is again due to Moser [46] but is rather different from the original one and is based on a quantitative analytic KAM Theorem (in the style of statement in Remark 7–(i) above) in conjunction with a characterization of differentiable functions in terms of functions, which are real-analytic on smaller and smaller complex strips; see [44] and, for an abstract functional approach, [65], [66]. By the way, this approach, suitably refined, leads to optimal differentiability assumptions (i. e., the Hamiltonian may be assumed to be C ` with ` > 2d); see, [50] and the beautiful exposition [59], which inspires the presentation reported here. Let us consider a Hamiltonian H D K C"P (as in (17)) with K a real-analytic Kolmogorov normal form as in (10) d with ! 2 D; and Q real-analytic; P is assumed to be ` d d a C (R ; T ) function with ` D `(d; ) to be specified later36 . Remark 11 The analytic KAM theorem, we shall refer to is the quantitative Kolmogorov Theorem as stated in Remark 7–(i) above, with (69) strengthened by including in the left hand side of (69) also37 k@( id)k and k@(Q Q )k (where “@” denotes, here, “Jacobian” with regard to (y; x) for ( id) and “gradient” for (Q Q )).
Kolmogorov–Arnold–Moser (KAM) Theory
The analytic characterization of differentiable functions, suitable for our purposes, is explained in the following two lemmata38 Lemma 2 (Jackson, Moser, Zehnder) Let f 2 C l (Rd ) with l > 0. Then, for any > 0 there exists a real-analytic function f : Xd :D fx 2 C d : jImx j j < g ! C such that 8 sup j f j ck f kC 0 ; ˆ ˆ ˆ Xd
0, k f kC l (Rd ) CA :
(126)
Finally, if the f i ’s are periodic in some variable xj then so is f . Now, denote by X D Xd T d C 2d and define (compare Lemma 2) P j :D P j ;
j :D
1 : 2j
(127)
Claim M: If j"j is small enough and if ` > C 1, then there exists a sequence of Kolmogorov symplectic transformations f˚ j g j0 , j"j-close to the identity, and a sequence of Kolmogorov normal forms K j such that H j ı ˚ j D K jC1 on W jC1
(128)
where H j :D K C "P j ˚0 D 0
and ˚ j :D ˚ j1 ı j ; ( j 1)
j : W jC1 ! W˛ j ; ˚ j1 : W˛ j ! X j ; 1 j 1 and ˛ :D p ; 2 sup x2T d
j˚ j (0; x) ˚ j1 (0; x)j constj"j2(` ) j :
jC1
(129)
The proof of Claim M follows easily by induction39 from Kolmogorov’s Theorem (compare Remark 11) and Lemma 2. From Claim M and Lemma 3 (applied to f (x) D ˚ j (0; x) ˚0 (0; x) and l D ` , which may be assumed not integer) it then follows that ˚ j (0; x) converges in the C1 norm to a C1 function : T d ! Rd T d , which is "-close to the identity, and, because of (128), (x C ! t) D lim ˚ j (0; x C ! t) D lim Ht ı ˚ j (0; x) D Ht ı (x)
(130)
showing that (T d ) is a C1 KAM torus for H (note that the map is close to the trivial embedding x ! (y; x)). Future Directions In this section we review in a schematic and informal way some of the most important developments, applications and possible future directions of KAM theory. For exhaustive surveys we refer to [9], Sect. 6.3 of [6] or [60]. 1. Structure of the Kolmogorov set and Whitney smoothness The Kolmogorov set (i. e., the union of KAM tori), in nearly-integrable systems, tends to fill up (in measure) the whole phase space as the strength of the perturbation goes to zero (compare Remark 7–(v) and Remark 10). A natural question is: what is the global geometry of KAM tori? It turns out that KAM tori smoothly interpolate in the following sense. For " small enough, there exists a C 1 symplectic diffeomorphism of the phase space M D B T d of the nearly-integrable, non-degenerate Hamiltonians H D K(y) C "P(y; x) and a Cantor set C B such that, for each y 0 2 C , the set 1 (fy 0 g T d ) is a KAM torus for H}; in other words, the Kolmogorov set is a smooth, symplectic deformation of the fiber bundle C T d . Still another way of describing this result is that there exists a smooth function K : B ! R such that (K C "P) ı and K agree, together with their derivatives, on C T d : we may, thus, say that, in general, nearly-integrable Hamiltonian systems are integrable on Cantor sets of relative big measure. Functions defined on closed sets which admit Ck extensions are called Whitney smooth; compare [64], where H. Whitney gives a sufficient condition, based on Taylor uniform approximations, for a function to be Whitney Ck . The proof of the above result – given, independently, in [50] and [19] in, respectively, the differentiable and the analytic case – follows easily from the following lemma40 :
825
826
Kolmogorov–Arnold–Moser (KAM) Theory
Lemma 4 Let C Rd a closed set and let f f j g, f0 D 0, be a sequence of functions analytic on Wj :D [ y2C D(y; r j ). P < 1. Then, f j Assume that j1 supW j j f j f j1 jrk j converges uniformly to a function f , which is Ck in the sense of Whitney on C . Actually, the dependence upon the angles x 0 of is analytic and it is only the dependence upon y 0 2 C which is Whitney smooth (“anisotropic differentiability”, compare Sect. 2 in [50]). For more information and a systematic use of Whitney differentiability, see [9]. 2. Power series expansions KAM tori T!;" D " (T d ) of nearly-integrable Hamiltonians correspond to quasi-periodic trajectories z(t; ; ") D " ( C! t) D Ht (z(0; ; ")); compare items (d) and (e) of Sect. “Introduction” and Remark 2–(i) above. While the actual existence of such quasi-periodic motions was proven, for the first time, only thanks to KAM theory, the formal existence, in terms of formal "-power series41 was well known in the nineteenth century to mathematicians and astronomers (such as Newcombe, Lindstedt and, especially, Poincaré; compare [49], vol. II). Indeed, formal power solutions of nearly-integrable Hamiltonian equations are not difficult to construct (see, e. g., Sect. 7.1 of [12]) but direct proofs of the convergence of the series, i. e., proofs not based on Moser’s “indirect” argument recalled in Remark 7–(iii) but, rather, based upon direct estimates on the kth "-expansion coefficient, are quite difficult and were carried out only in the late eighties by H. Eliasson [27]. The difficulty is due to the fact that, in order to prove the convergence of the Taylor–Fourier expansion of such series, one has to recognize compensations among huge terms with different signs42 . After Eliasson’s breakthrough based upon a semi-direct method (compare the “Postscript 1996” at p. 33 of [27]), fully direct proofs were published in 1994 in [30] and [18]. 3. Non-degeneracy assumptions Kolmogorov’s non-degeneracy assumption (70) can be generalized in various ways. First of all, Arnold pointed out in [2] that the condition Kyy Ky ¤0; (131) det Ky 0 (this is a (d C 1) (d C 1) symmetric matrix where last column and last row are given by the (d C 1)-vector (K y ; 0)) which is independent from condition (70), is also sufficient to construct KAM tori. Indeed, (131) may be used to construct iso-energetic KAM tori, i. e., tori on a fixed energy level43 E.
More recently, Rüssmann [57] (see, also, [58]), using results of Diophantine approximations on manifolds due to Pyartly [52], formulated the following condition (the “Rüssmann non-degeneracy condition”), which is essentially necessary and sufficient for the existence of a positive measure set of KAM tori in nearly-integrable Hamiltonian systems: the image !(B) Rd of the unperturbed frequency map y ! !(y) :D K y (y) does not lie in any hyperplane passing through the origin. We simply add that one of the prices that one has to pay to obtain these beautiful general results is that one cannot fix the frequency ahead of time. For a thorough discussion of this topic, see Sect. 2 of [60]. 4. Some physical applications We now mention a short (and non-exhaustive) list of important physical application of KAM theory. For more information, see Sect. 6.3.9 of [6] and references therein. 4.1. Perturbation of classical integrable systems As mentioned above (Remark 1–(iii)), one of the main original motivations of KAM theory is the perturbation theory for nearly-integrable Hamiltonian systems. Among the most famous classical integrable systems we recall: one-degree-of freedom systems; Keplerian two-body problem, geodesic motion on ellipsoids; rotations of a heavy rigid body with a fixed point (for special values of the parameters: Euler’s, Lagrange’s, Kovalevskaya’s and Goryachev–Chaplygin’s cases); Calogero–Moser’s system of particles; see, Sect. 5 of [6] and [47]. A first step, in order to apply KAM theory to such classical systems, is to explicitly construct actionangle variables and to determine their analyticity properties, which is in itself a technically non-trivial problem. A second problem which arises, especially in Celestial Mechanics, is that the integrable (transformed) Hamiltonian governing the system may be highly degenerate (proper degeneracies – see Sect. 6.3.3, B of [6]), as is the important case of the planetary n-body problem. Indeed, the first complete proof of the existence of a positive measure set of invariant tori44 for the planetary (n C 1) problem (one body with mass 1 and n bodies with masses smaller than ") has been published only in 2004 [29]. For recent reviews on this topic, see [16]. 4.2. Topological trapping in low dimensions The general 2-degree-of-freedom nearly-integrable Hamiltonian exhibits a kind of particularly strong stability: the phase space is 4-dimensional and the energy levels are 3-dimensional; thus KAM tori
Kolmogorov–Arnold–Moser (KAM) Theory
(which are two-dimensional and which are guaranteed, under condition (131), by the iso-energetic KAM theorem) separate the energy levels and orbits lying between two KAM tori will remain forever trapped in the invariant region. In particular the evolution of the action variables stays forever close to the initial position (“total stability”). This observation is originally due to Arnold [2]; for recent applications to the stability of threebody problems in celestial mechanics see [13] and item 4.4 below. In higher dimension this topological trapping is no longer available, and in principle nearby any point in phase space it may pass an orbit whose action variables undergo a displacement of order one (“Arnold’s diffusion”). A rigorous complete proof of this conjecture is still missing45 . 4.3. Spectral Theory of Schrödinger operators KAM methods have been applied also very successfully to the spectral analysis of the one-dimensional Schrödinger (or “Sturm–Liouville”) operator on the real line R L :D
d2 C v(t) ; dt 2
t 2R:
(132)
If the “potential” v is bounded then there exists a unique self-adjoint operator on the real Hilbert space L2 (R) (the space of Lebesgue square-integrable functions on R) which extends L above on C02 (the space of twice differentiable functions with compact support). The problem is then to study the spectrum (L) of L; for generalities, see [23]. If v is periodic, then (L) is a continuous band spectrum, as it follows immediately from Floquet theory [23]. Much more complicated is the situation for quasi-periodic potentials v(t) :D V (! t) D V (!1 t; : : : ; !n t), where V is a (say) real-analytic function on T n , since small-divisor problems appear and the spectrum can be nowhere dense. For a beautiful classical exposition, see [47], where, in particular, interesting connections with mechanics are discussed46 ; for deep developments of generalization of Floquet theory to quasi-periodic Schrödinger operators (“reducibility”), see [26] and [7]. 4.4. Physical stability estimates and break-down thresholds KAM Theory is perturbative and works if the parameter " measuring the strength of the perturbation is small enough. It is therefore a fundamental question: how small " has to be in order for KAM
results to hold. The first concrete applications were extremely discouraging: in 1966, the French astronomer M. Hénon [32] pointed out that Moser’s theorem applied to the restricted three-body problem (i. e., the motion of an asteroid under the gravitational influence of two unperturbed primary bodies revolving on a given Keplerian ellipse) yields existence of invariant tori if the mass ratio of the primaries is less than47 1052 . Since then, much progress has been made and very recently, in [13], it has been shown via a computer-assisted proof48 , that, for a restricted-three body model of a subsystem of the Solar system (namely, Sun, Jupiter and Asteroid Victoria), KAM tori exist for the “actual” physical values (in that model the Jupiter/Sun mass ratio is about 103 ) and, in this mathematical model – thanks to the trapping mechanism described in item 4.2 above – they trap the actual motion of the subsystem. From a more theoretical point of view, we notice that, (compare Remark 2–(ii)) KAM tori (with a fixed Diophantine frequency) are analytic in "; on the other hand, it is known, at least in lower dimensional settings (such as twist maps), that above a certain critical value KAM tori (curves) cannot exist ([39]). Therefore, there must exist a critical value "c (!) (“breakdown threshold”) such that, for 0 " < "c !, the KAM torus (curve) T!;" exists, while for " > "c (!) does not. The mathematical mechanism for the breakdown of KAM tori is far from being understood; for a brief review and references on this topic, see, e. g., Sect. 1.4 in [13]. 5. Lower dimensional tori In this item we consider (very) briefly, the existence of quasi-periodic solutions with a number of frequencies smaller than the number of degrees of freedom49 . Such solutions span lower dimensional (non Lagrangian) tori. Certainly, this is one of the most important topics in modern KAM theory, not only in view of applications to classical problems, but especially in view of extensions to infinite dimensional systems, namely PDEs (Partial Differential Equations) with a Hamiltonian structure; see, item 6 below. For a recent, exhaustive review on lower dimensional tori (in finite dimensions), we refer the reader to [60]. In 1965 V.K. Melnikov [41] stated a precise result concerning the persistence of stable (or “elliptic”) lower dimensional tori; the hypotheses of such results are, now, commonly referred to as “Melnikov conditions”. However, a proof of Melnikov’s statement was given only later by Moser [45] for the case n D d 1 and, in
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Kolmogorov–Arnold–Moser (KAM) Theory
the general case, by H. Eliasson in [25] and, independently, by S.B. Kuksin [37]. The unstable (“partially hyperbolic”) case (i. e., the case for which the lower dimensional tori are linearly unstable and lie in the intersection of stable and unstable Lagrangian manifolds) is simpler and a complete perturbation theory was already given in [45], [31] and [66] (roughly speaking, the normal frequencies to the torus do not resonate with the inner (or “proper”) frequencies associated with quasi-periodic motion). Since then, Melnikov conditions have been significantly weakened and much technical progress has been made; see [60], Sects. 5, 6 and 7, and references therein. To illustrate a typical situation, let us consider a Hamiltonian system with d D n C m degrees of freedom, governed by a Hamiltonian function of the form H(y; x; v; u; ) D K(y; v; u; ) C "P(y; x; v; u; ) ;
(133)
where (y; x) 2 T n Rn , (v; u) 2 R2m are pairs of standard symplectic coordinates and is a real parameter running over a compact set ˘ Rn of positive Lebesgue measure50; K is a Hamiltonian admitting the n-torus T0n () :D fy D 0g T n fv D u D 0g ;
2˘;
as invariant linearly stable invariant torus and is assumed to be in the normal form: m
K D E() C !() y C
1X ˝ j ()(u2j C v 2j ) : (134) 2 jD1
The Kt flow decouples in the linear flow x 2 T n ! x C !()t times the motion of m (decoupled) harmonic oscillators with characteristic frequencies ˝ j () (sometimes referred to as normal frequencies). Melnikov’s conditions (in the form proposed in [51]) reads as follows: assume that ! is a Lipschitz homeomorphism; let ˘ k;l denote the “resonant parameter set” f 2 ˘ : !() k C ˝ () D 0g and assume (
˝ i () > 0 ; ˝ i () ¤ ˝ j () ; 8 2 ˘ ; 8i ¤ j meas ˘ k;l D 0 ; 8k 2 Z n nf0g ; 8l 2 Z m : jlj 2 : (135) Under these assumptions and if j"j is small enough, there exists a (Cantor) subset of parameters ˘ ˘ of positive Lebesgue measure such that, to each 2 ˘ , there
corresponds a n-dimensional, linearly stable H-invariant torus T"n () on which the H flow is analytically conjugated to x ! x C ! ()t where ! is a Lipschitz homeomorphism of ˘ assuming Diophantine values and close to !. This formulation has been borrowed from [51], to which we refer for the proof; for the differentiable analog, see [22]. Remark 12 The small-divisor problems arising in the perturbation theory of the above lower dimensional tori are of the form !kl ˝ ;
jlj 2 ; jkj C jlj ¤ 0 ;
(136)
where one has to regard the normal frequency ˝ as functions of the inner frequencies ! and, at first sight, one has – in J. Moser words – a lack-of-parameter problem. To overcome this intrinsic difficulty, one has to give up full control of the inner frequencies and construct, iteratively, n-dimensional sets (corresponding to smaller and smaller sets of -parameters) on which the small divisors are controlled; for more motivations and informal explanations on lower dimensional small divisor problems, see, Sects. 5, 6 and 7 of [60]. 6. Infinite dimensional systems As mentioned above, the most important recent developments of KAM theory, besides the full applications to classical n-body problems mentioned above, is the successful extension to infinite dimensional settings, so as to deal with certain classes of partial differential equations carrying a Hamiltonian structure. As a typical example, we mention the non-linear wave equation of the form u t t u x x C V (x)u D f (u) ; f (u) D O(u2 ) ; 0 < x < 1 ; t 2 R :
(137)
These extensions allowed, in the pioneering paper [63], establishing the existence of small-amplitude quasi-periodic solutions for (137), subject to Dirichlet or Neumann boundary conditions (on a finite interval for odd and analytic nonlinearities f ); the technically more difficult periodic boundary condition case was considered later; compare [38] and references therein. A technical discussion of these topics goes far beyond the scope of the present article and, for different equations, techniques and details, we refer the reader to the review article [38].
Kolmogorov–Arnold–Moser (KAM) Theory
A The Classical Implicit Function Theorem Here we discuss the classical Implicit Function Theorem for complex functions from a quantitative point of view. The following Theorem is a simple consequence of the Contraction Lemma, which asserts that a contraction ˚ on a closed, non-empty metric space51 X has a unique fixed point, which is obtained as lim j!1 ˚ j (u0 ) for any52 u0 2 X. As above, D n (y0 ; r) denotes the ball in C n of center y0 and radius r. Theorem 3 (Implicit Function Theorem) Let F : (y; x) 2 D n (y0 ; r) D m (x0 ; s) C nCm ! F(y; x) 2 C n be continuous with continuous Jacobian matrix F y ; assume that F y (y0 ; x0 ) is invertible and denote by T its inverse; assume also that sup
D(y 0 ;r)D(x 0 ;s)
1 ; 2 r : sup jF(y0 ; x)j 2kTk D(x 0 ;s)
k1n TF y (y; x)k
(138)
sup jgj 2kTk sup jF(y0 ; )j : D(x 0 ;s)
Additions: (i)
If F is periodic in x or/and real on reals, then (by uniqueness) so is g; (ii) If F is analytic, then so is g (Weierstrass Theorem, since g is attained as uniform limit of analytic functions); (iii) The factors 1/2 appearing in the right-hand sides of (138) may be replaced by, respectively, ˛ and ˇ for any positive ˛ and ˇ such that ˛ C ˇ D 1. Taking n D m and F(y; x) D f (y) x for a given C 1 (D(y0 ; r); C n ) function, one obtains the Theorem 4 (Inverse Function Theorem) Let f : y 2 D n (y0 ; r) ! C n be a C1 function with invertible Jacobian f y (y0 ) and assume that sup k1n T f y k
Then, all solutions (y; x) 2 D(y0 ; r) D(x0 ; s) of F(y; x) D 0 are given by the graph of a unique continuous function g : D(x0 ; s) ! D(y0 ; r) satisfying, in particular, D(x 0 ;s)
which implies that y1 D g(x1 ) and that all solutions of F D 0 in D(y0 ; r) D(x0 ; s) coincide with the graph og g. Finally, (139) follows by observing that kg y0 k D k˚ (g) y0 k k˚(g) ˚(y0 )k C k˚ (y0 ) y0 k 1 2 kg y 0 k C kTkkF(y 0 ; )k, finishing the proof.
(139)
Proof Let X D C(D m (x0 ; s); D n (y0 ; r)) be the closed ball of continuous function from D m (x0 ; s) to D n (y0 ; r) with respect to the sup-norm k k (X is a non-empty metric space with distance d(u; v) :D ku vk) and denote ˚ (y; x) :D y TF(y; x). Then, u ! ˚(u) :D ˚ (u; ) maps C(D m (x0 ; s)) into C(C m ) and, since @ y ˚ D 1n TF y (y; x), from the first relation in (138), it follows that u ! ˚(u) is a contraction. Furthermore, for any u 2 C(D m (x0 ; s); D n (y0 ; r)), j˚(u) y0 j j˚(u) ˚(y0 )j C j˚(y0 ) y0 j 1 ku y0 k C kTkkF(y0 ; x)k 2 1 r r C kTk Dr; 2 2kTk showing that ˚ : X ! X. Thus, by the Contraction Lemma, there exists a unique g 2 X such that ˚(g) D g, which is equivalent to F(g; x) D 0 8x. If F(y1 ; x1 ) D 0 for some (y1 ; x1 ) 2 D(y0 ; r) D(x0 ; s), it follows that jy1 g(x1 )j D j˚(y1 ; x1 ) ˚(g(x1 ); x1 )j ˛jy1 g(x1 )j,
D(y 0 ;r)
1 ; 2
T :D f y (y0 )1 ;
(140)
then there exists a unique C1 function g : D(x0 ; s) ! D(y0 ; r) with x0 :D f (y0 ) and s :D r/(2kTk) such that f ı g(x) D id D g ı f . Additions analogous to the above also hold in this case. B Complementary Notes 1
2
Actually, the first instance of a small divisor problem solved analytically is the linearization of the germs of analytic functions and is due to C.L. Siegel [61]. The well-known Newton’s tangent scheme is an algorithm, which allows us to find roots (zeros) of a smooth function f in a region where the derivative f 0 is bounded away from zero. More precisely, if xn is an “approximate solution” of f (x) D 0, i. e., f (x n ) :D " n is small, then the next approximation provided by Newton’s tangent scheme is x nC1 :D x n f (x n )/ f 0 (x n ) [which is the intersection with x-axis of the tangent to the graph of f passing through (x n ; f (x n ))] and, in view of the definition of "n and Taylor’s formula, one has that "nC1 :D f (x nC1) D 1 00 0 2 2 2 f ( n )" n /( f n (x n ) (for a suitable n ) so that " nC1 D 2 2 O(" n ) D O("1 ) and, in the iteration, xn will converge (at a super-exponential rate) to a root x¯ of f . This type of extremely fast convergence will be typical in the analyzes considered in the present article.
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3
4
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The elements of T d are equivalence classes x D x¯ C 2Zd with x¯ 2 Rd . If x D x¯ C 2Zd and y D y¯ C 2Zd are elements of T d , then their distance d(x; y) is given by minn2Zd j x¯ y¯ C 2 nj where j j denotes the standard euclidean norm in Rn ; a smooth (analytic) function on T d may be viewed as (“identified with”) a smooth (analytic) function on Rd with period 2 in each variable. The torus T d endowed with the above metric is a real-analytic, compact manifold. For more information, see [62]. A symplectic form on an (even dimensional) manifold is a closed, non-degenerate differential 2-form. The symplectic form ˛ D dy ^ dx is actually exact symP plectic, meaning that ˛ D d( iD1 y i dx i ). For general information see [5]. For general facts about the theory of ODE (such as Picard theorem, smooth dependence upon initial data, existence times, . . . ) see, e. g., [23]. This terminology is due to that fact the the xj are “adimensional” angles, while analyzing the physical dimensions of the quantities appearing in Hamilton’s equations one sees that dim(y) dim(x) D dim H dim(t) so that y has the dimension of an energy (the Hamiltonian) times the dimension of time, i. e., by definition, the dimension of an action. This terminology is due to the fact that a classical mechanical system of d particles of masses m i > 0 and subject to a potential V(q) with q 2 A Rd is govP erned by a Hamiltonian of the form djD1 p2j /2m j C V (q) and d may be interpreted as the (minimal) number of coordinates necessary to physically describe the system. To be precise, (6) should be written as y(t) D v(T d (! t)), x(t) D T d (! t C u(T d (! t))) where T d denotes the standard projection of Rd onto T d , however we normally omit such a projection. As standard, U denotes the (d d) Jacobian matrix with entries (@U i )/(@ j ) D ı i j C (@u i )/(@ j ). For generalities, see [5]; in particular, a Lagrangian manifold L M which is a graph over T d admits a “generating function”, i. e., there exists a smooth function g : T d ! R such that L D f(y; x) : y D g x (x), x 2 T d g. Compare [54] and references therein. We remark that, if B(!0 ; r) denote the ball in Rd of radius r centered at ! 0 and fix > d 1, then one can prove that the d can be bounded by Lebesgue measure of B(y0 ; r)nD; d1 for a suitable constant cd depending only on d; cd r for a simple proof, see, e.g, [21]. The sentence “can be put into the form” means “there exists a symplectic diffeomorphism : (y; x) 2 M !
13
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15
16
17
(; ) 2 M such that H ı has the form (10)”; for multi-indices ˛, j˛j D ˛1 C C ˛d and @˛y D ˛ @˛y11 @ y dd ; the vanishing of the derivatives of a function f (y) up to order k in the origin will also be indicated through the expression f D O(jyj kC1 ). Notation: If A is an open set and p 2 N, then the Cp -norm of a function f : x 2 A ! f (x) is defined as k f kC p (A) : supj˛jp supA j@˛x f j. Notation: If f is a scalar function f y is a d-vector; f yy the Hessian matrix ( f y i y j ); f yyy the symmetric 3-tensor of third derivatives acting as follows: P f y y y a b c :D di; j;kD1 (@3 f )/(@y i @y j @y k )a i b j c k . Notation: If f is (a regular enough) function over d coefficients are defined as f np :D T R , its Fourier i nx dx/(2)d ; where, as usual, i D f (x)e 1 Td denotes imaginary unit; for general information about Fourier series see, e. g., [34]. The choice of norms on finite dimensional spaces (Rd , C d , space of matrices, tensors, etc.) is not particularly relevant for the analysis in this article (since changing norms will change d-depending constants); however for matrices, tensors (and, in general, linear operators), it is convenient to work with the “operator norm”, i. e., the norm defined as kLk D supu¤0 kLuk/kuk, so that kLuk kLkkuk, an estimate, which will be constantly be used; for a general discussion on norms, see, e. g., [36]. As an example, let us work out the first two estimates, i. e., the estimates on ks x k¯ and jbj: actually these estimates will be given on a larger intermediate domain, namely, Wı/3 , allowing to give the remaining bounds on the smaller domain W¯ (recall that W s denotes the complex domain D(0; s) Tsd ). Let f (x) :D P(0; x)hP(0; )i. By definition of kk and M, it follows that k f k kP(0; x)k CkhP(0; )ik 2M. By (P5) with p D 1 and 0 D ı/3, one gets 2M k 1 k 1 ks x k ı B¯ 1 ; 3 ı 3 which is of the form (53), provided c¯ (B¯ 1 2 3 k 1 )/ and ¯ k1 . To estimate b, we need to bound first jQ y y (0; x)j and jPy (0; x)j for real x. To do this we can use Cauchy estimate: by (P4) with p D 2 and, respectively, p D 1, and 0 D 0, we get kQ y y (0; )k0 mB2 C 2 mB2 Cı 2 ; kPy (0; x)k0 mB1 Mı
1
and
;
where m D m(d) 1 is a constant which depend on the choice of the norms, (recall also that ı < ). Putting these bounds together, one gets that jbj can be
Kolmogorov–Arnold–Moser (KAM) Theory
18
19
20
21
bounded by the r.h.s. of (53) provided c¯ m(B2 B¯ 1 2 3 k 1 1 C B1 ), 2 and ¯ k1 C2. The other bounds in (53) follow easily along the same lines. We sketch here the proof of Lemma 1. The defining relation " ı ' D id implies that ˛(x 0 ) D a(x 0 C "˛(x 0 )), where ˛(x 0 ) is short for ˛(x 0 ; ") and that equation is a fixed point equation for the non-linear operator f : u ! f (u) :D a(id C "u). To find a fixed point for this equation one can use a standard contraction Lemma (see [36]). Let Y denote the closed ball (with respect to the sup-norm) of continuous func¯ By (54), tions u : Td0 ! C d such that kuk 0 L. ¯ for any jIm(x 0 C "u(x 0 ))j < 0 C "0 L¯ < 0 C ı/3 D , u 2 Y, and any x 0 2 Td0 ; thus, k f (u)k 0 ;" kak¯ L¯ by (53), so that f : Y ! Y; notice that, in particular, this means that f sends periodic functions into periodic functions. Moreover, (54) implies also that f is a contraction: if u; v 2 Y, then, by the mean value the¯ orem, j f (u) f (v)j Lj"jjuvj (with a suitable choice of norms), so that, by taking the sup-norm, one has ¯ vk 0 < 1 ku vk 0 showk f (u) f (v)k 0 < "0 Lku 3 ing that f is a contraction. Thus, there exists a unique ˛ 2 Y such that f (˛) D ˛. Furthermore, recalling that the fixed point is achieved as the uniform limit limn!1 f n (0) (0 2 Y) and since f (0) D a is analytic, so is f n (0) for any n and, hence, by Weierstrass Theorem on the uniform limit of analytic function (see [1]), the limit ˛ itself is analytic. In conclusion, ' 2 B 0 and (55) holds. Next, for (y 0 ; x) 2 W¯ , by (53), one has jy 0 C "ˇ(y 0 ; x)j < ¯ C "0 L¯ < ¯ C ı/3 D so that (56) holds. Furthermore, since k"a x k¯ < "0 L¯ < 1/3 the matrix 1d C "a x is invertible with inverse given by the “Neumann series” (1d C "a x )1 D 1d C P1 k k kD1 (1) ("a x ) D: 1d C"S(x; "), so that (57) holds. The proof is finished. From (59), it follows immediately that h@2y 0 Q1 (0; )i D h@2y Q(0; )iC"h@2y 0 e Q(0; )i D T 1 (1d C"Th@2y 0 e Q(0; )i) 1 D: T (1d C "R) and, in view of (51) and (59), we see that kRk < L/(2C). Therefore, by (60), "0 kRk < 1/6 < 1/2, implying that (1C"R) is invertible P k k k and (1d C "R)1 D 1d C 1 kD1 (1) " R D: 1 C "D with kDk kRk/(1 j"jkRk) < L/C. In conclusion, T1 D (1 C "R)1 T D T C "DT D: T C "e T, ke Tk kDkC (L/C)C D L. Actually, there is quite some freedom in choosing the sequence f j g provided the convergence is not too fast; for general discussion, see, [56], or, also, [10] and [14]. In fact, denoting by B the real d-ball centered at 0 and of radius for 2 (0; 1), from Cauchy estimate (47) with D and 0 D , one has k
ˇ
22 23
24
25
idkC p (B T d ) D supB T d supj˛jCjˇ jp j@˛y @x ( ˇ id)j supj˛jCjˇ jp k@˛y @x ( id)k B p k idk 1/( ) p const p j"j with const p :D B p DBM1/( ) p . An identical estimate holds for kQ QkC p (B T d ) . Also very recently "-power series expansions have been shown to be a very powerful tool; compare [13]. A function f : A Rn ! Rn is Lipschitz on A if there exists a constant (“Lipschitz constant”) L > 0 such that j f (x) f (y)j Ljx yj for all x; y 2 A. For a general discussion on how Lebesgue measure changes under Lipschitz mappings, see, e. g., [28]. In fact, the dependence of on y¯ is much more regular, compare Remark 11. In fact, notice that inverse powers of appear through (48) (inversion of the operator D! ), therefore one sees that the terms in the first line of (53) may be replaced by c˜ 2 (in defining a one has to apply the 1 twice) but then in P (1) (see (26)) there operator D! appears kˇk2 , so that the constant c in the second line of (53) has the form (72); since < 1, one can replace in (53) c with cˆ 4 as claimed. Proof of Claim C Let H0 :D H, E0 :D E, Q0 :D Q, K0 :D K, P0 :D P, 0 :D and let us assume (inductive hypothesis) that we can iterate the Kolmogorov transformation j times obtaining j symplectic transformations iC1 : W iC1 ! W i , for 0 i j 1, and j i
Hamiltonians H iC1 D H i ı iC1 D K i C "2 Pi realanalytic on W i such that j!j; jE i j; kQ i k i ; kTi k < C ; j"j2 L i :D j"j2 cC ı0 2 i M i i
i
ıi ; 3
(*)
80 i j 1 : By ( ), Kolmogorov iteration (Step 2) can be applied to H i and therefore all the bounds described in paragraph Step 2 hold (having replaced H; E; : : : ; ; ı; H 0 ; E 0 ; : : : ; 0 with, respectively, H i ; E i ; : : : ; i ; ı i ; H iC1 ; E iC1 ; : : : ; iC1 ); in particular (see (61)) one has, for 0 i j 1 (and for any j"j "0 ), 8 2i ˆ ˆ jE iC1 j jE i j C j"j L i ; ˆ ˆ ˆ 2i < kQ k iC1 iC1 kQ i k i C j"j L i ; (C.1) ˆ 2i ˆ idk j"j L k ˆ iC1 i iC1 ˆ ˆ : M iC1 M i L i Observe that the definition of D, B and LI , j j 2j i j"j2 L j (3Cı 1 j ) D: DB j"j M j , so that L i < DB M i ,
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Kolmogorov–Arnold–Moser (KAM) Theory
thus by the second line in (C:1), for any 0 i j 1, iC1 i j"j2 M iC1 < DB i (M i j"j2 )2 , which iterated, yields (66) for 0 i j. Next, we show that, thanks to (65), ( ) holds also for i D j (and this means that Kolmogorov’s step can be iterated an infinite number of times). In fact, by ( ) and the definition of C P P j1 i in (64): jE j j jEj C iD0 "20 L i jEj C 13 i0 ı i < P jEjC 16 i1 2i < jEjC1 < C. The bounds for kQ i k and kTi k are proven in an identical manner. Now, j j 2j by (66) iD j and (65), j"j2 L j (3ı 1 j ) D DB j"j M j
30
j
26
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DB j (DB"0 M)2 /(DB jC1 ) 1/B < 1, which implies the second inequality in ( ) with i D j; the proof of the induction is finished and one can construct an infinite sequence of Kolmogorov transformations satisfying ( ), (C:1) and (66) for all i 0. To check (67), i i we observe that j"j2 L i D ı0 /(3 2 i )DB i j"j2 M i i (1/2 iC1 )(j"jDBM)2 (j"jDBM/2) iC1 and therefore P P 2i (j"jDBM/2) i j"jDBM. Thus, i0 j"j L i Pi1 2i ˜ kQ Q k i0 kQ i k i j"j L i j"jDBM; and analogously for jE E j and kT T k. To estimate k idk , observe that k˚ i idk i k˚ i1 ı i i i k i C k i idk i k˚ i1 idk i1 C j"j2 L i , P k which iterated yields k˚ i idk i ikD0 j"j2 L k j"jDBM: taking the limit over i completes the proof of (67) and the proof of Claim C. In fact, observe: (i) given any integer vector 0 ¤ n 2 Zd with d 2, one can find 0 ¤ m 2 Zd such n m D 0; (ii) the set ftn : t > 0 and n 2 Zd g is dense in Rd ; (iii) if U is a neighborhood of y0 , then K y (U) is a neighborhood of ! D K y (y0 ). Thus, by (ii) and (iii), in K y (U) there are infinitely many points of the form tn with t > 0 and n 2 Zd to which correspond points y(t; n) 2 U such that K y (y(t; n)) D tn and for any of such points one can find, by (i), m 2 Z such that m n D 0, whence K y (y(t; n)) m D tn m D 0. This fact was well known to Poincaré, who based on the above argument his non-existence proof of integral of motions in the general situation; compare Sect. 7.1.1, [6]. Compare (90) but observe, that, since Pˆ is a trigonometric polynomial, in view of Remark 9–(ii), g in (96) defines a real-analytic function on D(y0 ; r¯) Td0 with a suitable r¯ D r¯(") and 0 < . Clearly it is important to see explicitly how the various quantities depend upon "; this is shortly discussed after Proposition 2. P jnjı/2 Me(ı/4)N ˇ r;ı/2 M In fact: kPk jnj>N e P P jnjı/4 Me(ı/4)N jnj>0 ejnjı/4 jnj>N e const Me(ı/4)N ı d j"jM if (106) holds and N is taken as in (104).
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35
36
Apply the IFT of Appendix “A The Classical Implicit Function Theorem” to F(y; ) :D K y (y) C @ y P0 (y) K y (y0 ) defined on D d (y0 ; r¯) D1 (0; j"j): using the mean value theorem, Cauchy estimates and (114), k1d TF y k k1d TK y y k C j"jk@2y P0 k kTkkK y y y k¯r C kTkj"jk@2y P0 k C 2 2¯r/r C Cj"j4/r2 M 14 C 18 < 12 ; also: 2kTk kF(y0 ; k D 2kTkjj@ y P0 (y0 )k < 2Cj"jM2/r 2CM¯r1 j"j < 14 r¯ (where last inequality is due to (114)), showing that conditions (138) are fulfilled. Equation (111) comes from (139) and (113) follows easily by repeating the above estimates. Recall note 18 and notice that (1d C A)1 D 1d C D with kDk kAk/(1 kAk) 2kAk 20C 3 Mj"j, where last two inequalities are due to (113). Lemma 1 can be immediately extended to the y 0 -dependent case (which appear as a dummy parameter) as far as the estimates are uniform in y 0 (which is the case). By (118) and (54), j"jkg x kr¯;¯ j"jrL r/2 so that, by (116), if y 0 2 Dr¯/2 (y1 ), then y 0 C"g x (y 0 ; '(y 0 ; x 0 )) 2 Dr (y0 ). The first requirement in (123) is equivalent to require that r0 r, which implies that if r¯ is defined as the r.h.s. of (108), then r¯ r/2 as required in (110). Next, the first requirement in (114) at the ( jC1)th step of the iteration translates into 16C 2 r jC1 /r j 1, which is satisfied, since, by definition, r jC1 /r j D (1/(2 ))C1 (1/(2 ))2 D 1/(36C 2 ) < 1/(16C 2 ). The second condition in (114), which at the ( j C 1)th step, reads 2j 2j 2CM j r2 jC1 j"j is implied by j"j L j ı j /(3C) (corresponding to (54)), which, in turn, is easily controlled along the lines explained in note 25. An area-preserving twist mappings of an annulus A D [0; 1] S1 , (S1 D T 1 ), is a symplectic diffeomorphism f D ( f1 ; f2 ) : (y; x) 2 A ! f (y; x) 2 A, leaving invariant the boundary circles of A and satisfying the twist condition @ y f2 > 0 (i. e., f twists clockwise radial segments). The theory of area preserving maps, which was started by Poincaré (who introduced such maps as section of the dynamics of Hamiltonian systems with two degrees of freedom), is, in a sense, the simplest nontrivial Hamiltonian context. After Poincaré the theory of area-preserving maps became, in itself, a very rich and interesting field of Dynamical Systems leading to very deep and important results due to Herman, Yoccoz, Aubry, Mather, etc; for generalities and references, see, e. g., [33]. It is not necessary to assume that K is real-analytic, but it simplify a little bit the exposition. In our case, we shall see that ` is related to the number in (66). We
Kolmogorov–Arnold–Moser (KAM) Theory
37
38
39
40
41
recall the definition of Hölder norms: If ` D `0 C with `0 2 ZC and 2 (0; 1), then k f kC ` :D k f kC ` C supj˛jD`0 sup0<jxyj 0; c4 D 0, Eq. (131) admits a Weiertrass elliptic functions type solution p c3 4c1 4c0 ; ; : (144) () D } 2 c3 c3
If we substitute the following new transform into Eq. (149)
Let’s specifically see how the algebraic method works. For a given nonlinear differential equation, say, in two variables x; t
then Eq. (149) can be transformed into an ordinary differential equation
F(u; u t ; u x ; u x x ; u x t ; u t t ; : : :) D 0
(145)
where F is a polynomial function with respect to the indicated variables or some function which can be reduced to a polynomial function by using some transformations. By using the traveling wave transformation u D u(); D x t ;
(146)
Eq. (145) is reduced to an ordinary differential equation with constant coefficients G(U; U 0 ; U 00 ; U 000 ; : : :) D 0
(147)
A transformation was presented by Fan [5] in the form u(x) D A0 C
n X
A i i1 () ;
(148)
iD1
with the new variable () satisfying Eq. (131), where A0 ; A i ; d j are constants. Substituting (148) into (147) along with (131), we can determine the parameter n in (148). And then by substituting (148) with the concrete n into (147) and equating the coefficients of these terms ! i ! 0 j (i D 0; 1; 2; : : : ; j D 0; 1), we obtain a system of algebraic equations with respect to other parameters A0 ; A i ; d j ; . By solving the system, if available, we may determine these parameters. Therefore we establish a transformation (147) between (146) and (148). If we know the solutions of (148), then we can obtain the solutions of (147) (or (145)) by using (146).
c i D consts ; i D 1; 2; : : : ; r :
n
() D (g()) m ;
(149)
(150)
2 r X n(i2) d c i (g())2C m : g() D m2 n2 d
(151)
iD0
We give the proof of Theorem 4 [40] by using Maple as follows: Proof Step 1 Importing the following Maple program at the Maple Command Window eq :D diff(phi(x i); x i)2 sum(c[i] phi(x i) i ; i D 0::r); Eq. (149) is displayed at a computer screen (after implementing the above program) as follows: eq :D
2 X r d c i (()) i : () d iD0
Step 2 Importing the following Maple program at the Maple Command Window eq :D subs(phi(x i) D (g(x i))( n/m); eq) ; the following result is displayed at the screen (after running the above program): 2 n 2 d eq :D (g()) m n2 g() m2 (g())2 d r X n i c i (g()) m : iD0
855
856
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
Step 3 Importing the following Maple program at the Maple Command Window eq :D simplify(eq) ;
eq :D expand(eq) ; eq :D numer(eq) ;
the following result is displayed at the screen (after running the above program): 2 d n 2 g() eq :D (g()) m n2 d r X n i c i (g()) m m2 (g())2 : iD0
Step 4 Importing the following Maple program at the Maple Command Window eq :D subs(diff(g(x i); x i) D G(x i); g(x i) D g; eq) ; eq :D eq (g ( n/m))( 2) ;
C i n/m); i D 0::r)) m2 ; eq :D simplify(eq); the following result is displayed at the screen (after running the above program): r X
nm m
eq :D n2 (G())2
ci g
2nC2mCni m
m2 :
Step 5 Importing the following Maple program at the Maple Command Window eq :D subs(g D g(x i); G(x i) D diff(g(x i); x i); eq) ; the following result is displayed at the screen (after running the above program): eq :D n2
g(x i)( (2 nC2 mCn i)/m); i D 0::r) m2 ; Eq. (151) is displayed at a computer screen (after implementing the above program) as follows 2 X r 2nC2mCni d 2 m c i (g()) m2 g() eq :D n d iD0
eq621 :D subs(r D 6; m D 2; n D 1; eq) ;
iD0
Step 1 Import the following Maple program at the Maple Command Window
Step 2 Import the following Maple programs with some values of the degree r and parameter (n; m) in new transform (150) at the Maple Command Window. For example,
n i c i g m m2
iD0 r X
According to Theorem 4 in Subsect. “The the Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation”, in [40] we first presented the following mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree. The validity and reliability of our method are tested by its application to a firstorder nonlinear ordinary differential equation with six degrees [40]. Now, we simply describe our mechanization method as follows:
eq :D n2 diff(g(x i); x i)2 sum(c[i]
eq :D simplify(eq) ;
eq :D n2 G(x i)2 (sum(c[i] g ( (2 (n m)/m)
eq :D n2 (G())2 g 2
A New Mechanization Method to Find the Exact Solutions of a First-Order Nonlinear Ordinary Differential Equation with any Degree Using Maple and Its Application
2 X r 2nC2mCni d m c i (g()) m2 : g() d
eq621 :D simplify(eq621) ; the following result is displayed at the screen (after running the above program): eq621 :D
2 d g() 4c0 g() d
4c1 (g())3/2 4c2 (g())2 4c3 (g())5/2 4c4 (g())3 4c5 (g())7/2 4c6 (g())4 :
iD0
(153)
(152) We can reduce Eq. (152) to (151). Remark 6 The above transformation (150), Theorem 4, and its proof by means of Maple were first presented by us in Yu-Jie Ren’s Ph.D Dissertation of Dalian University of Technology [40].
Step 3 According to the output results in Step 2, we choose the coefficients of ci, i D 1; 2; : : : ; m(< n) to be zero. Then we import their Maple program. For example, we import the Maple program as follows: eq246 :D subs(c[0] D 0; c[1] D 0; c[3] D 0; c[5] D 0; eq621) ;
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
the following result is displayed at the screen (after running the above program):
4c6 c2 e4C 1
2 d g() d
eq246 :D
g4 () D 4 p
c 2 Ce4
p
p
c 2 (C 1C) c 2 p c 2 2e2 c 2 (C 1C) c
e2
4 Cc4
2 e4C 1
p
c2
:
(160)
4c2 (g())2 4c4 (g())3 4c6 (g())4 :
(154)
Step 4 Import the Maple program for solving the output equation in Step 3. For example, we import the Maple program for solving the output Eq. (154) as follows:
Step 5 We discuss the above solutions under different conditions in Step 3. For example, we discuss the solutions under different conditions c42 4c6 c2 < 0 or c42 4c6 c2 > 0 or c42 4c6 c2 D 0 or c2 < 0 or c2 > 0 or c2 D 0. When
dsolve(eq246; g(x i)); the following formal solutions of (154) are displayed at the screen (after running the above program):
g1 () D 1/2
g2 () D 1/2
c4
q
c42 4c6 c2
c6 q c4 C c42 4c6 c2
p
c6 2
p
;
(155)
;
2
p 2 p 2 g4 () D 4 e c 2 c2 eC 1 c 2 0 11 p 4 p 2 e c 2 e c 2 c4 B 2C @4c6 c2 p 4 C 2 p 2 c4 A : C c C c e 1 2 e 1 2 (158) Import the Maple program for reducing the above solutions of (154). For example, we import the Maple program for reducing the (157) and (158) as follows: g3 :D simplify(g3); g4 :D simplify(g4) ; the following results are displayed at a computer screen (after running the above program):
4c6 c2 e4
e2 c2
C e4C 1
p
p
we may import the following Maple program: assume(c[2] > 0; (4 c[2] c[6] c[4]2 ) < 0);
(156)
(157)
p
(161)
eq246 jie :D dsolve(eq246; g(x i));
g3 () D 4 eC 1 c 2 c2 e c 2 0 11 p 4 p 2 eC 1 c 2 e C 1 c 2 c4 B 2C @4c6 c2 C p 4 2 p 2 C c4 A ; c c e 2 e 2
g3 () D 4
c2 > 0; 4c6 c2 c42 < 0 ;
c 2 (C 1 C) c 2 p c 2 2e2 c 2 (C 1 C) c
4
C c42 e4
p
c2
;
(159)
six solutions of a first-order nonlinear ordinary differential equation with six degrees under condition (161), which includes two new solutions, are displayed at a computer screen (after running the above program). Here we omit them due to the length of our article. Importing the following Maple program for reducing the two new solutions above, eq246 jie1 :D subs(2 x i sqrt(c[2]) 2 C 1 sqrt(c[2]) D eta; eq246 jie1); eq246 jie2 :D subs(2 x i sqrt(c[2]) 2 C 1 sqrt(c[2]) D eta; eq246 jie2); the following results are displayed at a computer screen (after running the above program): (11 ())2 D c2 c4 2 C (tanh())2 c4 2 tanh() (162) p p c4 2 4c6 c2 c4 2 ((tanh())2 1) 2 ; 4c6 (tanh())2 c2 c4 2 c4 (12 ())2 D c2 c4 2 C (tanh())2 c4 2 C tanh() (163) p p c4 2 4c6 c2 c4 2 ((tanh())2 1) 2 : (4c6 (tanh())2 c2 c4 2 )c4
857
858
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
Importing the following Maple program for reducing (162) and (163): eq246 jie3 :D subs(sqrt(c[4]2 (tanh(eta)2 1)) D I abs(c[4] sech(eta)); c[4] C tanh(eta)2 c[4] D sech(eta)2 c[4]; eq246 jie3); eq246 jie4 :D subs(sqrt(c[4]2 (tanh(eta)2 1)) D I abs(c[4] sech(eta)); c[4] C tanh(eta)2 c[4] D sech(eta)2 c[4]; eq246 jie4); phi[1](x i) :D epsilon (eq246 jie3)( 1/2); phi[2](x i) :D epsilon (eq246 jie4)( 1/2); the following results are displayed at a computer screen (after running the above program): v u u 2c2 c42 sech2 () C ijc4 u u u sech()j tanh()p c 2 4c c u 4 6 2 1 () D "t ; c4 4c6 tanh2 ()c2 c4 2 (164) v u u 2c2 c4 2 sech2 () ijc4 u u u sech()j tanh()p c 2 4c c u 4 6 2 2 () D "t : c4 4c6 tanh2 ()c2 c4 2 (165) p
where D 2 c2 ( C1 ). Step 6 We need to further discuss the results found by sometimes adding new conditions so that the results are simpler in form. For example, we can import the following Maple program for getting rid of the absolute value sign in the above results: assume(c[4] sech(eta) < 0); eq246 jie3 :D 2 c[2] (c[4]2 sech(eta)2 C I
tanh(eta) sqrt(c[4]2 4 c[6] c[2])
abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2
c[2] c[4]2 )/c[4]; eq246 jie4 :D 2 c[2] (c[4]2 sech(eta)2 I
tanh(eta)sqrt(c[4]2 4 c[6] c[2]) abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2
c[2] c[4]2 )/c[4]; phi[1; 1](x i) :D e psilon
(eq246 jie3)( 1/2); (
phi[1; 2](x i) :D e psilon (eq246 jie4) 1/2);
the following results are displayed at a computer screen (after running the above program): 1;1 () D v u u 2c2 sech() c4 sech() i tanh()p c4 2 4c6 c2 t " ; 4c6 (tanh())2 c2 c4 2 (166) 1;2 () D v p u u 2c2 sech() c4 sech() C i tanh() c4 2 4c6 c2 t " : 4c6 (tanh())2 c2 c4 2 (167) Importing the following Maple program for reducing and discussing above results: assume(c[4] sech(eta) > 0); eq246 jie3 :D 2 c[2] (c[4]2 sech(eta)2 C I
tanh(eta) sqrt(c[4]2 4 c[6] c[2]) abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2
c[2] c[4]2 )/c[4]; eq246 jie4 :D 2 c[2] (c[4]2 sech(eta)2 I tanh(eta) sqrt(c[4]2 4 c[6] c[2]) abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2
c[2] c[4]2 )/c[4]; phi[2; 1](x i) :D e psilon (eq246 jie3)( 1/2); phi[2; 2](x i) :D e psilon (eq246 jie4)( 1/2); the following results are shown at a computer screen (after running the above program): 2;1 () D v p u u 2c2 sech() c4 sech() C i tanh() c4 2 4c6 c2 t " ; 4c6 (tanh())2 c2 c4 2 (168) 2;2 () D v p u u 2c2 sech() c4 sech() i tanh() c4 2 4c6 c2 t " : 4c6 (tanh())2 c2 c4 2 (169) By using this method, we obtained some new types of general solution of a first-order nonlinear ordinary dif-
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
ferential equation with six degrees and presented the following theorem in [40,43,44]. Theorem 5 The following nonlinear ordinary differential equation with six degrees d() D" d
s
c0 C c1 () C c2 2 () C c3 3 () ; Cc4 4 () C c5 5 () C c6 6 () (170)
admits many kinds of fundamental solutions which depend on the values and constraints between c i ; i D 0; 1; 2; 3; 4; 5; 6. Some of the solutions are listed in the following sections. Case 1. If c0 D c1 D c3 D c5 D 0, Eq. (170) admits the following constant solutions
1;2 () D " s 3;4 () D "
c4 C
c4
p
c4 2 4c6 c2 ; 2c6
(171)
v u u 7;8 () D 2"u u t
p
c4 2 4c6 c2 ; 2c6
(172)
e2
p
c 2 (CC 1 ) c 2 p p 4C c 1 2 4c c e C e4 c 2p p6 2 2e2 c 2 (CC 1 ) c4 C c4 2 e4C 1 c 2
e2
;
(173)
c2 > 0 ;
and the following tan-sec triangular type solutions v h u u 2c2 sec( ) c4 sec( ) u i p u u C" c4 2 4c6 c2 tan( ) t 5;6 () D " ; c2 < 0 ; 4c6 c2 tan2 ( ) C c4 2 (177)
7;8 () D v h i p u u 2c2 sec( ) c4 sec( ) " c4 2 4c6 c2 tan( ) t " ; 4c6 c2 tan2 ( ) C c4 2 c2 < 0 ;
c 2 (CC 1 ) c 2 p p 4 c 2 C e4C 1 c 2 4c c e 6 2 p p 2e2 c 2 (CC 1 ) c4 C c4 2 e4 c 2
:
(174)
Case 1.2. If 4c2 c6 c4 2 > 0, Eq. (170) admits the following sinh-cosh hyperbolic type solutions v h i p u u 2c2 c4 C " 4c2 c6 c4 2 sinh() t ; 9;10 () D " 4c2 c6 sinh2 () c4 2 cosh2 () c2 > 0 ;
4c6 c2 tanh2 () c4 2
;
11;12 () D "
c2 > 0 ; (175)
(179)
v h i p u u 2c2 c4 " 4c2 c6 c4 2 sinh() t 4c2 c6 sinh2 () c4 2 cosh2 () c2 > 0 ;
Case 1.1. If 4c2 c6 c4 2 < 0, Eq. (170) admits the following tanh-sech hyperbolic type solutions v h u u 2c sech() c4 sech() 2 u i p u 2 u C"i c4 4c6 c2 tanh() t
(178)
p
p
When we take different values and constraints of 4c2 c6 c4 2 , the solutions (5.43) and (5.44) can be written in different formats as follows:
1;2 () D "
;
where D 2 c2 ( C1 ) and C1 is any constant, D p 2 c2 ( C1 ) and C1 is any constant.
and the following exponential type solutions v u u 5;6 () D 2"u u t
4c6 c2 tanh2 () c4 2
(176)
c i D constant ; i D 0; 1; 2; 3; 4; 5; 6
s
3;4 () D "
v h u u 2c2 sech() c4 sech() u i p u u "i c4 2 4c6 c2 tanh() t
;
(180)
and the following sin-cos triangular type solutions v h i p u u 2c2 c4 C "i 4c2 c6 c4 2 sin( ) t ; 13;14 () D " 4c6 c2 sin2 ( ) C c4 2 cos2 ( ) c2 < 0
(181)
v h i p u u 2c2 c4 "i 4c6 c2 c4 2 sin( ) t 15;16 () D " ; 4c6 c2 sin2 ( ) C 4 2 cos2 ( ) c2 < 0 ;
(182)
859
860
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
p p where D 2 c2 ( C1 ), D 2 c2 ( C1 ) and C1 is any constant. Case 1.3. If 4c2 c6 c4 2 D 0, Eq. (170) admits the following tanh hyperbolic type solutions s 17;18;19;20() D " s D"
e"2
p
2c2 c 2 (C 1 )
c4
p 2c2 1 C " tanh c2 ( C1 ) p ; c2 > 0 ; 1 c4 "(1 C c4 ) tanh c2 ( C1 ) (183)
and the following tan triangular type solutions
F j (u1 ; : : : ; u n ; u1;t ; : : : ; u n;t ; u1;x 1 ; : : : ; u n;x m ; u1;t t ; : : : ; u n;t t ; u1;tx 1 ; : : : u n;tx m ; : : :) D 0 ; j D 1; 2; : : : ; n ;
(185)
where j D 1; 2; : : : ; n. By using the following more general transformation, which we first present here, u i (t; x1 ; x2 ; : : : ; x m ) D U i () ; D u i (t; x1 ; x2 ; : : : ; x m ) D U i () ; D ˛0 (t) C
m1 X
(186)
˛ i (x i ; x iC1 ; : : : ; x m ; t) ˇ i (x i ) ;
iD1
s 21;22;23;24() D "
in m C 1 independent variables t; x1 ; x2 ; : : : ; x m ,
p
2c2
e"2i c 2 (C 1 ) c4 s p 2c2 1 C " tan i c2 ( C1 ) p ; c2 < 0 D" 1 c4 "(1 C c4 ) tan i c2 ( C1 ) (184)
where C1 is any constant. Case 2. If c5 D c6 D 0, Eq. (170) admits the solutions in Theorem 6.1. Remark 7 By using our mechanization method using Maple to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree, we can obtain some new types of general solution of a firstorder nonlinear ordinary differential equation with r(D 7; 8; 9; 10; 11; 12; : : :) degree [40]. We do not list the solutions here in order to avoid unnecessary repetition. Summary of the Generalized Algebra Method In this section, based on a first-order nonlinear ordinary differential equation with any degree (149) and its the exact solutions obtained by using our mechanization method via Maple, we will develop the algebraic methods [50,51] for constructing the traveling wave solutions and present a new generalized algebraic method and its algorithms [40, 43,44]. The KdV equation is chosen to illustrate our algorithm so that more families of new exact solutions are obtained which contain both non-traveling solutions and traveling wave solutions. We outline the main steps of our generalized algebraic method as follows: Step 1. For given nonlinear differential equations, with some physical fields u i (t; x1 ; x2 ; : : : ; x m ), (i D 1; 2; : : : ; n)
where ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m 1 are functions to be determined later. For example, when n D 1, we may take D ˛0 (t) C ˛1 (x1 t) ˇ1 (x1 ) ; where ˛0 (t); ˛1 (x1 t) and ˇ1 (x1 ) are undetermined functions. Then Eq. (185) is reduced to nonlinear differential equations G j (U1 ; : : : ; U n ; U10 ; : : : ; U n0 ; U100 ; : : : ; U n00 ; : : :) D 0 ; j D 1; 2; : : : ; n ;
(187)
where G j ( j D 1; 2; : : : ; n) are all polynomials of U i (i D 1; 2; : : : ; n), ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t), ˇ i (x i ); i D 1; 2; : : : ; m 1 and their derivatives. If Gk of them is not a polynomial of U i (i D 1; 2; : : : ; n), ˛ i (x i ; x iC1 ; : : : ; x m ; t), ˇ i (x i ); i D 1; 2; : : : ; m 1, ˛0 (t) and their derivatives, then we may use new variable v i ()(i D 1; 2; : : : ; n) which makes Gk become a polynomial of v i ()(i D 1; 2; : : : ; n), ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m 1 and their derivatives. Otherwise the following transformation will fail to seek solutions of Eq. (185). Step 2. We introduce a new variable () which is a solution of the following ODE d() D" d
s
c0 C c1 () C c2 2 () C c3 3 () ; Cc4 4 () C C cr r () r D 0; 1; 2; 3; : : : :
(188)
Then the derivatives with respect to the variable become the derivatives with respect to the variable .
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
Step 3. By using the new variable , we expand the solution of Eq. (185) in the form:
U i D a i;0 (X) C
ni X
k
(a i;k (X) ((X))
kD1
C b i;k (X) k ((X))) :
(189)
where X D (x1 ; x2 ; : : : ; x m ; t), D (X), a i;0(X); a i;k (X); b i;k (X)(i D 1; 2; : : : ; n; k D 1; 2; : : : ; n i ) are all differentiable functions of X to be determined later. Step 4. In order to determine n i (i D 1; 2; : : : ; n) and r, we may substitute (188) into (187) and balance the highest derivative term with the nonlinear terms in Eq. (187). By using the derivatives with respect to the variable , we can obtain a relation for ni and r, from which the different possible values of ni and r can be obtained. These values lead to the series expansions of the solutions for Eq. (185). Step 5. Substituting (189) into the given Eq. (185) and k k collecting qPcoefficients of polynomials of ; , and r j i k jD1 c j (), with the aid of Maple, then setting each coefficient to zero, we will get a system of over-determined partial differential equations with respect to a i;0 (X); a i;k (X); b i;k (X); i D 1; 2; : : :, n; k D 1; 2; : : : ; n i c j ; j D 0; 1; : : : ; r˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m 1. Step 6. Solving the over-determined partial differential equations with Maple,then we can determine a i;0(X); a i;k (X); b i;k (X); i D 1; 2; : : : ; n; k D 1; 2; : : : ; n i )c j ; j D 0; 1; : : : ; r˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m 1 . Step 7. From the constants a i;0(X); a i;k (X); b i;k (X)(i D 1; 2; : : : ; n; k D 1; 2; : : : ; n i )c j ( j D 0; 1; : : : ; r), ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m 1 obtained in Step 6 to Eq. (188), we can then obtain all the possible solutions. Remark 8 When c5 D c6 D 0 and b i;k D 0, Eq. (170) and transformation (189) just become the ones used in our previous method [50,51]. However, if c5 ¤ 0 or c6 ¤ 0, we may obtain solutions that cannot be found by using the methods [50,51]. It should be pointed out that there is no method to find all solutions of nonlinear PDEs. But our method can be used to find more solutions of nonlinear PDEs, and with the exact solutions obtained by using our mechanization method via Maple, we will develop the algebraic methods [50,51] for constructing the travel-
ing wave solutions and present a new generalized algebraic method and its algorithms [40,43,44]. Remark 9 By the above description, we find that our method is more general than the method in [50,51]. We have improved the method [50,51] in five aspects: First, we extend the ODE with four degrees (131) into the ODE with any degree (188) and get its new general solutions by using our mechanization method via Maple [40,43,44]. Second, we change the solution of Eq. (185) into a more general solution (189) and get more types of new rational solutions and irrational solutions. Third, we replace the traveling wave transformation (146) in [50,51] by a more general transformation (186). Fourth, we suppose the coefficients of the transformation (186) and (189) are undetermined functions, but the coefficients of the transformation (146) in [50,51] are all constants. Fifth, we present a more general algebra method than the method given in [50,51], which is called the generalized algebra method, to find more types of exact solutions of nonlinear differential equations based upon the solutions of the ODE (188). This can obtain more general solutions of the NPDEs than the number obtained by the method in [50,51]. The Generalized Algebra Method to Find New Non-traveling Waves Solutions of the (1 + 1)-Dimensional Generalized Variable-Coefficient KdV Equation In this section, we will make use of our generalized algebra method and symbolic computation to find new nontraveling waves solutions and traveling waves solutions of the following (1C1) – dimensional Generalized Variable – Coefficient KdV equation [16]. Propagation of weakly nonlinear long waves in an inhomogeneous waveguide is governed by a variable – coefficient KdV equation of the form [15] u t (x; t) C 6u(x; t)u x (x; t) C B(t)u x x x (x; t) D 0 : (190) where u(x; t) is the wave amplitude, t the propagation coordinate, x the temporal variable and B(t) is the local dispersion coefficient. The applicability of the variable-coefficient KdV equation (190) arises in many areas of physics as, for example, for the description of the propagation of gravity-capillary and interfacial-capillary waves, internal waves and Rossby waves [15]. In order to study the propagation of weakly nonlinear, weakly dispersive waves in inhomogeneous media, Eq. (190) is rewritten as follows [16] u t (x; t)C6A(t)u(x; t)u x (x; t)CB(t)u x x x (x; t) D 0: (191)
861
862
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
which now has a variable nonlinearity coefficient A(t). Here, Eq. (191) is called a (1 C 1)-dimensional generalized variable-coefficient KdV (gvcKdV) equation. In order to find new non-traveling waves solutions and traveling waves solutions of the following (1 C 1)-dimensional gvcKdV equation (191) by using our generalized algebra method and symbolic computation, we first take the following new general transformation, which we first present here u(x; t) D u() ;
D ˛(x; t)ˇ(t) r(t) ;
(192)
where ˛(x; t); ˇ(t) and r(t) are functions to be determined later. By using the new variable D () which is a solution of the following ODE d() D" d
s
c0 C c1 () C c2 2 () C c3 3 () : (193) Cc4 4 () C c5 5 () C c6 6 ()
we expand the solution of Eq. (191) in the form [40,43]:
u D a0 (X)C
n X
(a i (X) i ((X))Cb i (X) i ((X))) (194)
iD1
where X D (x; t), a0 (X); a i (X); b i (X) (i D 1; 2; : : : ; n) are all differentiable functions of X to be determined later. Balancing the highest derivative term with the nonlinear terms in Eq. (191) by using the derivatives with respect to the variable , we can determine the parameter n D 2 in (194). In addition, we take a0 (X) D a0 (t); a i (X) D a i (t); b i (X) D b i (t); i D 1; 2 in (194) for simplicity, then substituting them and (192) into (194) along with n D 2, leads to: u(x; t) D a0 (t) C a1 (t)() C a2 (t)(())2 C
b1 (t) b2 (t) : C () (())2
(195)
where D ˛(x; t)ˇ(t) r(t), and ˛(x; t); ˇ(t); r(t); a0 (t); a i (t) and b i (t); i D 1; 2 are all differentiable functions of x or t to be determined later. By substituting (195) into the given Eq. (191) along with (193) and the derivatives of ,q and collecting coeffiP6 k j cients of polynomials of and jD1 c j (), with the aid of Maple, then setting each coefficient to zero, we will get a system of over-determined partial differential equations with respect to A(t); B(t); ˛(x; t); ˇ(t); r(t);
a0 (t); a i (t) and b i (t), i D 1; 2 as follows: 3 @ 21B(t)(ˇ(t))3 "a1 (t) ˛(x; t) c4 c6 C 18A(t) @x @ " ˛(x; t) ˇ(t)a1 (t)a2 (t)c6 3B(t)(ˇ(t))3 @x 3 @ "b1 (t) ˛(x; t) c6 2 D 0 ; @x d 12b2 (t) " r(t) c2 8B(t)(ˇ(t))3 dt 3 @ "b2 (t) ˛(x; t) c2 2 2b2 (t) @x @ " ˛(x; t) ˇ(t)c2 2B(t)ˇ(t) @t 3 @ ˛(x; t) c2 2b2 (t) "b2 (t) @x 3 d "˛(x; t) ˇ(t) c2 6A(t) dt @ " ˛(x; t) ˇ(t)(b1 (t))2 c2 @x @ 12A(t) " ˛(x; t) ˇ(t)a0 (t)b2 (t)c2 12A(t) @x @ " ˛(x; t) ˇ(t)(b2 (t))2 c4 D 0 ; @x d d b1 (t) " r(t) c6 a1 (t) " r(t) c4 dt dt @ C 6A(t) " ˛(x; t) ˇ(t)a2 (t)b1 (t)c4 @x d C a1 (t) "˛(x; t) ˇ(t) c4 b1 (t) dt d "˛(x; t) ˇ(t) c6 C 6A(t) dt @ " ˛(x; t) ˇ(t)a0 (t)a1 (t)c4 6A(t) @x @ " ˛(x; t) ˇ(t)a0 (t)b1 (t)c6 @x 3 @ 4B(t)(ˇ(t))3 "b1 (t) ˛(x; t) c2 c6 @x 3 @ C B(t)ˇ(t) "a1 (t) ˛(x; t) c4 @x 3 @ C a1 (t) " ˛(x; t) ˇ(t)c4 b1 (t) @t @ " ˛(x; t) ˇ(t)c6 6A(t) @t @ ˛(x; t) ˇ(t)a1 (t)b2 (t)c6 C 7B(t)(ˇ(t))3 " @x
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
3 @ "a1 (t) ˛(x; t) c2 c4 C 18A(t) @x @ " ˛(x; t) ˇ(t)a1 (t)a2 (t)c2 B(t)ˇ(t) @x 3 @ ˛(x; t) c6 D 0 ; "b1 (t) @x 3
Case 4 b2 (t) D 0 ;
A(t) D A(t) ;
˛(x; t) D F1 (t) ; (196)
::::::::::::
B(t) D B(t) ;
ˇ(t) D ˇ(t) ;
a2 (t) D C1 ;
a0 (t) D C4 ; a1 (t) D C2 ; Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C5 ; dt dt b1 (t) D C3 ;
because there are so many over-determined partial differential equations, only a few of them are shown here for convenience. Solving the over-determined partial differential equations with Maple, we have the following solutions.
where A(t), B(t), F1 (t), ˇ(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants.
Case 1
Case 5
(200)
A(t) D A(t) ;
B(t) D B(t) ;
˛(x; t) D F1 (t) ;
b2 (t) D 0 ;
ˇ(t) D ˇ(t) ;
a2 (t) D C1 ;
a1 (t) D C2 ;
B(t) D B(t) ;
b2 (t) D C3 ; a0 (t) D C5 ; Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C6 ; dt dt b1 (t) D C4 ; (197) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants.
A(t) D A(t) ;
˛(x; t) D F1 (t) ;
B(t) D B(t) ;
ˇ(t) D ˇ(t) ;
r(t) D C5 ;
a0 (t) D C4 ; b1 (t) D C3 ;
(201)
˛(x; t) D ˛(x; t) ; where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants. Case 6 C2 c4 ; ˇ(t) D 0 ; b2 (t) D 0 ; c6 A(t) D A(t) ; B(t) D B(t) ; a2 (t) D C1 ; a1 (t) D C2 ;
a2 (t) D C1 ;
a0 (t) D C4 ; a1 (t) D C2 ; Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C5 ; dt dt b2 (t) D C3 ; (198) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants. Case 3 ˇ(t) D 0 ;
a1 (t) D C2 ;
A(t) D A(t) ;
a2 (t) D C1 ;
b1 (t) D
Case 2 b1 (t) D 0 ;
ˇ(t) D 0 ;
A(t) D A(t) ;
B(t) D B(t) ;
a2 (t) D C1 ;
a1 (t) D C2 ;
b2 (t) D C3 ;
a0 (t) D C5 ;
b1 (t) D C4 ;
r(t) D C6 ;
(199)
˛(x; t) D ˛(x; t) ; where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants.
r(t) D C4 ;
(202)
a0 (t) D C3 ;
˛(x; t) D ˛(x; t) ; where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants. Case 7 B(t) D B(t) ; A(t) D 0 ;
b1 (t) D 0 ;
a0 (t) D C3 ;
b2 (t) D C2 ;
a1 (t) D 0 ; a2 (t) D 0 ; C1 ˛(x; t) D F1 (t)x C F2 (t) ; ˇ(t) D ; F1 (t) d d C1 ; F1 (t) dt F2 (t) dt F1 (t) Z F2 (t) C 4c2 B(t)C1 2 (F1 (t))2 r(t) D dt C C4 ; (F1 (t))2 (203) where F1 (t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants.
863
864
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
So we get new general forms of solutions of equations (191):
Case 8 B(t) D B(t) ;
b1 (t) D 0 ;
˛(x; t) D F1 (t) ;
ˇ(t) D ˇ(t) ;
A(t) D 0 ;
a1 (t) D 0 ;
u(x; t) D a0 (t) C a1 (t)(˛(x; t)ˇ(t) r(t))
a2 (t) D 0 ; Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C3 ; dt dt a0 (t) D C2 ;
b2 (t) D C1 ;
C a2 (t)((˛(x; t)ˇ(t) r(t)))2 C
b1 (t) b2 (t) : C (˛(x; t)ˇ(t) r(t)) ((˛(x; t)ˇ(t) r(t)))2 (208)
(204) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3 are arbitrary constants. Case 9 A(t) D A(t) ;
B(t) D B(t) ;
˛(x; t) D F1 (t) ; a2 (t) D C1 ;
b1 (t) D 0 ;
ˇ(t) D ˇ(t) ;
a0 (t) D C3 ;
b2 (t) D C2 ;
a1 (t) D 0 ; Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C4 ; dt dt
(205)
where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants. Case 10 A(t) D A(t) ;
B(t) D B(t) ;
˛(x; t) D F1 (t) ;
b1 (t) D 0 ;
ˇ(t) D ˇ(t) ;
a2 (t) D C1 ; b2 (t) D 0 ; a1 (t) D 0 ; Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C3 ; dt dt
where ˛(x; t), ˇ(t), r(t), a0 (t), a i (t), b i (t), i D 1; 2 satisfy (197)–(207) respectively, and the variable (˛(x; t)ˇ(t) r(t)) takes the solutions of the Eq. (193). For example, we may take the variable (˛(x; t)ˇ(t)r(t)) as follows: Type 1. If 4c2 c6 c4 2 < 0, corresponding to Eq. (193), (˛(x; t)ˇ(t) r(t)) is taken, we get the following four tanh-sech hyperbolic type solutions 1;2 ((˛(x; t)ˇ(t) r(t))) D s p 2c2 sech()[c4 sech() C "i c4 2 4c6 c2 tanh()] ; " 4c6 c2 tanh2 () c4 2 c2 > 0 ;
(209)
3;4 ((˛(x; t)ˇ(t) r(t))) D s p 2c2 sech()[c4 sech() "i c4 2 4c6 c2 tanh()] ; " 4c6 c2 tanh2 () c4 2 c2 > 0 ;
(210)
and the following tan-sec triangular type solutions
a0 (t) D C2 ; (206) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants.
c2 < 0 ;
Case 11 A(t) D A(t) ; a2 (t) D C1 ; ˇ(t) D 0 ;
5;6 ((˛(x; t)ˇ(t) r(t))) D s p 2c2 sec( )[c4 sec( ) C " c4 2 4c6 c2 tan( )] " ; 4c6 c2 tan2 ( ) C c4 2
B(t) D B(t) ; b2 (t) D 0 ;
b1 (t) D 0 ;
a1 (t) D 0 ;
r(t) D C3 ;
˛(x; t) D ˛(x; t) ;
(207)
a0 (t) D C2 ;
7;8 ((˛(x; t)ˇ(t) r(t))) D s p 2c2 sec( )[c4 sec( ) " c4 2 4c6 c2 tan( )] " ; 4c6 c2 tan2 ( ) C c4 2 c2 < 0 ;
where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2 are arbitrary constants. Because there are so many solutions, only a few of them are shown here for convenience.
(211)
(212)
p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and C1 is any p constant, D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and C1 is any constant.
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
Type 2. If 4c2 c6 c4 2 > 0, corresponding to Eq. (193), (˛(x; t)ˇ(t) r(t)) is taken, we get the following four sinh-cosh hyperbolic type solutions 9;10 ((˛(x; t)ˇ(t) r(t))) s p 2c2 [c4 C " 4c2 c6 c4 2 sinh()] D" ; 4c2 c6 sinh2 () c4 2 cosh2 ()
c2 > 0 ; (213)
11;12 ((˛(x; t)ˇ(t) r(t))) s p 2c2 [c4 " 4c2 c6 c4 2 sinh()] ; D" 4c2 c6 sinh2 () c4 2 cosh2 ()
Substituting (209)–(218) and (197)–(207) into (208), respectively, we get many new irrational solutions and rational solutions of combined hyperbolic type or triangular type solutions of Eq. (191). For example, when we select A(t); B(t); ˛(x; t); ˇ(t); r(t); a0 (t); a i (t) and b i (t); i D 1; 2 to satisfy Case 1, we can easily get the following irrational solutions of combined hyperbolic type or triangular type solutions of Eq. (191): u(x; t) D C5 C C2 (F1 (t)ˇ(t) r(t))
c2 > 0 ;
C C1 ((F1 (t)ˇ(t) r(t)))2 C4 C3 C ; C (F1 (t)ˇ(t) r(t)) ((F1 (t)ˇ(t) r(t)))2 (219)
(214) and the following sin-cos triangular type solutions 13;14 ((˛(x; t)ˇ(t) r(t))) s p 2c2 [c4 C "i 4c2 c6 c4 2 sin( )] D" ; 4c6 c2 sin2 ( ) C c4 2 cos2 ( )
c2 < 0 (215)
15;16 ((˛(x; t)ˇ(t) r(t))) s p 2c2 [c4 "i 4c6 c2 c4 2 sin( )] D" ; 4c6 c2 sin2 ( ) C 4 2 cos2 ( )
c2 < 0; (216)
p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ), D p 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ),and C1 is any constant. Type 3. If 4c2 c6 c4 2 D 0, corresponding to Eq. (193), (˛(x; t)ˇ(t) r(t)) is taken, we get the following two sech-tanh hyperbolic type solutions 4c2 c6 c4 2 D 0, Eq. (170) admits the following tanh hyperbolic type solutions 17;18;19;20(˛(x; t)ˇ(t) r(t)) D s p 2c2 f1 C " tanh[ c2 (˛(x; t)ˇ(t) r(t) C1 )]g ; " p 1 c4 "(1 C c4 ) tanh[ c2 (˛(x; t)ˇ(t) r(t) C1 )] c2 > 0 ;
(217)
and the following tan triangular type solutions 21;22;23;24(˛(x; t)ˇ(t) r(t)) D v u 2c f1 C " tan[i pc (˛(x; t)ˇ(t) r(t) C )]g 2 2 1 u "u ; t 1 c4 p "(1 C c4 ) tan[i c2 (˛(x; t)ˇ(t) r(t) C1 )] c2 < 0 ; where C1 is any constant.
where A(t); B(t); ˇ(t); F1 (t) are arbitrary functions of t, Ci , i D 1; 2; 3; 4; 5 are arbitrary constants, and Z r(t) D
F1 (t)
d ˇ(t) C dt
d F1 (t) ˇ(t)dt C C6 : dt
Substituting (209)–(218) into (219), respectively, we get the following new irrational solutions and rational solutions of combined hyperbolic type or triangular type solutions of Eq. (191). Case 1. If 4c2 c6 c4 2 < 0; c2 > 0, corresponding to (209), (210) and (197), Eq. (191) admits the following four tanh-sech hyperbolic type solutions u1;2 (x; t) D C5 C r "
C4
p
2c 2 sech()[c 4 sech() C"i c 42 4c 6 c 2 tanh()] 4c 6 c 2 tanh2 ()c 4 2
C3 (4c6 c2 tanh2 () c4 2 ) p 2c2 sech()[c4 sech() C "i c4 2 4c6 c2 tanh()] v u u 2c2 sech() u u p u [c4 sech() C "i c4 2 4c6 c2 tanh()] t C C2 " 4c6 c2 tanh2 () c4 2
C
C
p 2c2 C1 sech()[c4 sech() C "i c4 2 4c6 c2 tanh()] 4c6 c2 tanh2 () c4 2
;
(220)
(218) and
865
866
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
u3;4 (x; t) D C5 C3 (4c6 c2 tanh2 () c4 2 ) p 2c2 sech()[c4 sech() "i c4 2 4c6 c2 tanh()] C4 C r p sech()"i c 4 2 4c 6 c 2 tanh()] " 2c 2 sech()[c 44c 2 2 6 c 2 tanh ()c 4 v u u 2c2 sech() u p u [c4 sech() "i c4 2 4c6 c2 tanh()] t C C2 " 4c6 c2 tanh2 () c4 2 p 2c2 C1 sech()[c4 sech() "i c4 2 4c6 c2 tanh()] C ; 4c6 c2 tanh2 () c4 2 (221) C
p 2c2 C1 sec( )[c4 sec( ) " c4 2 4c6 c2 tan( )] C ; 4c6 c2 tan2 ( ) C c4 2 (223) p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and the rest of the parameters are the same as with Case 1. Case 3. If 4c2 c6 c4 2 > 0; c2 > 0, corresponding to (213), (214) and (197), Eq. (191) admits the following four sinh-cosh hyperbolic type solutions u9;10 (x; t) D C5 C C2 "
p
where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ), A(t); B(t); ˇ(t); F1 (t) are arbitrary functions of t, Ci , i D 1; 2; 3; 4; 5 are arbitrary constants, and Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C6 : dt dt Case 2. If 4c2 c6 c4 2 < 0; c2 < 0, corresponding to (211), (212) and (197), Eq. (191) admits the following four tan-sec triangular type solutions u5;6 (x; t) D C5 C r "
C4 p 2c 2 sec()[c 4 sec()C" c 4 2 4c 6 c 2 tan()] 2 4c 6 c 2 tan ()Cc 4 2
C3 (4c6 c2 tan2 ( ) C c4 2 ) p 2c2 sec( )[c4 sec( ) C " c4 2 4c6 c2 tan( )] s p 2c2 sec( )[c4 sec( ) C " c4 2 4c6 c2 tan( )] C C2 " 4c6 c2 tan2 ( ) C c4 2 p 2c2 C1 sec( )[c4 sec( ) C " c4 2 4c6 c2 tan( )] C ; 4c6 c2 tan2 ( ) C c4 2 (222) C
s
C
p 2c2 [c4 C " 4c2 c6 c4 2 sinh()]
4c2 c6 sinh2 () c4 2 cosh2 () p 2c2 C1 [c4 C " 4c2 c6 c4 2 sinh()]
4c2 c6 sinh2 () c4 2 cosh2 () C4 C r p 4c 2 c 6 c 4 2 sinh()] " 2c4c2 [cc4 C" 2 2 2 2 6 sinh ()c 4 cosh () p 2c2 C3 [c4 C " 4c2 c6 c4 2 sinh()] C ; 4c2 c6 sinh2 () c4 2 cosh2 ()
(224)
and u11;12 (x; t) D C5 C C2 " C
s
p 2c2 [c4 C " 4c2 c6 c4 2 sinh()]
4c2 c6 sinh2 () c4 2 cosh2 () p 2c2 C1 [c4 C " 4c2 c6 c4 2 sinh()]
4c2 c6 sinh2 () c4 2 cosh2 () C4 C r p 2 " 2c 2 [c 4 C" 24c 2 c 6 c24 sinh()] 4c 2 c 6 sinh ()c 4 cosh2 () p 2c2 C3 [c4 C " 4c2 c6 c4 2 sinh()] C ; 4c2 c6 sinh2 () c4 2 cosh2 ()
(225)
p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and the rest of the parameters are the same as with Case 1.
and u7;8 (x; t) D C5 C r "
C4 p 2c 2 sec()[c 4 sec()" c 4 2 4c 6 c 2 tan()] 2 4c 6 c 2 tan ()Cc 4 2
C3 (4c6 c2 tan2 ( ) C c4 2 ) p 2c2 sec( )[c4 sec( ) " c4 2 4c6 c2 tan( )] s p 2c2 sec( )[c4 sec( ) " c4 2 4c6 c2 tan( )] C C2 " 4c6 c2 tan2 ( ) C c4 2
C
Case 4. If 4c2 c6 c4 2 > 0; c2 < 0, corresponding to (215), (216) and (197), Eq. (191) admits the following four sin-cos triangular type solutions u13;14 (x; t) C4 D C5 C r p 2 4 C"i 4c 2 c 6 c 4 sin()] " 2c 2 [c 4c c sin2 ()Cc 2 cos2 () 6 2
4
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
C3 (4c6 c2 sin2 ( ) C c4 2 cos2 ( )) p 2c2 [c4 C "i 4c2 c6 c4 2 sin( )] s p 2c2 [c4 C "i 4c2 c6 c4 2 sin( )] C C2 " 4c6 c2 sin2 ( ) C c4 2 cos2 ( ) i h p 2c2 C1 c4 C "i 4c2 c6 c4 2 sin( ) C ; 4c6 c2 sin2 ( ) C c4 2 cos2 ( )
tan triangular type solutions
C
u21;22;23;24 (x; t)
v u u u D C 5 C C 2 "t
h i 2c2 1 C " tan i2 1 c4 "(1 C c4 ) tan i2 h i 2c2 C1 1 C " tan i2 1 c4 "(1 C c4 ) tan i2
(226)
and u15;16 (x; t) D C5 C r "
C4 i h p 2c 2 c 4 "i 4c 2 c 6 c 4 2 sin()
"
h
p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and the rest of the parameters are the same as with Case 1.
u17;18;19;20 (x; t)
Remark 10 We may further generalize (194) as follows: u D a0 (X)
C
m X
P
P
r i1 CCr in Di a r i1 ;:::;r ni (X) r i1 r ni G1i (1i (X)) : : : G ni ( ni (X))
l i1 CCl in Di b l i1 ;:::;l ni (X) l i1 l ni (1i (X)) : : : G ni ( ni (X)) G1i
;
i C c i (X) (230)
P where l i1 CCl in Di b2l i1 ;:::;l ni (X) C c 2i (X) ¤ 0, a0 (X), 1i (X); : : :, ni (X), a r i1 ;:::;r ni (X), b l i1 ;:::;l ni (X) and c i (X); i D 0; 1; 2; : : : ; m are all differentiable functions to be determined later, is a constant, G1i (1i (X)); : : :, G ni ( ni (X)) are all ( ki (X)) or 1 ( ki (X)) or some derivatives ( j) ( ki (X)); k D 1; 2; : : : ; n;i D 0; 1; 2; : : : ; m; j D ˙1; ˙2; : : :. We can get many new explicit solutions of Eq. (191).
s
2c2 1 C " tanh 2 D C5 C C2 " 1 c4 "(1 C c4 ) tanh 2 2c2 C1 1 C " tanh 2 C 1 c4 "(1 C c4 ) tanh 2 C4 C r 2c 1C" tanh( 2 ) " 1c 2"(1Cc ) tanh ( 2 ) 4 4 C3 1 c4 "(1 C c4 ) tanh 2 ; C 2c2 1 C " tanh 2
(229)
p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and the rest of the parameters are the same as with Case 1.
iD0
Case 5. If 4c2 c6 c4 2 D 0; c2 > 0, corresponding to (217) and (197), Eq. (191) admits the following four tanh hyperbolic type solutions
C4 h i 2c 2 1C" tan i 2 1c 4 "(1Cc 4 ) tan i 2
C3 1 c4 "(1 C c4 ) tan i2 h i ; 2c2 1 C " tan i2
4c 6 c 2 sin2 ()Cc 4 2 cos2 () C3 (4c6 c2 sin2 ( ) C c4 2 cos2 ( ))
i p 2c2 c4 "i 4c2 c6 c4 2 sin( ) v i h p u u 2c2 c4 "i 4c2 c6 c4 2 sin( ) t C C2 " 4c6 c2 sin2 ( ) C c4 2 cos2 ( ) h i p 2c2 C1 c4 "i 4c2 c6 c4 2 sin( ) C ; (227) 4c6 c2 sin2 ( ) C c4 2 cos2 ( )
C
s
C
(228)
p where D 2 c2 ((˛(x; t)ˇ(t) r(t)) C1 ) and the rest of the parameters are the same as with Case 1. Case 6. If 4c2 c6 c4 2 D 0; c2 < 0, corresponding to (218) and (197), Eq. (191) admits the following two sec-
A New Exp-N Solitary-Like Method and Its Application in the (1 + 1)-Dimensional Generalized KdV Equation In this section, in order to develop the Exp-function method [31], we present two new generic transformations, a new Exp-N solitary-like method, and its algorithm [40, 47]. In addition, we apply our method to construct new exact solutions of the (1 C 1)-dimensional classical generalized KdV(gKdV) equation.
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Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the
Summary of the Exp-N Solitary-Like Method In the following we would like to outline the main steps of our Exp-N solitary-like method [40,47]: Step 1. For a given NLEE system with some physical fields u m (t; x1 ; x2 ; : : : ; x n1 ), (m D 1; 2; : : : ; n) in n independent variables t; x1 ; x2 ; : : : ; x n1 , Fm (u1 ; u2 ; : : : ; u n ; u1;t ; u2;t ; : : : ; u n;t ; u1;x 1 ; u2;x 1 ; : : : ; u n;x 1 ; u1;t t ; u2;t t ; : : : ; u n;t t ; : : :) D 0; (m D 1; 2; : : : ; n) :
(231)
We introduce a new generic transformation 8 ˆ 0):
e c n en x x ! 1 : (8) n c n en x x ! C1 The direct scattering transform constructs the quantities fT(k); R(k); n ; c n g from a given potential function. The
important inversion formula were derived by Gel’fand and Levitan in 1955 [9]. These enable the potential u to be constructed out of the spectral or scattering data S D fR(k); n ; c n g. This is considerably more complicated than the Inverse Fourier Transform, involving the solution of a nontrivial integral equation, whose kernel is built out of the scattering data (see [10,11,12,13] for descriptions of this). To solve the KdV equation we first construct the scatting data S(0) from the initial condition u(x; 0). As a consequence of (4) (with an additional constant) with the given boundary conditions, the scatting data evolves in a very simple way. Indeed, we can give explicit formula: 3
R(k; t) D R(k; 0)e8i k t ;
3
c n (t) D c n (0)e4n t :
(9)
Using the inverse scattering transform on the scattering data S(t), we obtain the potential u(x; t) and thus the solution to the initial value problem for the KdV equation. This process cannot be carried out explicitly for arbitrary initial data, although, in this case, it gives a great deal of information about the solution u(x; t). However, whenever the reflection coefficient is zero, the kernel of Gel’fand-Levitan integral equation becomes separable and explicit solutions can be found. It is in this way that the N-soliton solution is constructed by IST from the initial condition: u(x; 0) D N(N C 1)sech2 x :
(10)
The general formula for the multi-soliton solution is given by: u(x; t) D 2(ln detM)x x ;
(11)
where M is a matrix built out of the discrete scattering data. Exact N-soliton Solutions of the KdV Equation Besides the IST, there are several analytical methods for obtaining solutions of the KdV equation, such as Hirota bilinear method [14,15,16], Bäcklund transformation [17], Darboux transformation [18], and so on. The existence of such analytical methods reflects a rich algebraic structure of the KdV equation. In Hirota’s method, we transform the equation into a bilinear form, from which we can get soliton solutions successively by means of a kind of perturbational technique. The Bäcklund transformation is also employed to obtain solutions from a known solution of the concerned equation. In what follows, we will mainly discuss the Hirota bilinear method to derive the N-soliton solutions of the KdV equation.
Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the
It is well known that Hirota developed a direct method for finding N-soliton solutions of nonlinear evolution equations. In particular, we shall discuss the KdV bilinear form D x D t C D3x f f D 0 ; (12) by the dependent variable transformation u(x; t) D 2(ln f )x x :
u(x; t) D 2(ln f2 )x x ; where f2 D 1 C exp(1 ) C exp(2 ) C exp(1 C 2 C A12 ) : (21)
(13) Frequently the N-soliton solutions are obtained as follows
Here the Hirota bilinear operators are defined by D xm D nt ab
For N D 2, the two-soliton solution for the KdV equation is similarly obtained from
m
n
D (@x @x0 ) (@ t @ t0 ) a(x; t)b(x0; t0)j x0Dx;t0Dt :
fN D
f (x; t) D 1 C f (1) " C f (2) "2 C C f ( j) " j C : (15) Substituting (15) into (12) and equating coefficients of powers of " gives the following recursion relations for the f (n) ":
f x(1) xxx
C
f x(1) t
D0;
(16)
1 (2) 4 (1) "2 : f x(2) f (1) ; x x x C f x t D (D x D t C D x ) f 2 (3) 4 (1) "3 : f x(3) f (2) ; x x x C fx t D Dx D t C Dx f
(17) (18)
and so on. N-soliton solutions of the KdV equation are found by assuming that f (1) has the form f (1) D
N X
exp( j );
j D k j x ! j t C x j0 ;
(19)
jD1
and k j ; ! j and x j0 are constants, provided that the series (15) truncates. For N D 1, we take
f
(n)
D 0;
Therefore we have !1 D k13 ;
and u(x; t) D
1 k12 sech2 k1 x k13 t C x10 : 2 2
jD1
1 j l A jl A ;
(22)
1 j 1:
(22)
D u n (x; t) C
n o
Lu n () C N u n () g () d ;
0
n 0;
where is the general Lagrange multiplier, which can be identified optimally by the variational theory, the sub script n denotes the nth approximation and u n considered
as a restricted variation, i. e. ı u n D 0. For linear problems, the exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified. In nonlinear problems, in order to determine the Lagrange multiplier in a simple manner, the nonlinear terms have to be considered as restricted variations. Consequently, the exact solution may be obtained by using u (x; t) D lim u n (x; t) : n!1
It should be emphasized that u m (x; t) for m 1 is governed by the nonlinear Eq. (20) with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation software such as Maple and Mathematica. The Variational Iteration Method (VIM) The VIM was first proposed by He [92,93,94,95,96]. This method has been employed to solve a large variety of linear and nonlinear problems with approximations converging rapidly to accurate solutions. The idea of VIM is to construct a correction functional by a general Lagrange multiplier. The multiplier in the functional should be chosen such that its correction solution is superior to its initial approximation (trial function) and is the best within the flexibility of the trial function, accordingly we can identify the multiplier by variational theory. The initial approximation can be freely chosen with possible unknowns, which can be determined by imposing the boundary/initial conditions. We consider the following general differential equation: Lu C Nu D g ;
(23)
where L and N are linear and nonlinear operators respectively, and g(t) is the source inhomogeneous term. According to the VIM, the terms of a sequence fu n g are constructed such that this sequence converges to the
(24)
(25)
This method has been employed to solve a large variety of linear and nonlinear problems with approximations converging to accurate solutions. This technique was used to find the solutions of partial differential equations, ordinary differential equations, boundary-value problems and integral equations by many authors [97,98,99,100, 101,102,103,104,105]. The Homotopy Perturbation Method (HPM) The HPM was developed by He [106,107,108,109]. In recent years, the applications of HPM in nonlinear problems has been devoted to scientists and engineers [110,111, 112,113]. We consider the following general nonlinear differential equation: A y f (r) D 0 ; r 2 ˝ ; (26) with boundary conditions B y; @y/@n D 0 ; r 2 ;
(27)
where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function and is the boundary of the domain ˝. The operator A can be generally divided into two parts L and N, where L is linear and N is nonlinear. Therefore, Eq. (26) can be written as follows: L y C N y f (r) D 0 : (28)
Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the
We construct a homotopy of Eq. (26) y r; p : ˝ [0; 1] ! R which satisfies H y; p D 1 p L y L y0 C p A y f (r) D0;
r 2 R; (29)
which is equivalent to
The ADM for Eq. (14) According to the ADM [64,65,66], Eq. (34) can be written in an operator form ( Lu D 6uu x u x x x ; p (35) c u (x; 0) D f (x) D 2c sec h2 2 x ; where the differential operator L is
H y; p D L y L y0 C pL y0 C p A y f (r)
@ ; @t
L
D0; (30) where p 2 [0; 1] is an embedding parameter and y0 is an initial approximation which satisfies the boundary conditions. It follows from Eqs. (29) and (30) that H y; 0 D L y L y0 D 0 and (31) H y; 1 D A y f (r) D 0 : Thus, the changing process of p from 0 to 1 is just that of y r; p from y0 (r) this is to y (r). In topology called deformation and L y L y0 and A y f (r) are called homotopic. Here the embedding parameter is introduced much more naturally, unaffected by artificial factors; further it can be considered as a small parameter for 0 p 1. So it is very natural to assume that the solution of (29) and (30) can be expressed as y (x) D u0 (x) C pu1 (x) C p2 u2 (x) C :
(32)
According to HPM, the approximate solution of (26) can be expressed as a series of the power of p, i. e.,
(36)
and the inverse operator L1 is an integral operator given by Z t L1 (ı) D (37) (ı) dt : 0
The ADM [64,65,66] assumes a series solution for the unknown function u (x; t) given by Eq. (3) and the nonlinear term Nu D uu x can be decomposed into a infinite series of polynomials given by Eq. (5). Operating with the integral operator (37) on both sides of (35) and using the initial condition, we get u (x; t) D f (x) C L1 (6uu x u x x x ) :
(38)
Substituting (3) and (5) into the functional Eq. (38) yields 1 X
u n (x; t)
nD0
"
D f (x) C L
1
6
1 X
! An
nD0
y D lim y (x) D u0 (x) C u1 (x) C u2 (x) C : (33)
1 X
!
Numerical Applications and Comparisons We now consider the initial value problem associated with the KdV equation: ( u t 6uu x C u x x x D0 ; p (34) c u (x; 0) D 2c sec h2 2 x ; with u (x; t) is a sufficiently smooth function and c is the velocity of the wavefront. In this section, we apply the above described four methods and the explicit finite difference method on the KdV Eq. (34) so that the comparisons are made numerically.
:
un
nD0
xxx
(39)
p!1
The convergence of series (33) has been proved by He in [106].
#
The ADM admits that the zeroth component u0 (x; t) be identified by all terms that arise from the initial conditions and from the source terms if exist, and as a result, the remaining components u n (x; t), n 1 can be determined by using the recurrence relation: p ( c u0 (x; t) D f (x) D 2c sec h2 2 x ; (40) u kC1 (x; t) D L1 6A k (u k ) x x x ; k 0 ; where Ak , k 0 are Adomian polynomials that represent the nonlinear term (uu x ) and given by 8 ˆ ˆ ˆA0 D u0 u0x ; ˆ g : (4) T2C
The quantity r(m; ) is called the “learning rate” function. Definition 1 The algorithm fA m g is said to be probably approximately correct (PAC) if r(m; ) ! 0 as m ! 1 for each > 0. The concept class C is said to be PAC learnable if there exists an algorithm that is PAC. Thus the algorithm fA m g is PAC if, for each target concept T, the distance d P (T; H m ) converges to zero in probability as m ! 1, and moreover, the rate of convergence is uniform with respect to the unknown target concept T.
Note that, if an algorithm is PAC, then for each pair of numbers ; ı > 0, there exists a corresponding integer m0 D m0 ( ; ı) such that P m fx 2 X m : d P [T; H m (T; x)] > g ı ; 8m m0 ; 8T 2 C :
(5)
In other words, if at least m0 i.i.d. samples are drawn and the algorithm is applied to the resulting labeled multisample, it can be said with confidence at least 1 ı that the hypothesis H m produced by the algorithm is within a distance of the true but unknown target concept T. In the above expression, the quantity is referred to as the “accuracy” of the algorithm, while ı is referred to as the “confidence.” The integer m0 ( ; ı) is called the “sample complexity” of the algorithm. In all but the simplest cases, one can only obtain upper bounds for m0 ( ; ı), but this is sufficient. The above statement provides some motivation for the nomenclature “PAC.” If the algorithm fA m g is PAC, the hypothesis H m produced by the algorithm need not exactly equal T; rather, H m is only an approximately correct version of T, in the sense that d P (H m ; T) . Even this statement is only probably true, with a confidence of at least 1 ı. The notion of PAC learnability is usually credited to Valiant (see [12]), who showed that a certain collection of Boolean formulas is PAC learnable; however, the term “PAC learnability” seems to be due to [1]. It should be mentioned that Valiant had some additional requirements in his formulation beyond the condition that r(m; ) ! 0 as m ! 1. In particular, he required that the mapping A m be computable in polynomial time. These additional requirements are, in some sense, what distinguish computational (PAC) learning theory from statistical (PAC) learning theory. The references on computational learning theory in the bibliography go into computational issues in greater depth. PAC learning is closely related to an older topic in probability theory known as empirical process theory, which is devoted to a study of the behavior of empirical means as the number of samples becomes large. One of the standard references in the field, namely [13], is actually about empirical process theory. In the book [15] PAC learning theory is treated side by side with the uniform convergence problem, and it is shown that each has something to offer to the other. However, in the present write-up we do not discuss the uniform convergence problem. Finally, the book [16] removes the assumption that the learning samples must be independent, and replaces it with the weaker assumption that the learning samples form a so-called mixing process.
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Conditions for PAC Learnability In this section we summarize the main results that provide necessary and sufficient conditions for a collection of sets to be PAC learnable. A central role is played by a combinatorial parameter known as the Vapnik–Chervonenkis (VC)-dimension. This notion is implicitly contained in the paper [14]. Definition 2 Let A be a collection of subsets of X. A set S D fx1 ; : : : ; x n g X is said to be shattered by A if, for every subset B S, there exists a set A 2 A such that S \ A D B. The Vapnik–Chervonenkis dimension of A, denoted by VC-dim(A), equals the largest integer n such that there exists a set of cardinality n that is shattered by A. One can think of the VC-dimension of A as a measure of the “richness” of the collection A. A set S is “shattered” by A if A is rich enough to distinguish between all possible subsets of S. Note that S is shattered by A if (and only if) one can “pick off” every possible subset of S by intersecting S with an appropriately chosen set A 2 A. In this respect, perhaps “completely distinguished” or “completely discriminated” would be a better term than “shattered.” However, the latter term has by now become standard in the literature. Note that, if A has finite VC-dimension, say d, then it is not rich enough to distinguish all subsets of any set containing d C 1 elements or more; but it is rich enough to distinguish all subsets of some set containing d elements (but not necessarily all sets of cardinality d). From the definition it is clear that, in order to shatter a set S of cardinality n, the concept class A must contain at least 2n distinct concepts. Thus an immediate corollary of the definition is that every finite collection of sets has finite VC-dimension, and that VC-dim(A) lg(jAj) : The importance of the VC-dimension in PAC learning theory arises from the following theorem, which is due to Blumer et al.; see [3]. In the statement of the theorem, we use the notion of a “consistent” algorithm. An algorithm is said to be “consistent” if it perfectly reproduces the training data. In other words, an algorithm is consistent if, for every labeled multisample ((x1 ; I T (x 1); : : : ; (x m ; I T (x m )), we have that I H m (x i ) D I T (x i )8i ; where H m is the hypothesis produced by the algorithm. Theorem 1 Suppose the concept class C has finite VC-dimension; then C is PAC-learnable for every probability measure over X. Let d denote the VC-dimension of C , and let fA m g be any consistent algorithm. Then the algorithm
is PAC; moreover, the quantity r(m; ) defined in (4) is bounded by 2em d m /2 r(m; ) 2 2 ; (6) d for every probability measure over X. Any consistent algorithm learns to accuracy and confidence ı provided at least
8d 8e 4 2 lg ; lg (7) m max ı samples are used. Thus Theorem 1 shows that the finiteness of the VC-dimension of the concept class C is a sufficient condition for so-called “distribution-free” PAC learnability. Moreover, (7) gives an estimate of the sample complexity, that is, the number of samples that suffice to achieve an accuracy of and a confidence of ı. Note that the sample complexity estimate is O(1/ log(1/ )) for fixed ı, and O(log(1/ı)) for fixed . This is often summarized by the statement “confidence is cheaper than accuracy.” There is a converse to Theorem 1. It can be shown that, in order for a concept class C to be PAC learnable for every probability measure on the set X, and for the learning rate to be uniformly bounded with respect to the underlying probability measure, the concept class C must have finite VC-dimension. Theorem 1 provides not only a condition for PAClearnability but also relates the learning rate to the VC-dimension of the concept class. Thus it is clearly worthwhile to determine upper bounds for the VC-dimension of various concept classes. Over the years, several researchers have studied this problem, using a wide variety of approaches, ranging from simple counting arguments to extremely sophisticated mathematical reasoning based on advanced ideas such as algebraic topology, model theory of real numbers, and so on. Since the list of relevant references would be too long, the reader is referred instead to Chap. 10 of [15,16] which contain a derivation of many of the important known results together with citations of the original sources. The Complexity of Learning In this section, we discuss briefly the complexity of a learning problem. Complexity can arise in two ways: sample complexity, and algorithmic complexity. Sample complexity refers to the rate at which the number m0 ( ; ı) grows with respect to 1/ and 1/ı, where m0 ( ; ı) is the smallest value of m0 such that (5) holds. Algorithmic complexity refers to the complexity of computing the hypothesis H m
Learning, System Identification, and Complexity
as a function of the number of samples of m, Each kind of complexity has its own interesting features.
complexity m0 ( ; ı)
Sample Complexity Let us begin by discussing the notions of covering numbers, and packing numbers. Given a concept class C consisting of subsets of a given set X and a probability measure P on X, let us observe that the quantity d P defines a pseudometric on the collection of (measurable) subsets of X, as follows: Given two subsets A; B X, define d P (A; B) D P(AB) ; where AB is the symmetric difference between the sets A and B. Note that d P is only a pseudometric because d P (A; B) D 0 whenever AB is a set of measure zero – it need not be the empty set (meaning that A D B). However, this is a minor technicality, and we can still refer to d P as defining a “distance” between subsets of X. Given the concept class C , the probability measure P (which in turn defines the pseudometric d P ), and a number , we say that a finite collection fC1 ; : : : ; C n g where each C i 2 C is an -cover of C if, for every A 2 C , we have d P (A; C i ) for at least one index i. To put it in other words, the collection fC1 ; : : : ; C n g is an -cover of C under the pseudometric d P if every set A 2 C is within a distance of from at least one of the sets C i . The -covering number N( ; C ; d P ) is defined to be the smallest number n such that there exists an -cover of cardinality n. Note that, in some cases, an -cover may not exist, in which case we take the -covering number to be infinity. The next, and related, notion is the -packing number of a concept class. A finite collection fC1 ; : : : ; C m g where each C i 2 C is said to be -separated if d P (C i ; C j ) > 8i ¤ j. In other words, the finite collection fC1 ; : : : ; C m g is -separated if the concepts are pairwise at a distance of more than from each other. The -packing number M( ; C ; d P ) is defined to be the largest number m such that there exists an -separated collection of cardinality m. It is not difficult to show that M(2 ; C ; d P ) N( ; C ; d P ) M( ; C ; d P ) 8 > 0 : See for instance Lemma 2.2 of [16]. The next two theorems, due to Benedek and Itai [2], are fundamental results on sample complexity. Theorem 2 Suppose C is a given concept class, and let > 0 be specified. Then C is PAC-learnable to accuracy provided the covering number N( /2; C ; d P ) is finite. Moreover, there exists an algorithm that is PAC with sample
32 N( /2; C ; d P ) ln : ı
Thus a concept class with a finite covering number N( /2; C ; d P ) is learnable to accuracy . The next theorem provides a kind of converse. Theorem 3 Suppose C is a given concept class, and let > 0 be specified. Then any algorithm that is PAC to accuracy requires at least lg M(2 ; C ; d P ) samples, where M(2 ; C ; d P ) denotes the 2 -packing number of the concept class C with respect to the pseudometric d P . One consequence of the theorem is that a concept class C is learnable to accuracy only if M(2 ; C ; d P ) is finite. This fact is used to construct examples of “non-learnable” concept classes; see for instance Example 6.10 in [16]. However, the question of interest here is whether there exists a concept class that is learnable, but for which the sample complexity m0 ( ; ı) grows faster than any polynomial in 1/ . In Sect. 3.3 in [7], the author constructs a concept class with the property that M(1/n; C ; d P ) 22
cn 1/4
;
where c is some constant. Thus it follows from Theorem 3 that the sample complexity satisfies 1/4
m0 ( /2; ı) 2c(1/ )
;
8 :
So it is after all possible to have a situation where a concept class is learnable, but the sample complexity grows faster than any polynomial in the reciprocal accuracy 1/ . Algorithmic Complexity We have seen from Theorem 1 that if a concept class C has finite VC-dimension, then every consistent algorithm is PAC. In the original paper [12], one studies not a single concept class C , but rather an indexed family of concept classes fCn g, for every integer n. The objective is to determine whether there exists an algorithm that PAC-learns each concept class, such that the computational complexity of learning Cn is polynomial in n. Note that we are not referring here to the sample complexity. In most problems in computer science, each concept class Cn is finite and therefore automatically has finite VC-dimension. Rather, we are referring to the difficulty of constructing a consistent hypothesis, and the computational complexity of this task as a function of n. In the original paper [12], it is shown that the problem of learning 3-CNF (3 conjunctive
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normal forms) is PAC learnable in this more restrictive sense. Note that 3-CNF in n Boolean variables consists of all conjunctions of clauses in which each clause is a disjunction of no more than three Boolean variables or their negations. For instance, if n D 5, then (x1 _ :x2 _ x5 ) ^ (x2 _ x3 _ :x4 ) ^ (:x1 _ :x3 ) ^ (:x2 _ :x3 _ :x5 ) belongs to 3-CNF. In this instance, Valiant shows that it is possible to construct a consistent hypothesis in polynomial time in n. Shortly after the publication of this paper, a very simple example was produced of a learning problem in which it is NP-hard to construct a consistent hypothesis. Specifically, consider the family of all conjunctions of at most k clauses, where each clause is a disjunction of various literals or their negations (without limit). For instance, if n D 5, then (x1 _ :x2 _ :x3 _ x5 ) ^ (:x1 _ x3 _ :x4 _ :x5 ) ^ (:x1 _ x2 _ x4 _ :x5 ) is an example of this kind of formula. In this case, it can be shown that, for each fixed k and n, the VC-dimension of the concept class is polynomial in k and n. Thus it follows from Theorem 1 that the sample complexity of learning is polynomial. However, it is shown in [11] that the computational complexity of finding a consistent hypothesis is intractable unless P D N P. In contrast, if each such Boolean formula is expressed as a 3-DNF, then it is possible to construct a consistent hypothesis in polynomial time with respect to n. System Identification as a Learning Problem The Need for a Quantitative Identification Theory By tradition, identification theory is asymptotic in nature. In other words, the theory analyzes what happens as the number of samples approaches infinity. However, there are situations where it is desirable to have finite time, or nonasymptotic, estimates of the rate at which the output of the identification process converges to the best possible model. For instance, in “indirect” adaptive control, one first carries out an identification of an unknown system, and after a finite amount of time has elapsed, designs a controller based on the current model of the unknown system. If one can quantify just how close the identified model is to the true but unknown system after a finite number of samples, then these estimates can be combined with robust control theory to ensure that indirect adaptive control leads to closed-loop stability.
Among the first papers to state that the derivation of finite-time estimates is a desirable property in itself are Weyer, Williamson and Mareels; see [20,21]. In those papers, it is assumed that the time series to which the system identification algorithm is applied consists of so-called “finitely-dependent” data; in other words, it is assumed that the time series can be divided into blocks that are independent. In a later paper [19], the author replaced the assumption of finite dependence by the assumption that the time series is ˇ-mixing. However, he did not derive conditions under which a time series is ˇ-mixing. In a still later paper [4], the authors study the case where the data is generated by general linear systems (Box–Jenkins models) driven by i.i.d. Gaussian noise, and the model family also belongs to the same class. In this paper, the authors say that they are motivated by the observation that “signals generated by dynamical systems are not ˇ-mixing in general”. This is why their analysis is limited to an extremely restricted class. However, their statement that dynamical systems do not generate ˇ-mixing sequences is simply incorrect. Specifically, the all the systems studied in [4] are themselves ˇ-mixing! In another paper [18], the present author has shown that practically any exponentially stable system driven by i.i.d. noise with bounded variance is ˇ-mixing. Hence, far from being restrictive, the results of [19] have broad applicability, though he did not show it at the time. Problem Formulation System identification viewed as a learning problem can be stated as follows: One is given a time series f(u t ; y t )g, where u t denotes the input to the unknown system at time t, and y t denotes the output at time t. We have that u t 2 U R l ; y t 2 Y R k for all t, where for simplicity it is assumed that both U and Y are compact sets. One is also given a family of models fh( ); 2 g parametrized by a parameter vector belonging to a set . Usually is a subset of a finite-dimensional Euclidean space. At time t, the data available to the modeler consists of all the past measurements until time t 1. Based on these measurements, the modeler chooses a parameter t , with the objective of making the best possible prediction of the next measured output y t . The method of choosing t is called the identification algorithm. To make this formulation a little more precise, let us Q introduce some notation. Define U :D 1 1 U, and define Y analogously. Note that the input sequence fu t g belongs to the set U, while the output sequence belongs to Y . Let U01 denote the one-sided infinite cartesian product Q0 U01 :D 1 U, and for a given two-sided infinite se-
Learning, System Identification, and Complexity
quence u 2 U, define ut :D (u t1 ; u t2 ; u t3 ; : : : ) 2
U01
:
Thus the symbol ut denotes the infinite past of the input signal u at time t. The family of models fh( ); 2 g consists of a collection of maps h( ); 2 , where each h( ) maps U01 into Y. Thus, at time t, the quantity h( ) ut D: yˆ t ( ) is the “predicted” output if the model parameter is chosen as . The quality of this prediction is measured by a “loss function” ` : Y Y ! [0; 1]. Thus `(y t ; h( ) u t ) is the loss we incur if we use the model h( ) to predict the output at time t, and the actual output is y t . Since we are dealing with a time series, all quantities are random. Hence, to assess the quality of the prediction made using the model h( ), we should take the expected value of the loss function `(y t ; h( )ut ) with respect to the law of the time series f(u t ; y t )g, which we denote by P˜u;y . Thus we define the objective function J( ) :D E `(y t ; h( ) u t ); P˜u;y : (8) Note that J( ) depends solely on and nothing else. A key observation at this stage is that the probability measure P˜u;y is unknown. This is because, if the statistics of the time series are known ahead of time, then there is nothing to identify! Definition 3 System Identification Problem: Given the time series f(u t ; y t )g with unknown law P˜u;y , construct if possible an iterative algorithm for choosing t as a function of t, in such a way that
from time 1 to time t. Note that the least squares error is in general unbounded, whereas the loss function ` is supposed to map into the bounded interval [0; 1]. To circumvent this technical difficulty, we can choose t 1X Jˆt ( ) D k y i yˆ i ( ) k2 ; t iD1
where () is any kind of saturating nonlinear function, for instance (s) D tanh(s). This corresponds to the choice `(y; z) D tanh(k y z k2 ). ˆ ) Note that, unlike the quantity J( ), the function J( can be computed on the basis of the available data. Hence, in principle at least, it is possible to choose t so as to minimize the “approximate” (but computable) objective funcˆ ) in the hope that, by doing so, we will somehow tion J( minimize the “true” (but uncomputable) objective function J( ). The next theorem gives some sufficient conditions for this approach to work. Specifically, if the estimates Jˆt ( ) converge uniformly to the correct values J( ) as t ! 1, where the uniformity is with respect to 2 , then the above approach works. The importance of this uniform convergence property was recognized very early on in identification theory. In particular, Caines [5,6] uses ergodic theory to establish this property, whereas Ljung [8] takes a more direct approach. Theorem 4 At time t, choose t so as to minimize Jˆt ( ); that is, t D arg min Jˆt ( ) :
J( t ) ! inf J( ) :
2
2
Let A General Result Since the law P˜u;y of the time series is not known, the objective function J( ) cannot be computed exactly. To circumvent this difficulty, one replaces the “true” objective function J() by an “empirical approximation,” as defined next. For each t 1 and each 2 , define the empirical error t
1X Jˆt ( ) :D `[y i ; h( ) ui ] : t
(9)
2
Define the quantity q(t; ) :D P˜u;y f sup j Jˆt ( ) J( )j > g :
k2 ,
then
t
1X k y i yˆ i ( ) k2 Jˆt ( ) D t iD1
would be the average cumulative mean-squared error between the actual output y i and the predicted error yˆ i ( ),
(10)
2
Suppose it is the case that q(t; ) ! 0 as t ! 1; 8 > 0. Then P˜u;y fJ( t ) > J C g ! 0
iD1
If we could choose `(y; z) Dk y z
J :D inf J( ) :
as t ! 1; 8 > 0 : (11)
In other words, the quantity J( t ) converges to the optimal value J in probability, with respect to the measure P˜u;y . Corollary 1 Suppose that q(t; ) ! 0 as t ! 1; 8 > 0. Given ; ı > 0, choose t0 D t0 ( ; ı) such that q(t; ) < ı8t t0 ( ; ı) :
(12)
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Learning, System Identification, and Complexity
Then P˜u;y fJ( t )
> J C g < ı8t t0 ( /3; ı) :
(13)
Proofs of the theorem and corollary can be found in [17]. A Result on the Uniform Convergence of Empirical Means The property of Theorem 4 does indeed hold in the commonly studied case where y t is the output of a “true” system corrupted by additive noise, and the loss function ` is the squared error, provided the model family and the true system satisfy some fairly reasonable conditions, namely: each system is BIBO (bounded input, bounded output) stable, all systems in the model family have exponentially decaying memory, and finally, an associated collection of input-output maps has finite P-dimension. Note that the P-dimension is a generalization of the VC-dimension to the case of maps assuming values in a bounded interval [0; 1]. By identifying a set with its support function, we can think of the VC-dimension as a combinatorial parameter associated with binary-valued maps. The P-dimension is defined in Chap. 4 of [15,16] Define the collection of functions H mapping U into R as follows: g( ) :D u 7!k ( f h( )) u0 k2 : U ! R ; G :D fg( ) : 2 g :
Now the various assumptions are listed.
Now we can state the main theorem. Theorem 5 Define the quantity q(t; ) as in (10) and suppose Assumptions A1 through A3 are satisfied. Given an > 0, choose k( ) large enough that k /4 for all k k( ). Then for all t k( ) we have
32e 32e ln q(t; ) 8k( )
d(k( ))
exp(bt/k( )c 2 /512M 2 ) ;
(15)
where bt/k( )c denotes the largest integer part of t/k( ). Theorem 6 Let all notation be as in Theorem 5. Then, in order to ensure that the current estimate t satisfies the inequality J( t ) J C (i. e., is -optimal) with confidence 1 ı, it is enough to choose the number of samples t large enough that " 24k( ) 32e 512M 2 bt/k( )c ln C d(k( )) ln 2 ı # 32e C d(k( )) ln ln : (16) Bounds on the P-Dimension
A1. There exists a constant M such that jg( ) u0 j M; 8 2 ; u 2 U : This assumption can be satisfied, for example, by assuming that the true system and each system in the family fh( ); 2 g is BIBO stable (with an upper bound on the gain, independent of ), and that the set U is bounded (so that fu t g is a bounded stochastic process). A2. For each integer k 1, define g k ( ) ut :D g( ) (u t1 ; u t2 ; : : : ; u tk ; 0; 0; : : : ) :
sentially means that each of the systems in the model family has decaying memory (in the sense that the effect of the values of the input at the distant past on the current output becomes negligibly small). A3. Consider the collection of maps G k D fg k ( ) : 2 g, viewed as maps from U k into R. For each k, this family G k has finite P-dimension, denoted by d(k).
In order for the estimate in Theorem 5 to be useful, it is necessary for us to derive an estimate for the P-dimension of the family of functions defined by G k :D fg k ( ) : 2 g ;
(17)
where g k ( ) : U k ! R is defined by g k ( )(u) :Dk ( f h( )) u k k2 ; where u k :D (: : : ; 0; u k ; u k1 ; : : : ; u1 ; 0; 0; : : : ) :
(14)
With this notation, define k :D sup sup j(g( ) g k ( )) u0 j : u2U 2
Then the assumption is that k is finite for each k and approaches zero as k ! 1. This assumption es-
Note that, in the interests of convenience, we have denoted the infinite sequence with only k nonzero elements as u k ; : : : ; u1 rather than u0 ; : : : ; u1k as done earlier. Clearly this makes no difference. In this section, we state and prove such an estimate for the commonly occurring case where each system model h( ) is an ARMA model where the parameter enters linearly. Specifically, it is
Learning, System Identification, and Complexity
supposed that the model h( ) is described by x tC1 D
l X
i i (x t ; u t ); y t D x t ;
(18)
3.
iD1
where D ( 1 ; : : : ; l ) 2 R l , and each i (; ) is a polynomial of degree no larger than r in the components of x t ; u t .
4. 5.
Theorem 7 With the above assumptions, we have that P-dim(G k ) 9l C 2l lg[2(r kC1 1)/(r 1)] 9l C 2l k lg(2r)
if
r > 1:
6.
(19)
In case r D 1 so that each system is linear, the above bound can be simplified to P-dim(G k ) 9l C 2l lg(2k) :
(20)
It is interesting to note that the above estimate is linear in both the number of parameters l and the duration k of the input sequence u, but is only logarithmic in the degree of the polynomials i . In the practically important case of linear ARMA models, even k appears inside the logarithm.
7. 8. 9. 10. 11. 12.
Future Directions
13.
At present, system identification theory is a mature, widely used subject. Statistical learning theory is newer and still needs to find its feet. In particular, the various estimates for sample complexity given here are deemed to be “too conservative” to be of practical use. Thus considerable research needs to be carried out on the issue of sample complexity. The interplay between system identification theory and statistical learning theory perhaps offers the greatest scope for interesting new work. At present, statistical learning theory allows us to derive finite time estimates of the disparity between the current estimates of a system, and the true but unknown system. However, these estimates do not always translate readily into a metric “distance” between the true but unknown system and the current estimate. In order to be useful in the context of robust controller design, it is necessary to derive estimates for some kind of metric distance. This is an important topic for future research.
14.
Acknowledgment The author thanks Erik Weyer for his careful reading and comments on an earlier draft of this article. Bibliography Primary Literature 1. Angluin D (1987) Queries and concept learning. Mach Learn 2:319–342 2. Benedek GM, Itai A (1988) Learnability by fixed distributions.
15. 16.
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21.
First Workshop on Computational Learning Theory. MorganKaufmann, San Mateo, pp 80–90 Blumer A, Ehrenfeucht A, Haussler D, Warmuth M (1989) Learnability and the Vapnik–Chervonenkis dimension. J ACM 36(4):929–965 Campi MC, Weyer E (2002) Finite sample properties of system identification methods. IEEE Trans Auto Control 47:1329–1334 Caines PE (1976) Prediction error identification methods for stationary stochastic processes. IEEE Trans Auto Control AC21:500–505 Caines PE (1978) Stationary linear and nonlinear system identification and predictor set completeness. IEEE Trans Auto Control AC-23:583–594 Hammer B (1999) Learning recursive data, Math of Control. Signals Syst 12(1):62–79 Ljung L (1978) Convergence analysis of parametric identification methods. IEEE Transa Auto Control AC-23:77–783 Ljung L (1999) System Identification: Theory for the User. Prentice-Hall, Englewood Cliffs Pintelon R, Schoukens J (2001) System Identification: A frequency domain approach. IEEE press, Piscataway Pitt L, Valiant L (1988) Computational limits on learning from examples. J ACM 35(4):965–984 Valiant LG (1984) A theory of the learnable. Comm ACM 27(11):1134–1142 Vapnik VN (1982) Estimation of Dependences Based on Empirical Data. Springer, Heidelberg Vapnik VN, Chervonenkis AY (1971) On the uniform convergence of relative frequencies to their probabilities. Theory Prob Appl 16(2):264–280 Vidyasagar M (1997) A Theory of Learning and Generalization. Springer, London Vidyasagar M (2003) Learning and Generalization: With Applications to Neural Networks and Control Systems. Springer, London Vidyasagar M, Karandikar RL (2001) System identification: A learning theory approach. Proc IEEE Conf on Decision and Control, Orlando, pp 2001–2006 Vidyasagar M, Karandikar RL (2001) A learning theory approach to system identification and stochastic adaptive control. In: Calafiore G, Dabbene F (eds) Probabilistic and Randomized Algorithms for Design Under Uncertainty. Springer, Heidelberg, pp 265–302 Weyer E (2000) Finite sample properties of system identification of ARX models under mixing conditions. Automatica 36(9):1291–1299 Weyer E, Williamson RC, Mareels I (1996) Sample complexity of least squares identification of FIR models. Proc 13th World Congress of IFAC, San Francisco, pp 239–244 Weyer E, Williamson RC, Mareels I (1999) Finite sample properties of linear model identification. IEEE Trans Auto Control 44(7):1370–1383
Books and Reviews Anthony M, Biggs N (1992) Computational Learning Theory. Cambridge University Press, Cambridge Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recognition. Springer, New York Haussler D (1992) Decision theoretic generalizations of the PAC model for neural net and other learning applications. Info Comput 100:78–150
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Kearns M, Vazirani U (1994) Introduction to Computational Learning Theory. MIT Press, Cambridge Kulkarni SR, Vidyasagar M (2001) Learning decision rules under a family of probability measures. IEEE Trans Info Thy Natarajan BK (1991) Machine Learning: A Theoretical Approach. Morgan-Kaufmann, San Mateo van der Vaart AW, Wallner JA (1996) Weak convergence and Empirical Processes. Springer, New York
Vapnik AN, Chervonenkis AY (1981) Necessary and sufficient conditions for the uniform convergence of means to their expectations. Theory Prob Appl 26(3):532–553 Vapnik VN (1995) The Nature of Statistical Learning Theory. Springer, New York Vapnik VN (1998) Statistical Learning Theory. Wiley, New York Weyer E, Campi MC (2002) Non-asymptotic confidence ellipsoids for the least squares error estimate. Automatica 38:1529–1547
Lyapunov–Schmidt Method for Dynamical Systems
Lyapunov–Schmidt Method for Dynamical Systems ANDRÉ VANDERBAUWHEDE Department of Pure Mathematics and Computer Algebra, Ghent University, Gent, Belgium Article Outline Glossary Definition of the Subject Introduction Lyapunov–Schmidt Method for Equilibria Lyapunov–Schmidt Method in Discrete Systems Lyapunov–Schmidt Method for Periodic Solutions Lyapunov–Schmidt Method in Infinite Dimensions Outlook Bibliography Glossary Bifurcation A change in the local or global qualitative behavior of a system under the change of one or more parameters; the critical parameter values at which the change takes place are called bifurcation values. Saddle-node bifurcation A bifurcation where two equilibria, usually one stable and the other unstable, merge and then disappear. A similar bifurcation can also happen with periodic orbits. Pitchfork bifurcation A bifurcation where an equilibrium loses stability and at the same time two new stable equilibria emerge; this type of bifurcation frequently appears in systems with symmetry. Period-doubling bifurcation A bifurcation where a fixed point of a mapping (discrete system) loses stability and a stable period-two point emerges. In a continuous system this corresponds to a periodic orbits losing stability while a new periodic orbit, with approximately twice the period of the original one, bifurcates. Equivariant system A system with symmetries; these symmetries form a group of (usually linear) transformations commuting with the vectorfield or the mapping generating the system. Reversible system A symmetric system where some of the symmetries involve a reversal of the time. Definition of the Subject The mathematical description of the state and the evolution of systems used to model all kind of applications requires in many cases a large or even an infinite number of
variables. In contrast, a number of basic phenomena observed in such systems (in particular bifurcation phenomena) are known to depend only on a very small number of variables. A simple example is the Hopf bifurcation, where the system changes from an equilibrium state to a periodic regime when the parameters cross a certain critical boundary, a transition which in a typical system essentially only involves two of the state variables. There are several ways in which the original system can be reduced to a much smaller system which captures the phenomenon one wants to analyze. There is a dynamical approach, usually in the form of some kind of center manifold reduction, where one tries to find a subsystem (invariant under the full system flow) which contains the core of the bifurcation under consideration. The Lyapunov–Schmidt method takes a rather different point of view, by concentrating purely on the objects under study, such as equilibria or periodic solutions, setting up an equation for these objects in an appropriate space, and then performing a reduction on this equation without any further reference to the original dynamics. This is done by splitting both the unknown and the equation in two parts, solving one of the resulting partial equations and substituting the solutions in the second equation. The outcome is typically a low-dimensional set of algebraic equations – usually called bifurcation equations – the solutions of which correspond in a one-to-one way with the chosen solutions of the original system. The analysis of these bifurcation equations (using appropriate techniques) allows to prove existence and multiplicity results and to describe bifurcation scenarios. The Liapunov–Schmidt reduction method has a long history, going back to the early 20th century. It has been used in many different forms, and sometimes under different names, to a large variety of problems both within and outside the strict domain of dynamical systems and bifurcation theory: continuous and discrete systems, ordinary and partial differential equations, functional differential equations, integral equations, variational problems, and so on. Within the theory of dynamical systems the method has been widely used (and remains to be used) for the study of bifurcations of equilibria and periodic orbits. Introduction The Lyapunov–Schmidt method is, as its name suggests, a method, a technique or procedure which can be applied to a wide class of problems and equations, ranging from local bifurcation problems for equilibria and periodic solutions in finite-dimensional systems to global existence and multiplicity results for solutions of infinite-dimensional problems. The common theme in each of these applica-
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Lyapunov–Schmidt Method for Dynamical Systems
tions is that of a reduction: the original equation is partly solved, and substitution of the partial solution in the remaining equations gives a reduced equation of a lower dimension than the original one. In many cases – but there are notorious exceptions – the reduced problem is finitedimensional, with low dimension (one, in the best case). Lyapunov–Schmidt Method – An Overview The basic idea behind the Lyapunov–Schmidt reduction method is a rather simple splitting algorithm which can in very general terms be described as follows. The starting point is an equation F (x) D 0
(1)
for an unknown x belonging to some space X, and depending on some parameter . The mapping F takes the space X into a space Y; in simple cases Y D X, and typically X is a subspace of Y (although this is not necessary for the method to work). For example, X D Y may be the phase space, and (1) the condition for x 2 X to be an equilibrium of a given dynamical system on X. Or X and Y may be spaces of periodic mappings, while (1) represents the condition for such periodic mapping to be a solution of a given system. The main ingredients for the splitting are two projections: a projection P of the space X onto a subspace U, and a projection Q of the space Y onto a subspace W; in most applications the subspaces U and W are finite-dimensional. The projection P is used to split the variable x: x D Px C (I P)x D u C v ; with u D Px 2 U and v D (I P)x 2 V D (I P)(X). The projection Q allows to split the Eq. (1) into a set of two equations, (I Q)F (x) D 0 and QF (x) D 0. Bringing the two splittings together shows that (1) is equivalent to the system (I Q)F (u C v) D 0 ;
QF (u C v) D 0 :
(2)
Of course these equations heavily depend on the choices for the projections P and Q. The main hypothesis which allows to perform a reduction is that the projections P and Q should be such that the first equation in (2) has for each (u; ) in a certain domain a unique solution v D v (u). Depending on the particular problem this solution v (u) will be obtained by different techniques, but mostly some form of the contraction principle or the implicit function theorem is used. Substituting the solution of the first equation of (2) into the second equation results in the following
reduced equation, usually called the bifurcation or determining equation: G (u) D QF u C v (u) D 0 :
(3)
Under the condition that both U and W are finite-dimensional this represents a finite set of equations for a finite number of unknowns. Every solution (u; ) of (3) corresponds to a solution (x; ) D (u C v (u); ) of (1), and conversely, every solution (x; ) of (1) (in the appropriate domain) can be written as x D uCv (u) for some solution (u; ) of (3). The main difficulty in analyzing the Eq. (3) consists in the fact that typically the solution v (u) of the first equation of (2) is not explicitly known, and therefore also the mapping G (which takes U into W) is not explicitly known. It is therefore important that the set-up (i. e. the projections P and Q used in the reduction) should be such that at least some qualitative properties of v and G , such as continuity and differentiability, can be obtained; when the method is used to study local bifurcation problems also a good approximation of these mappings, such as for example a few terms in their Taylor expansion, is required to obtain relevant conclusions. When the method is used to study (local) bifurcation phenomena one generally assumes that a particular solution (x0 ; 0 ) of (1) is known, and one wants to obtain the full solution set of (1) close to this given solution. In such case a convenient choice for the projections P and Q is related to the linearization L0 D DF0 (x0 ) of the mapping F : the projection P should be such that U D N(L0 ) D fx 2 X j L0 x D 0g, while Q should be a projection of Y onto a subspace W which is complementary to the range R(L0 ) D fA0 x j x 2 Xg of L0 , i. e. (I Q)(Y) D R(L0 ). With such choice, and under appropriate smoothness conditions for the mapping F , the existence of a unique solution v D v (u) of the first equation in (2) is guaranteed by the implicit function theorem for all (u; ) near (u0 ; 0 ) D (Px0 ; 0 ). Also the approximation of v (u) and G (u) by Taylor expansions works quite well in such case. Lyapunov–Schmidt Method – An Example As a simple but illustrative example on how the method works in practice consider the second order scalar ordinary differential equation y¨ C ˛ y˙ C y C g(y; y˙) D 0 ;
(4)
with ˛ 2 R a parameter and g(y; y˙) a smooth function of two variables such that g(y; y˙) D O((jyj C j y˙j)2 ) as (y; y˙) ! (0; 0). This equation has the equilibrium solution
Lyapunov–Schmidt Method for Dynamical Systems
y(t) 0 for all values of the parameter; the linearization at this equilibrium, y¨ C ˛ y˙ C y D 0 ;
For each ( ; v; ) 2 R V (2) R the term g( cos t C v(t); (1 C )( sin t C v˙(t))) appearing in (6) is decomposed as
(5) A( ; v; ) cos t C B( ; v; ) sin t C g˜( ; v; )(t) ;
has for ˛ D 0 a two-dimensional space of periodic solutions given by fy(t) D a cos t C b sin t j (a; b) 2 R2 g. One may then ask which of these periodic solutions survive when the linear Eq. (5) is replaced by the non-linear Eq. (4). Since (5) is an approximation of (4) only for small (y; y˙), and since (5) has periodic solutions only for ˛ D 0, the question can be stated more precisely as follows: determine, for all small values of ˛, all small periodic solutions of (4). An immediate problem which arises is that different periodic solutions may have different periods; indeed, the determination of the period is part of the problem. In the Lyapunov–Schmidt approach this is handled by setting y(t) D x((1 C )t), resulting in the new equation (1 C )2 x¨ C ˛(1 C )x˙ C x C g(x; (1 C )x˙ ) D 0 ; (6) 2-periodic solutions of (6) correspond to 2 (1C )1 -periodic solutions of (4). The new problem is then to determine, for all small values of (; ˛), all small 2-periodic solutions of (6). Observe that this problem is symmetric with respect to phase shifts: if x(t) is a 2-periodic solution, then so is x(t C ), for each . To solve this problem a possible 2-periodic solution x(t) is written in the form x(t) D a cos t C b sin t C v(t) D cos(t C ) C v(t) ; with v(t) a twice continuously differentiable function such that Z 2 Z 2 v(t) cos tdt D v(t) sin tdt D 0 : (7) 0
0
Taking the symmetry with respect to phase shifts into account one can set D 0, such that x(t) takes the form x(t) D cos t C v(t) ;
with v 2 V (2) ;
(8)
here V (2) denotes the space of 2-periodic C2 -functions satisfying (7), while V (0) will denote the space of continuous such functions. It is an easy exercise to verify that for each v˜ 2 V (0) the non-homogeneous linear equation v¨ C v D v˜(t) has a unique solution v 2 Z
(9) V (2) ,
t
sin(t s)˜v (s)ds C
v(t) D 0
given by 1 2
Z
2
sin(t s)s v˜(s)ds : 0
with 1 A( ; v; ) D
Z
2
g( cos t C v(t); 0
(1 C )( sin t C v˙(t))) cos tdt ; B( ; v; ) D
1
Z
2
g( cos t C v(t); 0
(1 C )( sin t C v˙(t))) sin tdt ; and g˜( ; v; ) 2 V (0) . The Eq. (6) then splits into two equations, namely [(2 C 2 ) C A( ; v; )] cos t C ˛(1 C ) C B( ; v; ) sin t D 0
(10)
and (1 C )2 v¨ C ˛(1 C )˙v C v C g˜( ; v; ) D 0 :
(11)
Both equations are satisfied for ( ; v) D (0; 0). Using the solvability property of (9) mentioned above in combination with the implicit function theorem proves that the Eq. (11) has for each sufficiently small ( ; ; ˛) a unique small solution v D v ( ; ; ˛) 2 V (2) , with v (0; ; ˛) D 0. Bringing this solution in (10) gives two scalar equations (2 C 2 ) C A( ; v ( ; ; ˛); ) D 0 and ˛(1 C ) C B( ; v ( ; ; ˛); ) D 0 :
(12)
It is easy to verify that A( ; v ( ; ; ˛); ) and B( ; v ( ; ; ˛); ) are of the order O( 2 ) as ! 0, such that ˜ ; ˛) A( ; v ( ; ; ˛); ) D A( ; ˜ and B( ; v ( ; ; ˛); ) D B( ; ; ˛) ; ˜ ; ˛) D B(0; ˜ ; ˛) D 0, and (12) reduces for with A(0; non-trivial solutions (i. e. solutions with ¤ 0) to ˜ ; ˛) D 0 (2 C 2 ) C A( ; ˜ and ˛(1 C ) C B( ; ; ˛) D 0 :
(13)
For small ( ; ; ˛) these equations can be solved by the implicit function theorem for (; ˛) D ( ( ); ˛ ( )), with ( (0); ˛ (0)) D (0; 0). This means that for each small one can find a parameter value ˛ D ˛ ( ) such that (4) has a periodic solution with “amplitude” equal to j j and
939
940
Lyapunov–Schmidt Method for Dynamical Systems
given by x ( )(t) D cos((1 C ( ))t) C v ( ; ( ); ˛ ( ))((1 C ( ))t) ; the minimal period of this solution is 2(1 C ( ))1 . In fact, the periodic orbits corresponding to and coincide, as can be seen from the following. The phase shift symmetry together with cos t D cos(t C ) and the uniqueness of the solution v ( ; ; ˛) of (11) implies that v ( ; ; ˛)(t) D v ( ; ; ˛)(t C ), from which one can easily deduce that A( ; v ( ; ; ˛); ) D A( ; v ( ; ; ˛); ), B( ; v ( ; ; ˛); ) D B( ; v ( ; ; ˛); ) and ( ( ); ˛ ( )) D ( ( ); ˛ ( )). Hence x ( )(t) D x ( )(t C (1 C ( ))1 ) : A further consequence is that ( ) and ˛ ( ) are in fact functions of 2 , i. e. ( ) D ˆ ( 2 ) D c 2 C h.o.t. and ˆ 2 ) D d 2 C h.o.t. ˛ ( ) D ˛( It is interesting to notice the difference in treatment between the auxiliary equation (11) and the bifurcation equations (12). The auxiliary equation (11) is solved by a direct application of the implicit function theorem, while the same implicit function theorem can only be used to solve the bifurcation equations after some manipulation, which in this example simply consists in dividing by to “eliminate” the trivial solution branch. A further remark is that when the nonlinearity g in (4) does not depend on y˙ (i. e. g D g(y)) then ˛ ( ) 0 and we have what is usually called a vertical bifurcation. For the example this means that the equation y¨ C ˛ y˙ C y C g(y) D 0 can have periodic solutions only for ˛ D 0, and that all small solutions of y¨ C y C g(y) D 0 are indeed periodic. To see how this well known fact fits into the Lyapunov–Schmidt reduction observe that the equation y¨ C y C g(y) D 0 is reversible, in the sense that if y(t) is a solution then so is y(t). Using this reversibility one can show that v ( ; ; 0)(t) D v ( ; ; 0)(t); therefore g( cos t C v ( ; ; 0)(t)) is an even function of t, and B( ; v ( ; ; 0); ) D 0 for all ( ; ). This immediately implies that ˛ ( ) 0, as claimed. The foregoing example forms a very simple case of a Hopf bifurcation; this bifurcation will be treated in a more general context in Subsect. “Hopf Bifurcation”. Lyapunov–Schmidt Method – Historical Background The Lyapunov–Schmidt method has been initiated in the early 20th century by A.M. Lyapunov [48] and E. Schmidt [55] in their (independent) studies of nonlinear integral equations. Later the method was used in diverse
forms in the work of (among many others) Cacciopoli [8], Cesari [9,10,11], Shimizu [56], Cronin [19], Bartle [5], Hale [26], Lewis [46], Vainberg and Trenogin [59], Antosiewicz [2], Rabinowitz [51], Bancroft, Hale and Sweet [4], Mawhin [49], Hall [33,34], Crandall and Rabinowitz [17], etc. Expositions on the method were given by Friedrichs [21], Nirenberg [50], Krasnosel’skii [43,44], Hale [27,28,30], Vainberg and Trenogin [60], and others. In particular the work of Cesari and Hale, who used the name alternative method, has put the method on a strong functional analytic basis. Apparently the name Lyapunov– Schmidt method was first introduced in the paper [59] of Vainberg and Trenogin; it has been used systematically for this type of approach since the seventies. In the seventies and the eighties the method was used extensively in a wide variety of bifurcation studies, not only in finitedimensional systems but also in delay equations [29], reaction-diffusion equations [18,20], boundary value problems [17,52], and so on. In particular it forms together with group representation theory and singularity theory the main cornerstone for equivariant bifurcation theory, the study of bifurcation problems in the presence of symmetry (see e. g. the monographs by Sattinger [54], Vanderbauwhede [61], Golubitsky, Stewart and Schaeffer [23,24], and Chossat and Lauterbach [13], and the more recent theory of bifurcations in cell networks initiated by Golubitsky, Pivato and Stewart [25]). Later the method was extended and adapted to study the bifurcation of periodic orbits in Hamiltonian systems [63], reversible systems [41,42] and mappings [15,62], and there have been several numerical implementations [3,6,38]. Lin’s method to study bifurcations near homoclinic and heteroclinic connections, based on the work of Lin [47] and developed by Sandstede [53] and Knobloch [40], is maybe from a technical point of view not a pure example of a Lyapunov–Schmidt reduction, but it certainly has all the flavors from it. In a different direction the method was used in combination with topological methods, in particular degree theory, to prove all kind of existence and multiplicity results and to obtain more global bifurcation results; a few references where one can read more about this are [22,37] and [39]. A recent account of the use of the Lyapunov–Schmidt method by mainly Russian authors is given in [57]. In recent years the Lyapunov–Schmidt method has been used to study (among many other topics) the existence of transition layers in singularly perturbed reaction-diffusion equations [31,32,58], and, in combination with a Newton scheme to solve small divisor problems, the existence of periodic and quasi-periodic solutions of nonlinear Schrödinger equations in one or more space dimensions [7,16].
Lyapunov–Schmidt Method for Dynamical Systems
The subsequent sections give a more detailed account of the method as it is used in the local bifurcation theory for equilibria and periodic orbits of finite-dimensional systems. Lyapunov–Schmidt Method for Equilibria The easiest example to illustrate the Lyapunov–Schmidt method is to consider the equilibria of a finite-dimensional parameter-dependent system of the form x˙ D f (x; ) ; Rn
(14) Rn
Rk
Rn
with x 2 and f : ! a smooth mapping. These equilibria are given by the solutions of the equation f (x; ) D 0 :
(15)
Let (x0 ; 0 ) be any solution of this equation; by a translation one can without loss of generality assume that (x0 ; 0 ) D (0; 0), and hence f (0; 0) D 0. The problem is to describe all solutions of (15) which are close to the given solution (0; 0). Let A0 be the Jacobian (n n)-matrix D x f (0; 0). If A0 is invertible, that is, if the nullspace N(A0 ) is trivial, then by the implicit function theorem the Eq. (15) has for each small parameter value a unique small solution x D x () 2 Rn . This means that the equilibrium x D 0 for D 0 can be continued for all nearby parameter values . Therefore one is mainly interested in the case where A0 is not invertible and N(A0 ) nontrivial, and that is when the Lyapunov–Schmidt method can be used to reduce the Eq. (15), as follows. Let P be the orthogonal projection in Rn onto N(A0 ). This projection allows to split a vector x 2 Rn as x D Px C (I P)x D u C v, with u D Px 2 N(A0 ) and v D (I P)x 2 V D N(A0 )? (the orthogonal complement of N(A0 )). It is known from elementary algebra that R(A0 ) D N(AT0 )? , where AT0 is the transpose of A0 , and that dim N(A0 ) D dim N(AT0 ). Hence, if Q is the orthogonal projection of Rn onto W D N(AT0 ), then (I Q)(Rn ) D R(A0 ) and Rn D W ˚ R(A0 ). Bringing the splitting x D u C v into (15) and using Q to split (15) into Q f (x; ) D 0 and (I Q) f (x; ) results in the following system of two equations, equivalent to (15):
and
f˜(u; v; ) D Q f (u C v; ) D 0 fˆ(u; v; ) D (I Q) f (u C v; ) D 0 :
(16)
Clearly f˜(0; 0; 0) D 0 and fˆ(0; 0; 0) D 0; also, Dv fˆ(0; 0; 0) is equal to the restriction of (I Q)A0 to V, which is an invertible linear operator from V onto R(A0 ). From the implicit function theorem one can then conclude that
the auxiliary equation fˆ(u; v; ) D 0 has, for each sufficiently small (u; ) 2 U R k , a unique small solution v D v (u; ) 2 V, with v : U R k ! V a smooth mapping such that v (0; 0) D 0. Moreover, differentiating the identity fˆ(u; v (u; ); ) D 0 and using the fact that A0 u D 0 for all u 2 U gives (I Q)A0 Du v (0; 0) D 0, and hence Du v (0; 0) D 0. Substitution of the solution v D v (u; ) in the first equation of (16) gives the bifurcation equation g(u; ) D f˜(u; v (u; ); ) D Q f (u C v (u; ); ) D 0 :
(17)
The bifurcation mapping g : U R k ! W is smooth, and direct calculation shows that g(0; 0) D 0 and Du g(0; 0) D 0 (just use the fact that QA0 x D 0 for all x 2 Rn ). In order to come to conclusions about the equilibria of (14) one has now to study the bifurcation Eq. (17). Two remarks about this equation are in order here. First, since the solution v D v (u; ) is only implicitly defined, the mapping v (u; ) and hence also g(u; ) are usually not explicitly known; however, it is in principle possible to obtain the Taylor expansions of these mappings around the point (u; ) D (0; 0). Second, since Du g(0; 0) D 0 it is not possible to use the implicit function theorem to solve (17) for u (or even part of u) as a function of the parameter . Lyapunov–Schmidt Reduction at a Simple Zero Eigenvalue In the case of just one single parameter (i. e. k D 1 and 2 R) one can argue as follows. If at some parameter value one has an equilibrium at which the linearization is invertible, then, as explained before, this equilibrium can be continued for nearby parameter values. This leads to one-parameter branches of equilibria of the form f(x (); ) j 2 Ig, with I R some interval. This continuation stops at parameter values D 0 where the linearization A D D x f (x (); ) becomes singular, i. e. when zero is an eigenvalue of A0 . For one-parameter families of matrices, such as fA j 2 Ig, this typically happens in such a way that zero will be a simple eigenvalue of A0 , which means that dim N(A0 ) D 1 and Rn D N(A0 ) ˚ R(A0 ). Shifting (x (0 ); 0 ) to the origin and using an appropriate linear transformation this motivates the following hypothesis: f (0; 0) D 0
and
A0 x D (0; Aˆ 0 v);
8x D (u; v) 2 R Rn1 D Rn ;
(18)
where (as before) A0 D D x f (0; 0) and where Aˆ 0 is an invertible (n 1) (n 1)-matrix. With this set-up the
941
942
Lyapunov–Schmidt Method for Dynamical Systems
Lyapunov–Schmidt reduction is straightforward: both P and Q are the projections in Rn onto the first component, the decomposition x D (u; v) used in (18) is precisely the one needed to apply the reduction, and the resulting bifurcation equation is a single scalar equation g(u; ) D f1 (u; v (u; ); ) D 0 :
(19)
As in the general case g(0; 0) D 0 and @g/@u(0; 0) D 0. Depending on some further hypotheses this bifurcation equation leads to different bifurcation scenario’s which are discussed in the next subsections. It should be noted that under appropriate conditions the bifurcation function g(u; ) also contains some stability information for the equilibria of (14) corresponding to solutions (u; ) of (19) (see e. g. [23]): such equilibrium is stable if @g/@u(u; ) < 0, and unstable if @g/@u(u; ) > 0. Saddle-Node Bifurcation
can write the bifurcation function g(u; ) in the form g(u; ) D u g˜(u; ) ; with g˜(u; ) D g˜(u; ) and g˜(0; 0) D 0 ;
hence the non-trivial solutions of (19) (these are solutions with u ¤ 0) are obtained by solving the equation g˜(u; ) D 0 :
g˜(u; ) D u2 C ı C h.o.t. ;
D
1 @3 f1 (0; 0; 0) ; 6 @u3
1 @2 f1 (0; 0; 0) ; 2 @u2
ˇD
@ f1 (0; 0; 0) : @
(20)
Assuming that ˇ ¤ 0 one can solve (19) for as a function of u to obtain D (u) D ˛ˇ 1 u2 C h.o.t.
ıD
@2 f1 (0; 0; 0) : @u@
(24)
Under the condition ı ¤ 0 it is possible to solve (23) for as a function of u, giving
Just keeping the lowest order terms in u and the bifurcation function g(u; ) can be written as
˛D
(23)
Further calculations show that
D (u) ;
g(u; ) D ˛u2 C ˇ C h.o.t. ;
(22)
with (0) D 0; (u) D (u) and (u) D ı 1 u2 C h.o.t.
(25)
When also ¤ 0 this results in the following bifurcation scenario when ı > 0 and < 0: the zero equilibrium is stable for < 0 and becomes unstable for > 0; at D 0 two branches of symmetrically related non-trivial equilibria bifurcate from the trivial branch of equilibria; these nontrivial equilibria exist for > 0 and are stable. For other signs of and ı similar scenario’s hold. This type of bifurcation is known as a pitchfork bifurcation.
(21)
If also ˛ ¤ 0 then one has, in case ˛ˇ > 0, for < 0 two equilibria, one stable and one unstable, which merge for D 0 and then disappear for > 0; in case ˛ˇ < 0 one has the opposite scenario. This is called a saddle-node or limit point bifurcation. Pitchfork Bifurcation Suppose that the vectorfield f (x; ) appearing in (14) has some symmetry properties, for example f (x; ) D f (x; ). Then clearly one has for each parameter value 2 R the trivial equilibrium x () D 0; setting (as before) A D D x f (0; ) one will typically find isolated parameter values where zero is a simple eigenvalue of A . Assuming that D 0 is such a critical parameter value, and working through the Lyapunov–Schmidt reduction, it is not hard to see that v (u; ) D v (u; ), g(u; ) D g(u; ) and g(0; ) D 0. Then clearly both coefficients ˛ and ˇ appearing in (20) will be zero, and the conditions for a saddle-node bifurcation are not satisfied. Instead, one
Transcritical Bifurcation The arguments used to obtain the pitchfork bifurcation remain partly valid in the following situation which is not necessarily related to symmetry. Suppose that the system (14) has a given equilibrium x () for each 2 R; such equilibrium may be a consequence of symmetry properties (see the case of a pitchfork bifurcation) or of some other property of the system. For example, in population models one always has the zero equilibrium; and in oscillation systems there is usually some basic equilibrium which persists for all parameter values. By a parameter dependent translation one can always arrange that x () D 0 and hence f (0; ) D 0 for all . Assuming as before that zero is a simple eigenvalue of A0 D D x f (0; 0) and applying the Lyapunov–Schmidt method one finds that v (0; ) D 0;
g(0; ) D 0 ; 8 2 R ) g(u; ) D u g˜(u; ) ;
(26)
with g˜(u; ) a smooth function such that g˜(0; 0) D 0.
Lyapunov–Schmidt Method for Dynamical Systems
So again non-trivial equilibria are obtained by solving the Eq. (23), in which now the function g˜(u; ) can be written as
1 @2 f1 (0; 0; 0) ; 2 @u2
The discrete analogue of the continuous system (14) has the form x jC1 D f (x j ; ) ;
g˜(u; ) D u C ı C h.o.t. ; D
Lyapunov–Schmidt Method in Discrete Systems
ıD
@2 f1 (0; 0; 0) : @u@
(27)
Assuming that ı ¤ 0 one can solve (23) for as a function of u, resulting in
j D 0; 1; 2; : : : ;
again with x j 2 Rn ( j 2 N) and f : Rn R k ! Rn smooth. For a given value of the parameter the equilibria of (29) are given by the fixed points of f D f (; ), i. e. by the solutions of the equation f (x; ) D x :
D (u) D ı
1
u C h.o.t.
(28)
When also ¤ 0 this represents a transcritical bifurcation: for each ¤ 0 there exists next to the trivial equilibrium at zero also a non-trivial equilibrium which approaches zero as tends to zero. For the same parameter value ¤ 0 the trivial and the nontrivial equilibria have opposite stability properties, and there is an exchange of stability at D 0. For example, if ı > 0 then the trivial equilibrium is stable for < 0 and unstable for > 0, while the non-trivial equilibrium is unstable for < 0 and stable for > 0. A remark about the condition ı ¤ 0 used to obtain the pitchfork and the transcritical bifurcation is in order here. In both cases it was assumed that zero is a simple eigenvalue of A0 D AD0 , where A D D x f (0; ). This simple eigenvalue can be continued for small values of , i. e. the matrix A has, for sufficiently small values of , a simple real eigenvalue () which depends smoothly on and is such that (0) D 0. It is then not very hard to prove that the condition ı ¤ 0 is equivalent to the transversality condition 0 (0) ¤ 0 which states that the simple eigenvalue () crosses zero with non-zero speed when crosses zero. Similar transversality conditions are needed to obtain most of the elementary bifurcations which appear in continuous systems such as (14) or in discrete systems such as considered in the next section. A further remark has to do with the condition that zero should be a simple eigenvalue of A0 D D x f (0; 0). In certain cases this condition can be satisfied by restricting a larger system to an appropriate invariant subspace. This happens in particular when one studies bifurcations in equivariant systems; in many such systems the simple zero eigenvalue can be obtained by restricting the full system to invariant subsystems consisting of solutions which are invariant under certain isotropy subgroups. Usually the symmetry also forces the origin to be an equilibrium. Pitchfork or transcritical bifurcations observed for such invariant subsystems form then special cases of the Equivariant Branching Lemma [12,61]; see [61] and [24] for more details.
(29)
(30)
The continuation and bifurcation of such fixed points under a change of the parameter can be studied by the same techniques as the ones used in Sect. “Lyapunov–Schmidt Method for Equilibria”: it is sufficient to replace f (x; ) by F(x; ) D f (x; ) x. At a given solution (x0 ; 0 ) 2 Rn R k of (29) the linearization L0 D D x F(x0 ; 0 ) of F0 D F(; 0 ) is given by L0 D A0 I n , where A0 D D x f (x0 ; 0 ) and I n is the identity matrix on Rn . Therefore, if 1 is not an eigenvalue of A0 then the fixed point x0 can be continued for all sufficiently close to 0 , and if 1 is a simple eigenvalue of A0 (and k D 1) then, depending on some further conditions, one may have at D 0 a saddlenode, a pitchfork bifurcation or a transcritical bifurcation of fixed points. Discrete systems such as (29) arise for example frequently in mathematical biology and mathematical economics. A special situation arises when the mapping f in (29) is a Poincaré map constructed near a given periodic orbit 0 of a continuous parameter-dependent autonomous (n C 1)-dimensional system (see e. g. [45]); in that case a fixed point of f corresponds to a periodic orbit of the original continuous system, usually with the origin corresponding to 0 . The periodic orbit 0 has always 1 as a Floquet multiplier; if this multiplier is simple, then 1 is not an eigenvalue of the linearization at the origin of the Poincaré map, and the periodic orbit 0 can be continued for nearby parameter values. If k D 1 (a single parameter) and the multiplier 1 of 0 has multiplicity two, then 1 will be a simple eigenvalue of the linearization of the Poincaré map, and one will typically obtain a saddlenode of fixed points of the Poincaré map, corresponding to a saddle-node of periodic orbits in the original continuous system. In particular cases such as the presence of symmetry one may also see a transcritical or pitchfork bifurcation of periodic orbits. Periodic Points in Discrete Systems Next to the bifurcation of fixed points for (29) one can also study the bifurcation of periodic points from fixed points.
943
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Lyapunov–Schmidt Method for Dynamical Systems
For any 2 R k and any integer q 1 a q-periodic point of f D f (; ) is a point x 2 Rn such that the orbit fx j j j 2 Ng of (29) starting at x0 D x has the property that x jCq D x j for all j 2 N. In case f is a Poincaré map associated to a periodic solution with minimal period T0 > 0 of a continuous system then such q-periodic point corresponds to a periodic orbit of the continuous system with minimal period near qT 0 . Such periodic solution is called a subharmonic solution (when q 2); the bifurcation of periodic points from fixed points for discrete systems such as (29) is therefore strongly related to subharmonic bifurcation in continuous systems. The periodicity condition x jCq D x j (for all j 2 N) is equivalent to the condition that x q D x0 D x, or to q
f (x) D x;
q
f D f ı f ı ı f
q
(32) This orbit space is finite-dimensional, with dimension qn. Any map g : Rn ! Rn can be lifted to a map gˆ : Oq ! Oq by setting 8z D (x j ) j2Z 2 Oq :
The shift operator is the linear mapping : Oq ! Oq defined by ( z) j D x jC1 ;
8j 2 Z ;
fˆ (z) D z :
(33)
The advantage of this equation over the equivalent Eq. (31) is that (33) is Z q -equivariant: both sides of the equation commute with , and if (z; ) 2 Oq R k is a solution, then so is ( j z; ) for any j 2 Z. This equivariance plays an important role in the bifurcation analysis for q-periodic points of (29) since it puts strong restrictions on the possible form of the bifurcation equation. The Z q -equivariance of (33) also fits nicely together with the Lyapunov– Schmidt reduction, as will be shown next.
(q times) : (31)
Hence a q-periodic point of f is a fixed point of f (the q-th iterate of f ), and therefore one could in principle study the bifurcation of q-periodic points of f by q studying the bifurcation of fixed points of f . Observe that q a fixed point of f is automatically also a fixed point of f , for any q 1. However, there is a better approach which takes explicitly into account an important property of (31), namely that if x is a solution then so are all the other points of the j orbit fx j D f (x) j j 2 Ng generated by x. The basic idea is the following: instead of considering all orbits of (29) and then finding all those which are q-periodic, one considers all q-periodic sequences with elements in Rn and then tries to determine which ones are actually orbits of the system (29). To work this out one has to introduce (for the chosen value of q) the space ˚ Oq D z D (x j ) j2Z j x j 2 Rn ; x jCq D x j ; 8 j 2 Z :
gˆ(z) D (g(x j )) j2Z ;
A q-periodic sequence z D (x j ) j2Z 2 Oq forms an orbit under the iteration of f if and only if f (x j ) D x jC1 for all j 2 Z, that is, if and only if
8z D (x j ) j2Z 2 Oq :
Observe that q D IOq , and hence generates a Z q -action on Oq (Z q is the cyclic group of order q, i. e. Z q D Z/qZ). Also gˆ ı D ı gˆ for any g : Rn ! Rn . And finally, identifying Oq with (Rn )q and using the standard scalar product it is easily seen that h z1 ; z2 i D hz1 ; z2 i for all z1 ; z2 2 Oq , i. e. is orthogonal and T D 1 .
Lyapunov–Schmidt Reduction for Periodic Points Assume that (30) has a solution (x0 ; 0 ) 2 Rn R k ; as before one can without loss of generality translate this solution to the origin and assume that f (0; 0) D 0. Consider then, for a given integer q 1, the following problem: (Pq ) Determine, for all small values of the parameter , all small q-periodic points of f . The bifurcation equation for this problem can be obtained by a Lyapunov–Schmidt reduction near the solution (z; ) D (0; 0) of (33), as follows. Linearization of (33) at (0; 0) gives the equation Aˆ 0 z D z ;
A0 D D x f (0; 0) ;
(34)
and the corresponding linear operator L0 D Aˆ 0 : Oq ! Oq . The nullspace N(L0 ) consists of the q-periodic orbits of the linear discrete system x jC1 D A0 x j and can be described as follows. Let q U D N A0 I n ;
ˇ S D A0 ˇU : U ! U
and : U ! Oq ; u 7! (u) D (S j u) j2Z :
(35)
Then N(L0 ) D (U), and is an isomorphism between U and N(L0 ). Observe that the subspace U of Rn is the sum of the eigenspaces of A0 corresponding to all eigenvalues 2 C of A0 for which q D 1, i. e. all eigenvalues which are q-th roots of unity. A further observation is that the linear operator S generates a Z q -action on U (since S q D I U by definition of U and S), and that (Su) D (u) ;
8u 2 U :
(36)
The description of R(L0 ) D N(L0T )? and the necessary projections can be somewhat simplified by assuming that all eigenvalues of A0 which are q-th roots of unity are semi-
Lyapunov–Schmidt Method for Dynamical Systems
q
simple, an assumption which implies that Rn D N(A0 q I n ) ˚ R(A0 I n ) D U ˚ Y; writing x 2 Rn as x D (u; y), with u 2 U and y 2 Y, and constructing an appropriate basis of Rn one can then put A0 in the following (diagonal) form: A0 x D A0 (u; y) D (Su; A˜0 y) ; 8x D (u; y) 2 U Y D Rn ;
(37)
where S is orthogonal, i. e. S T S D I U , and where A˜ 0 : Y ! q Y is such that A˜ 0 I Y is invertible. Under these assumptions (see [62] for the general case) it is easily seen that N(L0T ) D N(b A0T T ) is equal to N(L0 ) (use the fact that T 1 D and S T D S 1 ). Therefore R(L0 ) is the orthogonal complement in Oq D (Rn )q of N(L0 ) D (U), and every z 2 Oq has a unique decomposition of the form z D (u)C v ;
with u 2 U and v 2 V D R(L0 ): (38)
In particular, going back to (33) and replacing x by (u) Cv, one can write fˆ ( (u) C v) D (g(u; v; )) C h(u; v; ) ; with g : U V R k ! U and h : U V R k ! V smooth mappings such that g(0; 0; 0) D 0, h(0; 0; 0) D 0, Du g(0; 0; 0) D ˇS, Dv g(0; 0; 0) D 0, Du h(0; 0; 0) D 0, Dv h(0; 0; 0) D Aˆ 0 ˇV , and g(Su; v; ) D Sg(u; v; ) ; h(Su; v; ) D h(u; v; ) :
(39)
This last property is a consequence of (36), the Zq -equivariance of fˆ , and the fact that leaves the decomposition Oq D (U) ˚ V invariant since commutes with L0 . It follows from the foregoing that the Eq. (33) can be split in the two equations g(u; v; ) D Su
h(u; v; ) D v : (40) ˇ Since h(0; 0; 0) D 0, Dv h(0; 0; 0) D Aˆ 0 ˇV and L0 D Aˆ 0 is invertible on V D R(L0 ), the second equation in (40) can, locally near (u; v; ) D (0; 0; 0), be solved by the implicit function theorem for v D v (u; ); the mapping v : U R k is smooth, with v (0; 0) D 0 and Du v (0; 0) D 0. Also, (39) and the uniqueness of solutions given by the implicit function theorem imply that and
v (Su; ) D v (u; ) :
g (Su; ) D Sg (u; ) ;
8(u; ) 2 U R k :
(43)
This last property ensures that the bifurcation equation (42) is Z q -equivariant, i. e. commutes with the Z q -action generated by S on U, and if (u; ) 2 U R k is a so˜ ) for any u˜ belonging to the Zq -orbit lution then so is (u; fS j u j j 2 Zg generated by u. Period-Doubling Bifurcation Suppose that f (0; 0) D 0 and that A0 D D x f (0; 0) has 1 as a simple eigenvalue, while +1 is not an eigenvalue. Then, taking q D 2 in the foregoing, dim U D 1, Su D u for all u 2 U, and the bifurcation equation (42) is a scalar equation: g (u; ) D u ;
@g (0; 0) D 1 @u and g (u; ) D g (u; ) : with
(44)
Due to the symmetry the discussion of this bifurcation equations runs completely analogous to the one in Subsect. “Pitchfork Bifurcation” and leads, for k D 1 and under appropriate genericity conditions, to a pitchfork bifurcation. However, the interpretation of the bifurcation diagram is different in this case: the trivial solution u D 0 of (42) corresponds to a fixed point x0 () of f which changes stability as crosses zero, while the symmetric ˆ pairs ˙u() of nontrivial solutions which exist either for < 0 or for > 0 correspond to the two points xˆ () and f (xˆ ()) of a period-two orbit of f . Such bifurcation is known as a period-doubling bifurcation or a flip bifurcation. In case f represents a Poincaré map near a given (nontrivial) periodic orbit with minimal period T 0 of a continuous system such period-doubling bifurcation means the following: the given periodic orbit persists for nearby parameter values, with minimal period T near T 0 , but when the parameter crosses the critical value there bifurcates a branch of periodic solutions with minimal period near 2T0 ; so also in the continuous system one sees a period-doubling.
(41)
Substituting the solution v D v (u; ) into the first equation of (40) gives the bifurcation equation g (u; ) D Su ; with g (u; ) D g(u; v (u; ); ) :
This is an equation on the subspace U of Rn ; the mapping g : U R k ! U is smooth, with g (0; 0) D 0; Du g (0; 0) D S and
(42)
Bifurcation of q-Periodic Points for q 3 Next let q 3 and assume that f (0; 0) D 0 and that A0 has a pair of simple eigenvalues exp(˙i 0 ) ;
with 0 D
2 p ; q
gcd(p; q) D 1 : (45)
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Lyapunov–Schmidt Method for Dynamical Systems
Assume also that A0 has no other eigenvalues with q D 1; in particular +1 and 1 are no eigenvalues. Then dim U D 2 and the linear operator S in (37) has the matrix form cos 0 sin 0 SD : sin 0 cos 0 Identifying u D (u1 ; u2 ) 2 U D R2 with z D u1 Ciz2 2 C the expression (37) for A0 takes the more explicit form A0 x D A0 (z; y) D ei 0 z; A˜0 y ; 8x D (z; y) 2 C Y D Rn :
(46)
The Lyapunov–Schmidt reduction for q-periodic points as explained in Subsect. “Lyapunov–Schmidt Reduction for Periodic Points” leads to a complex bifurcation equation g (z; ) D ei 0 z ;
with g (z; 0) D ei 0 z C O(jzj2 ) and g ei 0 z; D ei 0 g (z; ) : (47)
It can be shown (see e. g. [24] or [62]) that the Z q -equivariance of g implies that g (z; ) D e i 0 C 1 (jzj2 ; ) z C 2 (jzj2 ; )¯z q1 C O(jzj qC1 );
(48)
with 1 and 2 complex valued functions and 1 (0; 0) D 0. The bifurcation equation (47) has for each 2 R k the trivial solution z D 0, corresponding to the continuation of the fixed point x D 0 at D 0 (remember that it was assumed that +1 is not an eigenvalue of A0 ). To obtain nontrivial solutions one can multiply (47) by z¯, set z D e i and divide by 2 ; the resulting equation has the form g˜( ; ; ) D 1 ( 2 ; ) C 2 ( 2 ; ) q2 ei q C O( q ) D 0 ;
(49)
with g˜( ; C 0 ; ) D g˜( ; ; ) and g˜( ; C ; ) D g˜( ; ; ). The standard way to handle (49) is to assume k D 2 (i. e. D (1 ; 2 ) 2 R2 ) together with the transversality condition @( 0) if x0 2 N(eA 0 T0 I n ). In order to exclude a possible bifurcation of equilibria one should require that zero is not an eigenvalue of A0 , which entails that without further loss of generality one can also assume that f (0; ) D 0 for all . By a time rescale one can put T0 D 2. The nullspace N(e2 A 0 I n ) is then given by the direct sum of the (real) eigenspaces corresponding to all pairs of eigenvalues of the form ˙ik (k D 1; 2; : : :). A further simplification in the formulation of the Lyapunov–Schmidt reduction occurs if one assumes that all such eigenvalues are semisimple, leading to the following hypotheses about (14) which will be assumed further on: (H) f (0; ) D 0 for all , A0 D D x f (0; 0) is invertible, and 1 is a semisimple eigenvalue of e2 A 0 , i. e. Rn D N(e2 A 0 I n ) ˚ R(e2 A 0 I n ) D U ˚ Y, where U D N(e2 A 0 I n ) and Y D R(e2 A 0 I n ). By an appropriate choice of basis one can then more explicitly assume that A0 x D A0 (u; y) D (Su; A˜0 y) ; 8x D (u; y) 2 U Y D Rn ;
(52)
with S : U ! U and A˜0 : Y ! Y linear and invertible, S anti-symmetric (S T D S), e2 S D I U and ˜ N(e2 A 0 I Y ) D f0g. The space of 2-periodic solutions of (51) is then given by fe S t u j u 2 Ug. The question one wants to answer is then which of these periodic solutions survive when we turn on the nonlinearities and change parameters. More precisely, the problem can be formulated as follows: (P) Determine, for all (T; ) 2 R R k near (2; 0), all small T-periodic solutions of (14). Lyapunov–Schmidt Reduction for the Problem (P) Analogically to the treatment for discrete systems in Sect. “Lyapunov–Schmidt Method in Discrete Systems” the first step should be to reformulate (P) in an appropriate orbit space; to do so one has to fix the period, which is done as
follows. Let xˆ (t) be a T-periodic solution of (14), define 2 R by (1 C )T D 2, and set x˜ (t) D xˆ ((1 C )1 t). ˜ is a 2-periodic solution of the equation Then x(t) (1 C )x˙ D f (x; ) :
(53)
Conversely, if x˜ (t) is a 2-periodic solution of (53) then xˆ (t) D x˜ ((1 C )t) is a T-periodic solution of (14), with T D 2(1 C )1 . Therefore the problem (P) is equivalent to (P* ) Determine, for all small (; ) 2 R R k , all small 2-periodic solutions of (53). 0 of all conThe orbit space for this problem is the space C2 n tinuous 2-periodic mappings x : R ! R ; this is a Banach space when one uses the norm kxk02 D supfkx(t)k j 0 one can define a natural S1 -action (where t 2 Rg. On C2 1 S D R/2R is the circle group) by the phase shift operator 0 ˙ : S 1 C2 ! C20 ; (; x) 7! ˙ (x) ;
with ˙ (x)(t) D x(t C ) :
(54)
S1 -action
The infinitesimal generator of this is defined 1 of continuously differentiable on the dense subspace C2 2-periodic mappings x : R ! Rn and is given by ˇ d ˇˇ 1 0 ! C2 ; x 7! D t x D ˙ (x) ; D t : C2 d ˇD0 1 (D t x)(t) D x˙ (t); 8(x; t) 2 C2 R:
(55)
1 is The subspace C2 1 the norm kxk2 D
itself a Banach space when one uses 1 kxk02 C kD t xk02 ; the space C2 0 is continuously imbedded in C2 . Also, any continuous 0 mapping g : Rn ! Rn can be lifted to a mapping on C2 by defining 0 0 gˆ : C2 ! C2 ; x 7! gˆ(x) ;
with gˆ(x)(t) D g(x(t)) ;
0 R: 8(x; t) 2 C2
Moreover, such a lifting commutes with the shift operator: gˆ ı ˙ D ˙ ı gˆ for all 2 S 1 . Using the foregoing definitions the problem (P ) 1 translates into finding all small solutions (x; ; ) 2 C2 k R R of the equation b f (x) D (1 C )D t x :
(56)
(Observe the analogy and the differences with (33)). This equation is S1 -equivariant: both sides commute with ˙ for all 2 S 1 ; therefore, if (x; ; ) is a solution, then so is (˙ (x); ; ) for all 2 S 1 . 1 R R k ! C 0 by F(x; ; ) D Next define F : C2 2 b f (1 C )D t ; then F(0; ; ) D 0, and the linearization 1 ! C 0 is given by L D b L0 D D x F(0; 0; 0) : C2 A0 D t . 0 2 1 is the space of all 2-periodic The nullspace N(L0 ) C2
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Lyapunov–Schmidt Method for Dynamical Systems
solutions of (51) which was described before; hence there exists an isomorphism : U ! N(L0 ) given by (u)(t) D eS t u ;
8u 2 U; 8t 2 R :
(57) ˇ It follows from the definition of U and S D A0 ˇU that S is the infinitesimal generator of an S1 -action on U given by (; u) 2 S 1 U 7! e S u 2 U : Also, (e S u) D ˙ (u) and (Su) D D t (u) for all (; u) 2 S 1 U. In order to describe R(L0 ) one has to find those z 2 0 for which the non-homogeneous linear 2-periodic C2 differential equation x˙ D A0 x C z(t)
(58)
has a 2-periodic solution. The Fredholm alternative for this problem (see e. g. [28] or [61]) states that this will be the case if and only if Z 2 1 hz(t); x(t)idt D 0 ; 8x 2 C2 : x˙ D AT0 x : 0
0 Stated differently: if we introduce on C2 a scalar product h; iC 0 by 2
Z hx; xˆ iC 0 D
2
ˆ hx(t); x(t)idt ;
2
0
0 8x ; xˆ 2 C2 ;
1 ! C0 and if we define the adjoint operator L0 : C2 2 T ? b by L0 D A0 C D t , then R(L0 ) D N(L0 ) . It follows easily from (52) that N(L0 ) D N(L0 ), and therefore 0 D N(L0 ) ˚ R(L0 ). Moreover, R(L0 ) D N(L0 )? and C2
N(L0 ) is finite-dimensional and R(L0 ) is a closed sub0 ; these properties imply that L : C 1 ! C 0 space of C2 0 2 2 is a Fredholm operator of index zero (see the next section for the definition). 0 One can associate the orthogonal splitting C2 D N(L0 ) ˚ R(L0 ) D (U) ˚ V (with V D R(L0 )) to a con0 ! C 0 such that R(P) D (U) tinuous projection P : C2 2 ˜ with P˜ : C 0 ! U and N(P) D V; explicitly P D ı P, 2 given by 1 ˜ P(x) D 2
Z
2
eSs u(s)ds ;
0 0 : 8x D x() D (u(); y()) 2 C2
a similar way b f ( (u) C v) D (g(u; v; )) C h(u; v; ), with g(u; v; ) D P˜ fˆ ( (u) C v) 2 U and h(u; v; ) D (IP) fˆ ( (u)Cv) 2 V; the mappings g : U V R k ! U and h : U V R k ! V are smooth, with g(0; 0; ) D 0, h(0; 0; ) D 0, and satisfy the equivariance relations g e S u; ˙ v; D eS g(u; v; ) and h e S u; ˙ v; D ˙ (h(u; v; )) ; 8 2 S 1 : Using the foregoing decompositions the Eq. (56) splits into two equations g(u; v; ) D (1 C )Su and
(60)
h(u; v; ) D (1 C )D t v ;
1 )RR k . to be solved for small (u; v; ; ) 2 U (V \C2 Since h(0; 0; ) D 0 and the restriction of L0 to 1 is an isomorphism of V \ C 1 onto V the secV \ C2 2 ond equation in (60) can be solved for v D v (u; ; ), 1 a smooth mapping, with v : U R R k ! V \ C2 v (0; ; ) D 0, Du v (0; 0; 0) D 0 and v (e S u; ; ) D ˙ (v (u; ; )) for all 2 S 1 . Bringing this solution into the first equation of (60) gives the bifurcation equation
g (u; ; ) D (1 C )Su ; with g (u; ; ) D g(u; v (u; ; ); ) :
(61)
The mapping g : U R R k ! U is smooth, with g (0; ; ) D 0, Du g (0; ; 0) D S and g (e S u; ; ) D eS g (u; ; ) ;
8 2 S 1 :
(62)
This S1 -equivariance of g implies that for each solution (u; ; ) 2 U RR k of (62) also (e S u; ; ) is a solution, for each 2 S 1 . Solving (61) near the origin and working back through the reduction allows to solve the problems (P ) and (P). The method outlined in this subsection can be adapted for special types of systems, such as Hamiltonian or reversible systems, and can also be generalized to the case where the linearization A0 has some nilpotencies (nonsemisimple eigenvalues); see e. g. [41,63] and [42] for more details. In these papers there is also established a link between the Lyapunov–Schmidt reduction and normal form theory.
(59)
Observe that P˜ ı ˙ D e S ı P˜ and P ı ˙ D ˙ ı P 0 D (U) ˚ for every 2 S 1 , and that the splitting C2 1 V is invariant under the S -action; as a consequence the 1 infinitesimal generator Dt of the S1 -action maps V \ C2 1 into V. Every x 2 C2 can then be written as x D (u)Cv, 1 ; in ˜ with u D P(x) 2 U and v D (I P)x 2 V \ C2
Hopf Bifurcation The simplest possible case in which the reduction of Subsect. “Lyapunov–Schmidt Reduction for the Problem (P)” can be applied is when A0 has ˙i as simple eigenvalues, while there are no other eigenvalues of the form ˙ki, with k 2 N (this is called a non-resonance condition). Then
Lyapunov–Schmidt Method for Dynamical Systems
dim U D 2 and S can be brought into the matrix form 0 1 SD : 1 0 Identifying u D (u1 ; u2 ) 2 U D R2 with z D u1 C iu2 2 C the linear operator A0 gets the form (compare with (52)) A0 x D A0 (z; y) D (iz; A˜0 y) ; 8x D (z; y) 2 C Rn2 D Rn :
(63)
The bifurcation equation is then a single complex equation g (z; ; ) D (1 C )iz ;
(64)
with g : C R R k ! C such that g (0; ; ) D 0, g (z; ; 0) D iz C O(jzj2 ) and g (ei z; ; ) D ei g (z; ; ) ;
8 2 S 1 :
(65)
This equivariance property of g implies (see e. g. [24]) that one can write g (z; ; ) D g˜(jzj2 ; ; )z ;
(67)
The solutions of this equation come in circles f(ei ; ; ) j 2 S 1 g; periodic solutions of (14) corresponding to different points on such circle are related to each other by phase shift, and therefore each circle of solutions of (67) corresponds to a unique periodic orbit of (14). The standard way to handle (67) is to assume k D 1 (i. e. 2 R) and R
@ g˜ (0; 0; 0) ¤ 0 : @
˛ 0 (0) ¤ 0 ;
(69)
this transversality condition means that as crosses zero the pair of eigenvalues ˛() ˙ iˇ() of the linearization A at the equilibrium x D 0 crosses the imaginary axis with non-zero transversal speed. Similarly as for equilibria the requirement for Hopf bifurcation to have a pair of simple eigenvalues crossing the imaginary axis can sometimes be achieved by restricting (56) to periodic loops with some particular space-time symmetries. Working out this idea leads to the Equivariant Hopf Bifurcation Theorem. There exists a massive literature developing and applying this theorem; for more details see the basic reference [24]. Lyapunov–Schmidt Method in Infinite Dimensions
with g˜(0; ; 0) D i : (66)
The Eq. (64) has for each (; ) the trivial solution z D 0, corresponding to the equilibrium x D 0 of (14) Using (66) the nontrivial solutions of (64) are given by the solutions of the equation g˜(jzj2 ; ; ) D (1 C )i :
Since the eigenvalues ˙i of A0 were assumed to be simple they can be continued to eigenvalues ˛() ˙ iˇ() of A D D x f (0), for small values of . It is not hard to prove that the condition (68) is equivalent to
(68)
Then (67) can be split in real and imaginary parts and solved by the implicit function theorem for (; ) D ( (jzj2 ); (jzj2 )), with ( (0); (0)) D (0; 0). For each sufficiently > 0 there is a parameter value D ( 2 ) such that the vectorfield f has a small periodic orbit of “amplitude” and with period 2(1 C ( 2 ))1 . Such bifurcation of periodic orbits from an equilibrium is known as a Hopf bifurcation, or Andronov–Hopf bifurcation, or even Poincaré–Andronov–Hopf bifurcation. The papers [35] and [1] contain some of the original work on this topic; an extensive discussion on the history of Hopf bifurcation together with many more references can be found in Chap. 9 of [14].
The Hopf bifurcation problem as explained in the preceding section leads to an Eq. (56) in the infinite-dimensional 0 orbit space C2 . Also bifurcation problems associated to boundary value problems for partial differential equations typically lead to equations in infinite-dimensional spaces (see e. g. [39] for some examples). The aim of this section is to briefly sketch a basic abstract setting which, under the appropriate conditions, allows to reduce these infinite-dimensional problems to finite-dimensional ones. Consider an equation F(x; ) D 0 ;
(70)
with F : X R k ! Y a smooth parameter-dependent mapping between two Banach spaces X and Y. Assuming that x D 0 is a solution for D 0, one can ask the question to what extend this solution persists for nearby values of , i. e. one wants to find all solutions (x; ) of (70) near (0; 0). The first step in solving this local problem consists in considering the linearized equation A0 x D 0 ;
with A0 D D x F(0; 0) ;
(71)
and to determine the nullspace and range of A0 . The following hypothesis about A0 allows to proceed with the Lyapunov–Schmidt reduction: (F) The continuous linear operator A0 : X ! Y is a Fredholm operator: (i) the nullspace N(A0 ) is finite-dimensional; (ii) the range R(A0 ) is closed in Y; (iii) the range R(A0 ) has finite codimension in Y.
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Lyapunov–Schmidt Method for Dynamical Systems
The index of such Fredholm operator is given by the number dim N(A0 ) codimR(A0 ); in many applications this index is zero, i. e. codimR(A0 ) D dim N(A0 ). Under the condition (F) there exist continuous linear projections P : X ! X and Q : Y ! Y such that R(P) D N(A0 ) and N(Q) D R(A0 ) ;
(72)
resulting in the topological decompositions X D N(A0 ) ˚ N(P) D U ˚V and Y D R(Q)˚R(A0 ) D W˚R(A0 ), with U D N(A0 ), V D N(P) and W D R(Q). The restriction of A0 to V is an isomorphism from V onto R(A0 ). Replacing x 2 X by x D Px C (I P)x D u C v in (70), and splitting the equation itself using Q results in the following system equivalent to (70): QF(u C v; ) D 0 ;
(I Q)F(u C v; ) D 0 :
(73)
The second of these equations is sometimes called the auxiliary equation; the corresponding mapping Faux : U V R k ! R(A0 ) defined by Faux (u; v; ) D (I Q)F(u Cv; ) is such that Faux (0; 0; 0) D ˇ0, Du Faux (0; 0; 0) D 0 and Dv Faux (0; 0; 0) D (I Q)A0 ˇV . Since this last operator is an isomorphism from V onto R(A0 ) one can apply the implicit function theorem to solve the auxiliary equation in the neighborhood of (u; v; ) D (0; 0; 0) for v D v (u; ). The mapping v : U R k ! V is smooth, with v (0; 0) D 0 and Du v (0; 0) D 0. Substituting v D v (u; ) into the first equation of (73) we obtain the bifurcation equation
G(u; ) D QF(u C v (u; ); ) D 0 :
(74)
Rk
! W is smooth, The bifurcation mapping G : U with G(0; 0) D 0, Du G(0; 0) D 0 and D G(0; 0) D QD F(0; 0). The further analysis of the bifurcation Eq. (74) will of course strongly depend on the particularities of the problem, such as the dimensions of U and W, possible symmetries, etc. A well known and widely applied particular case is the by now classical theorem of Crandall and Rabinowitz [17] on bifurcation from simple eigenvalues. The hypotheses for this result are the following: (CR) F : X R ! Y is smooth, with F(0; ) D 0 for all 2 R, and: (i) the nullspace of D x F(0; 0) is one-dimensional; (ii) the range of D x F(0; 0) is closed and has codimension one in Y; (iii) D D x F(0; 0) u0 62 R(D x F(0; 0)), where u0 ¤ 0 belongs to N(D x F(0; 0)). Under these hypotheses dim U D dim W D 1 in the foregoing, and therefore U D f u0 j 2 Rg and W D fw0 j 2 Rg for some non-zero elements u0 2 N(D x F(0; 0)) and w0 62 R(D x F(0; 0)). Also v (0; ) D 0 and G(0; ) D 0
for all 2 R; writing G( u0 ; ) D g( ; )w0 it follows that also g(0; ) D 0 for all . Hence Z g( ; ) D g˜( ; ) ;
1
with g˜( ; ) D 0
@g (s ; )ds : @
@g Clearly g˜(0; 0) D 0, since g˜(0; 0)w0 D @ (0; 0)w0 D Du G(0; 0) u0 and Du G(0; 0) D 0. Moreover,
@ g˜ @2 g (0; 0)w0 D (0; 0)w0 D D Du G(0; 0) u0 @ @@ D QD D x F(0; 0) u0 ¤ 0 ; @ g˜
by the hypothesis (CR)-(iii), and so @ (0; 0) ¤ 0. Nonzero solutions of the bifurcation equation g( ; ) D 0 are given by the solutions of g˜( ; ) D 0, and using the implicit function theorem this last equation can be solved for D ( ), with j j sufficiently small and (0) D 0. This gives, under the hypotheses (CR), a unique branch f(x ( ); ( )) Dj 0 < j j < 0 g ; with x ( ) D u0 C v ( u0 ; ( )) ; of nontrivial solutions of (70) bifurcation from the trivial branch f(0; ) j 2 Rg at D 0. Outlook The Liapunov–Schmidt method is a well established reduction technique which can and has been used on a wide variety of problems. The preceding sections just give a small glimpse of its use in local bifurcation theory; the references given in the introduction will point the reader to a whole plethora of other applications. The basics of the method are rather simple, but depending on the mappings and the spaces one is using a number of technical conditions (sometimes easy, sometimes more difficult) need to be verified before the reduction can be carried out. In view of the steady increase in applications which the method has seen over the last half century one can safely say that the Lyapunov–Schmidt method will remain a valuable tool for future research in bifurcation theory, dynamical systems theory and nonlinear analysis. Bibliography Primary Literature 1. Andronov A, Leontovich FA (1938) Sur la théorie de la variation de la structure qualitative de la division du plan en trajectoires. Dokl Akad Nauk 21:427–430 2. Antosiewicz HA (1966) Boundary value problems for nonlinear ordinary differential equations. Pacific J Math 17:191–197
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3. Ashwin P, Böhmer K, Mei Z (1993) A numerical Lyapunov– Schmidt method for finitely determined systems. In: Allgower E, Georg K, Miranda R (eds) Exploiting Symmetry in Applied and Numerical Analysis. Lectures in Appl Math, vol 29. AMS, Providence, pp 49–69 4. Bancroft S, Hale JK, Sweet D (1968) Alternative problems for nonlinear functional equations. J Differ Equ 4:40–56 5. Bartle R (1953) Singular points of functional equations. Trans Amer Math Soc 75:366–384 6. Böhmer K (1993) On a numerical Lyapunov–Schmidt method for operator equations. Computing 53:237–269 7. Bourgain J (1998) Quasi-periodic Solutions of Hamiltonian Perturbations of 2D Linear Schrödinger Equations. Ann Math 148(2):363–439 8. Cacciopoli R (1936) Sulle correspondenze funzionali inverse diramata: teoria generale e applicazioni ad alcune equazioni funzionali non lineari e al problema di Plateau. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur Sez I 24:258–263, 416–421 9. Cesari L (1940) Sulla stabilità delle soluzioni dei sistemi di equazioni differenziali lineari a coefficienti periodici. Atti Accad Ital Mem Cl Fis Mat Nat 6(11):633–692 10. Cesari L (1963) Periodic solutions of hyperbolic partial differential equations. In: Intern. Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, New York, pp 33–57 11. Cesari L (1964) Functional analysis and Galerkin’s method. Michigan Math J 11:385–414 12. Cicogna C (1981) Symmetry breakdown from bifurcations. Lett Nuovo Cimento 31:600–602 13. Chossat P, Lauterbach R (2000) Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol 15. World Scientific, Singapore 14. Chow S-N, Hale JK (1982) Methods of Bifurcation Theory. Grundlehren der Mathematischen Wissenschaften, vol 251. Springer, New York 15. Ciocco M-C, Vanderbauwhede A (2004) Bifurcation of Periodic Points in Reversible Diffeomorphisms. In: Elaydi S, Ladas G, Aulbach B (eds) New Progress in Difference Equations – Proceedings ICDEA2001, Augsburg. Chapman & Hall, CRC Press, pp 75–93 16. Craig W, Wayne CE (1993) Newton’s method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math 46:1409–1498 17. Crandall MG, Rabinowitz PH (1971) Bifurcation from simple eigenvalues. J Funct Anal 8:321–340 18. Crandall MG, Rabinowitz PH (1978) The Hopf bifurcation theorem in infinite dimensions. Arch Rat Mech Anal 67:53–72 19. Cronin J (1950) Branch points of solutions of equations in Banach spaces. Trans Amer Math Soc 69:208–231; (1954) 76:207– 222 20. Da Prato G, Lunardi A (1986) Hopf bifurcation for fully nonlinear equations in Banach space. Ann Inst Henri Poincaré 3:315– 329 21. Friedrichs KO (1953–54) Special Topics in Analysis. Lecture Notes. New York University, New York 22. Gaines R, Mawhin J (1977) Coincidence Degree, and Nonlinear Differential Equations. Lect Notes in Math, vol 568. Springer, Berlin 23. Golubitsky M, Schaeffer DG (1985) Singularities and Groups in Bifurcation Theory, vol 1. Appl Math Sci, vol 51. Springer, New York
24. Golubitsky M, Stewart IN, Schaeffer DG (1988) Singularities and Groups in Bifurcation Theory, vol 2. Appl Math Sci, vol 69. Springer, New York 25. Golubitsky M, Pivato M, Stewart I (2004) Interior symmetry and local bifurcation in coupled cell networks. Dyn Syst 19:389–407 26. Hale JK (1954) Periodic solutions of non-linear systems of differential equations. Riv Mat Univ Parma 5:281–311 27. Hale JK (1963) Oscillations in nonlinear systems. McGraw-Hill, New York 28. Hale JK (1969) Ordinary Differential Equations. Wiley, New York 29. Hale JK (1977) Theory of Functional Differential Equations. Appl Math Sciences, vol 3. Springer, New York 30. Hale JK (1984) Introduction to dynamic bifurcation. In: Salvadori L (ed) Bifurcation Theory and Applications. Lecture Notes in Math, vol 1057. Springer, Berlin, pp 106–151 31. Hale JK, Sakamoto K (1988) Existence and stability of transition layers. Japan J Appl Math 5:367–405 32. Hale JK, Sakamoto K (2005) A Lyapunov–Schmidt method for transition layers in reaction-diffusion systems. Hiroshima Math J 35:205–249 33. Hall WS (1970) Periodic solutions of a class of weakly nonlinear evolution equations. Arch Rat Mech Anal 39:294–332 34. Hall WS (1970) On the existence of periodic solutions of the 2p equation Dtt u C (1)p Dx u D "f (t; x; u). J Differ Equ 7:509– 526 35. Hopf E (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber Math Phys Kl Sachs Acad Wiss Leipzig 94:1–22. English translation in: Marsden J, McCracken M (1976) Hopf Bifurcation and its Applications. Springer, New York 36. Iooss G, Joseph D (1981) Elementary Stability and Bifurcation Theory. Springer, New York 37. Ize J (1976) Bifurcation theory for Fredholm operators. Mem AMS 174 38. Jepson A, Spence A (1989) On a reduction process for nonlinear equations. SIAM J Math Anal 20:39–56 39. Kielhöfer H (2004) Bifurcation Theory, An Introduction with Applications to PDEs. Appl Math Sciences, vol 156. Springer, New York 40. Knobloch J (2000) Lin’s method for discrete dynamical systems. J Differ Eq Appl 6:577–623 41. Knobloch J, Vanderbauwhede A (1996) A general reduction method for periodic solutions in conservative and reversible systems. J Dyn Diff Eqn 8:71–102 42. Knobloch J, Vanderbauwhede A (1996) Hopf bifurcation at k-fold resonances in conservative systems. In: Broer HW, van Gils SA, Hoveijn I, Takens F (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel, pp 155–170 43. Krasnosel’skii MA (1963) Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford 44. Krasnosel’skii MA, Vainiko GM, Zabreiko PP, Rutitskii YB, Stesenko VY (1972) Approximate solutions of operator equations. Wolters-Noordhoff, Groningen 45. Kuznetsov YA (1998) Elements of Applied Bifurcation Theory, 2nd edn. Appl Math Sci, vol 112. Springer, New York 46. Lewis DC (1956) On the role of first integrals in the perturbations of periodic solutions. Ann Math 63:535–548 47. Lin XB (1990) Using Melnikov’s method to solve Silnikov’s problems. Proc Roy Soc Edinb 116A:295–325
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48. Lyapunov AM (1906) Sur les figures d’équilibre peu différentes des ellipsoï, des d’une masse liquide homogène dotées d’un mouvement de rotation. Zap Akad Nauk St Petersb, pp 1–225; (1908), pp 1–175; (1912), pp 1–228; (1914), pp 1–112 49. Mawhin J (1969) Degré topologique et solutions périodiques des systèmes différentiels non linéaires. Bull Soc Roy Sci Liège 38:308–398 50. Nirenberg L (1960–61) Functional Analysis. Lecture Notes. New York University, New York 51. Rabinowitz PH (1967) Periodic solutions of nonlinear hyperbolic partial differential equations. Comm Pure Appl Math 20:145–206 52. Rabinowitz PH (1976) A Survey of Bifurcation Theory. In: Cesari L, Hale JK, LaSalle J (eds) Dynamical Systems, An International Symposium, vol 1. Academic Press, New York, pp 83–96 53. Sandstede B (1993) Verzweigungstheorie homokliner Verdopplungen. Ph D Thesis, University of Stuttgart, IAAS Report No. 7 54. Sattinger DH (1979) Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Math, vol 762. Springer, Berlin 55. Schmidt E (1908) Zur Theorie des linearen und nichtlinearen Integralgleichungen, 3 Teil. Über die Auflösung der nichtlinearen Integralgleichungen und die Verzweigung ihrer Lösungen. Math Ann 65:370–399 56. Shimizu T (1948) Analytic operations and analytic operational equations. Math Japon 1:36–40 57. Sidorov N, Loginov B, Sinitsyn A, Falaleev M (2002) Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications. Mathematics and Its Applications, vol 550. Kluwer, Dordrecht 58. Taniguchi M (1995) A Remark on Singular Perturbation Meth-
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Maximum Principle in Optimal Control
Maximum Principle in Optimal Control VELIMIR JURDJEVIC University of Toronto, Toronto, Canada Article Outline Glossary Definition of the Subject Introduction The Calculus of Variations and the Maximum Principle Variational Problems with Constraints Maximum Principle on Manifolds Abnormal Extrema and Singular Problems Future Directions Bibliography Glossary Manifolds Manifolds M are topological spaces that are covered by a set of compatible local charts. A local chart is a pair (U; ) where U is an open set on M and is a homeomorphism from U onto an open set in Rn . If is written as '(x) D (x1 ; : : : ; x n ) then x1 ; : : : ; x n are called coordinates of a point x in U. Charts are said to be compatible if for any two charts (U; ) and (V; ) the mapping ' 1 : (U \ V) ! (U \ V ) is smooth, i. e. admits derivatives of all orders. Such manifolds are also called smooth. Manifolds on which the above mappings ' 1 are analytic are called analytic. Tangent and cotangent spaces The vector space of tangent vectors at a point x in a manifold M will be denoted by Tx M. The tangent bundle TM of M is the union [fTx M : x 2 Mg. The cotangent space at x, consisting of all linear functions on Tx M will be denoted by Tx M. The cotangent bundle of M, denoted by T M is equal to the union [fTx M : x 2 Mg. Lie groups Lie groups G are analytic manifolds on which the group operations are compatible with the manifold structure, in the sense that both (g; h) ! gh from G G onto G and g ! g 1 from G onto G are analytic. The set of all n n non-singular matrices is an n2 dimensional Lie group under matrix multiplication, and so is any closed subgroup of it. Absolutely continuous curves A parametrized curve x : [0; T] ! Rn is said to be absolutely continuous if each derivative R T (dx i )/(dt)(t) exists almost everywhere in [0;RT] 0 j(dx i )/(dt)(t)jdt < 1, and t x(t2 ) x(t1 ) D t12 (dx)/(dt)(t)dt for almost all points
t1 and t2 in [0; T]. This notion extends to curves on manifolds by requiring that the above condition holds in any system of coordinates. Control systems and reachable sets A control system is any system of differential equations in Rn or a manifold M of the form dx dt (t) D F(x(t); u(t)); where u(t) D (u1 (t); : : : ; u m (t)) is a function, called control function. Control functions are assumed to be in some specified class of functions U, called the class of admissible controls. For purposes of optimization U is usually assumed to consist of functions measurable and bounded on compact intervals [t1 ; t2 ] that take values in some a priori specified set U in Rm . Under the usual existence and uniqueness assumptions on the vector fields F, each control u(t) : R ! Rm and each initial condition x0 determine a unique solution x(x0 ; u; t) that passes through x0 at t D 0. These solutions are also called trajectories. A point y is said to be reachable from x0 at time T if there exists a control u(t) defined on the interval [0; T] such that x(x0 ; u; T) D y. The set of points reachable from x0 at time T is called the reachable set at time T. The set of points reachable from x0 at any terminal time T is called the reachable set from x0 . Riemannian manifolds A manifold M together with a positive definite quadratic form h ; i : Tx M Tx M ! R that varies smoothly with base point x is called Riemannian. If x(t), t 2 [0; T] is a curve on q M then its Riemannian length is given by RT RT 0 0 h(dx)/(dt); (dx)/(dt)i dt. Riemannian metric is a generalization of the Euclidean metric in Rn given P by hx; yi D niD1 x i y i . Definition of the Subject The Maximum Principle provides necessary conditions that the terminal point of a trajectory of a control system belongs to the boundary of its reachable set or to the boundary of its reachable set at a fixed time T. In practice, however, its utility lies in problems of optimization Rt in which a given cost functional t01 f (x(t); u(t))dt is to be minimized over the solutions (x(t); u(t)) of a control system dx dt D F(x(t); u(t)) that conform to some specified boundary conditions at the end points of the inoptimal trajectory terval [t0 ; t1 ]. For then, the extended Rt ¯ ¯ ¯ (x¯0 (t); x(t); u(t)) with x¯0 (t) D t0 f (x¯ (t); u(t))dt must be on the boundary of the reachable set associated with extended control system dx0 D f (x(t); u(t)) ; dt
dx D F(x(t); u(t)) : dt
(1)
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In the original publication [24] the Maximum Principle is stated for control systems in Rn in which the controls u(t) D (u1 (t); : : : ; u m (t)) take values in an arbitrary subset U of Rm and are measurable and bounded on each interval [0; T], under the assumptions that the cost functional f and each coordinate F i of the vector field F in i Eq. (1) together with the derivatives @F are continuous @x j ¯ where U¯ denotes the topological closure of U. on Rn U, ¯ and the corresponding trajecA control function u(t) ¯ tory x¯ (t) of dx D F(x(t); u(t)) each defined on an interval dt [0; T] are said to be optimal relative to the given bound¯ ary submanifolds S0 and S1 of Rn if x(0) 2 S0 ; x¯ (T) 2 S1 , and Z T Z S ¯ f (x¯ (t); u(t))dt f (x(t); u(t))dt (2) 0
0
for any other solution x(t) of dx dt D F(x(t); u(t)) that satisfies x(0) 2 S0 and x(S) 2 S1 . Then Proposition 1 (The Maximum Principle (MP)) Suppose ¯ ¯ that (u(t); x(t)) is an optimal pair on the interval [0; T]. Then there exist an absolutely continuous curve p(t) D (p1 (t); : : : ; p n (t)) on the interval [0; T] and a multiplier p0 0 with the following properties: P 1. p20 C niD1 p2i (t) > 0, for all t 2 [0; T], 2.
dp i dt (t)
@H
¯ D @x pi 0 (x¯ (t); p(t); u(t)), i D 1; : : : ; n, a. e. P in [0; T] where H p 0 (x; p; u) D p0 f (x; u) C niD1 p i F i (x; u).
¯ D M(t) a. e. in [0; T] where M(t) 3. H p 0 (x¯ (t); p(t); u(t)) denotes the maximum value of H p 0 (x¯ (t); p(t); u) relative to the controls u 2 U. Furthermore, 4. M(t) D 0 for all t 2 [0; T]. 5. hp(0); vi D 0, v 2 Tx(0) ¯ S0 and hp(T); vi D 0, v 2 Tx(T) S . These conditions, known as the transversality ¯ 1 conditions, become void when the manifolds S0 and S1 reduce to single points x0 and x1 . The maximum principle is also valid for optimal problems in which the length of the interval [0; T] is fixed, ¯ satisfies RinT the sense that theR Toptimal pair (x¯ (t); u(t)) ¯ ¯ f ( x (t); u(t))dt f (x(t); u(t))dt for any other tra0 0 jectory (x(t); u(t)) with x(0) 2 S0 and x(T) 2 S1 . In this context the maximum principle asserts the existence of a curve p(t) and the multiplier p0 subject to the same conditions as stated above except that the maximal function M(t) must be constant in the interval [0; T] and need not be necessarily equal to zero. These two versions of the (MP) are equivalent in the sense that each implies the other [1].
Pairs (x(t); p(t)) of curves that satisfy the conditions of the maximum principle are called extremal. The extremal curves that correspond to the multiplier p0 ¤ 0 are called normal while the ones that correspond to p0 D 0 are called abnormal. In the normal case it is customary to reduce the multiplier p0 to p0 D 1. Since the original publication, however, the maximum principle has been adapted to control problems on arbitrary manifolds [12] and has also been extended to more general situations in which the vector fields that define the control system are locally Lipschitz rather than continuously differentiable [9,27]. On this level of generality the Maximum Principle stands out as a fundamental principle in differential topology that not only merges classical calculus of variations with mechanics, differential geometry and optimal control, but also reorients the classical knowledge in two major ways: 1. It shows that there is a natural “energy” Hamiltonian for arbitrary variational problems and not just for problems of mathematical physics. The passage to the appropriate Hamiltonians is direct and bypasses the Euler–Lagrange equation. The merits of this observation are not only limited to problems with inequality constraints for which the Euler-equation is not applicable; they also extend to the integration procedure of the extremal equations obtained through the integrals of motion. 2. The Hamiltonian formalism associated with (MP) further enriched with geometric control theory makes direct contact with the theory of Hamiltonian systems and symplectic geometry. In this larger context, the maximum principle brings fresh insights to these classical fields and also makes their theory available for problems of optimal control. Introduction The Maximum Principle of Pontryagin and his collaborators [1,12,24] is a generalization of C. Weierstrass’ necessary conditions for strong minima [29] and is based on the topological fact that an optimal solution must terminate on the boundary of the extended reachable set formed by the competing curves and their integral costs. An important novelty of Pontryagin’s approach to the calculus of variations consists of liberating the variations along an optimal trajectory of the constricting condition that they must terminate at the given boundary data. Control theoretic context induces a natural class of variations that generates a cone of directions locally tangent to the reachable set at the terminal point defined by the optimal trajectory. As a consequence of optimality, the direction of
Maximum Principle in Optimal Control
the decreasing cost cannot be in the interior of this cone. This observation leads to the separation theorem, a generalization of the classic Legendre transform in the calculus of variations, that ultimately produces the appropriate Hamiltonian function. The maximum principle asserts that the Hamiltonian that corresponds to the optimal trajectory must be maximal relative to the completing directions, and it also asserts that each optimal trajectory is the projection of an integral curve of the corresponding Hamiltonian field. The methodology used in the original publication extends the maximum principle to optimal control problems on arbitrary manifolds where, combined with Lie theoretic criteria for reachable sets, it stands out as a most important tool of optimal control available for problems of mathematical physics and differential geometry. Much of this article, particularly the selection of the illustrating examples is devoted to justifying this claim. The exposition begins with comparisons between (MP) and the classical theory of the calculus of variations in the absence of constraints. Then it proceeds to optimal control problems in Rn with constraints amenable by the (MP) stated in Proposition 1. This section, illustrated by two famous problems of classical theory, the geodesic problem on the ellipsoid of C.J.G. Jacobi and the mechanical problem of C. Newman–J. Moser is deliberately treated by control theoretic means, partly to illustrate the effectiveness of (MP), but mostly to motivate extensions to arbitrary manifolds and to signal important connections to the theory of integrable systems. The exposition then shifts to the geometric version of the maximum principle for control problems on arbitrary manifolds, with a brief discussion of the symplectic structure of the cotangent bundle. The maximum principle is first stated for extremal trajectories (that terminate on the boundary of the reachable sets) and then specialized to optimal control problems. The passage from the first to the second clarifies the role of the multiplier. There is brief discussion of canonical coordinates as a bridge that connects geometric version to the original formulation in Rn and also leads to left invariant adaptations of the maximum principle for problems on Lie groups. Left invariant variational control problems on Lie groups make contact with completely integrable Hamiltonian systems, Lax pairs and the existence of spectral parameters. For that reason there is a section on the Poisson manifolds and the symplectic structure of the coadjoint orbits of a Lie group G which is an essential ingredient of the theory of integrable systems. The exposition ends with a brief discussion of the abnormal and singular extremals.
The Calculus of Variations and the Maximum Principle The simplest problems in the calculus of variations can be formulated as optimal control problems of the form U D Rn , dx dt (t) D u(t), S0 D fx0 g and S1 D fx1 g, with one subtle qualification connected with the basic terminology. In the literature on the calculus of variations one usu¯ defined on an inally assumes that there is a curve x(t) terval [0; T] that provides a “local minimum” for the inteRT gral 0 f (x(t); dx (t))dt in the sense that there is a “neighdt borhood” N in the space of curves on [0; T] such that RT RT d x¯ dx ¯ 0 f (x (t); dt (t))dt 0 f (x(t); dt (t))dt for any other curve x(t) 2 N that satisfies the same boundary conditions as x¯ (t). There are two distinctive topologies, strong and weak on the space of admissible curves relative to which optimality is defined. In strong topology admissible curves consist of absolutely continuous curves with bounded derivatives on [0; T] in which an neighborhood N consists of all admissible curves x(t) such that kx¯ (t) x(t)k < for all t 2 [0; T]. In this setting local minima are called strong minima. For weak minima admissible curves consist of continuously differentiable curves on [0; T] with ¯ defined by an neighborhood of x(t) dx¯ dx (t) < for all t 2 [0; T] : kx¯ (t) x(t)k C (t) dt dt (3) Evidently, any strong local minimum that is continuously differentiable is also a weak local minimum. The converse, however, may not hold (see p. 341 in [12]). The Maximum Principle is a necessary condition for local strong minima x¯ (t) under suitable restriction of the state space. It is a consequence of conditions (1) and (3) in Proposition 1 that the multiplier p0 can not be equal to 0. Then it may be normalized to p0 D 1 and (MP) can be rephrased in terms of the excess function of Weierstrass [7,29] as
d x¯ f (x¯ (t); u) f x¯ (t); (t) dt n X d x¯ @f x¯ (t); (t) (u¯ i u i ); for all u 2 Rn ; @u i dt iD1
(4) because the critical points of H (x¯ (t); p(t); u) D f (x¯ (t); P u) C niD1 p i (t)u i relative to u 2 Rn are given by p i (t) D @f ¯ yields the maximum of (x¯ (t); u). Since ddtx¯ (t) D u(t) @u i
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H (x¯ (t); p(t); u) it follows that
p i (t) D
@f ¯ (x¯ (t); u(t)) : @u i
(5)
Combining Eq. (5) with condition 2 of the maximum principle yields the Euler–Lagrange equation in integrated R t @f @f ¯ ¯ 0 dx i (x¯ (); u()) d D c, with c form @u i (x¯ (t); u(t)) a constant, which under further differentiability assumptions can be stated in its usual form @f d @f ¯ ¯ (x¯ (t); u(t)) (x¯ (); u()) d D 0 : (6) dt @u i dx i As a way of illustration consider Example 2 (The harmonic oscillator) The problem of minRT imizing 0 21 (mu2 kx 2 )dt over the trajectories of dx dt D u leads to the family of Hamiltonians Hu D 12 (mu2 kx 2 ) C pu. According to the Maximum Principle every optimal trajectory x(t) is the projection of a curve p(t) that dp Hu D kx(t), subject to the maximality satisfies dt D @@x condition that 1 (mu(t)2 kx(t)2 ) C p(t)u(t) 2 1 (mv 2 kx(t)2 ) C p(t)v 2
(7)
for any choice of v. That implies that the optimal control that generates x(t) is of the form u(t) D
1 p(t) m
(8)
which then further implies that optimal solutions are the integral curves of a single Hamiltonian function H D 1 2 1 2 2m p C 2 kx . This Hamiltonian is equal to the total en2
ergy of the oscillator. The Euler–Lagrange equation ddtx2 D du k dt D m x(t) is easily obtained by differentiating, but there is no need for it since all the information is already contained in the Hamiltonian equations. It follows from the above that the projections of the q qextremal curves are given by x(t) D A cos t mk CB sin t mk for arbitrary constants A and B. It can be shown, by a separate argument q[12], that the preceding curves are optimal
if and only if t mk . This example also illustrates that the Principle of Least Action in mechanics may be valid only on small time intervals [t0 ; t1 ] (in the sense that it yields the least action). Variational Problems with Constraints The early applications of (MP) are best illustrated through time optimal problems for linear control systems
dx dt (t) D Ax(t) C Bu(t) with control functions u(t) D (u1 (t); : : : ; u r (t)) taking values in a compact neighborhood U of the origin in Rr . Here A and B are matrices of appropriate dimensions such that the “controllability” matrix [B; AB; A2 B; : : : ; An1 B] is of rank n. In this situation if (x(t); u(t)) is a time optimal pair then the corresponding Hamiltonian H p 0 (x; p; u) D p0 C hp(t); Ax(t) C Bu(t)i defined by the Maximum Principle is equal to 0 almost everywhere and is also maximal almost everywhere relative to the controls u 2 U. The rank condition together with the fact that p(t) is the solution of a linear differendp tial equation dt D AT p(t) easily implies that the control u(t) cannot take value in the interior of U for any convergent sequence of times ft n g otherwise, lim p(t n ) D 0, and therefore p0 D 0, which in turn contradicts condition 1 of Proposition 1. It then follows that each optimal control u(t) must take values on the boundary of U for all but possibly finitely many times. This fact is known as the Bang-Bang Principle, since when U is a polyhedral set then optimal control “bangs” from one face of U to another. In general, however, optimal controls may take values both in the interior and on the boundary of U, and the extremal curves could be concatenations of pieces generated by controls with values in the interior of U and the pieces generated by controls with values on the boundary. Such concatenations may exhibit dramatic oscillations at the juncture points, as in the following example.
RT Example 3 (Fuller’s problem) Minimize 12 0 x12 (t)dt over dx 2 1 the trajectories of dx dt D x2 ; dt D u(t) subject to the constraint that ju(t)j 1. Here T may be taken fixed and sufficiently large that admits an optimal trajectory x(t) D (x1 (t); x2 (t)) that transfers the initial point a to a given terminal point b in T units of time. Evidently, the zero trajectory is generated by zero control and is optimal relative to a D b D 0. The (MP) reveals that any other optimal trajectory is a concatenation of this singular trajectory and a bang-bang trajectory. At the point of juncture, whether leaving or entering the origin, optimal control oscillates infinitely often between the boundary values ˙1 (in fact, the oscillations occur in a geometric sequence [12,19]). This behavior is known as Fuller’s phenomenon. Since the original discovery Fuller’s phenomena have been detected in many situations [18,30]. The Maximum Principle is the only tool available in the literature for dealing with variational problems exhibiting such chattering behavior. However, even for problems of geometry and mechanics which are amenable by the classical methods the (MP) offers certain advantages as the following examples demonstrate.
Maximum Principle in Optimal Control
Example 4 (Ellipsoidal Geodesics) This problem, initiated and solved by C.G. Jacobi in 1839 [23] consists of finding the curves of minimal length on a general ellipsoid hx; A1 xi D
x12 x2 x2 C 22 C C 2n D 1 : 2 an a1 a2
(9)
Recall that the length of a curve x(t) on an interRT dx val [0; T] is given by 0 k dx dt (t)kdt, where k dt (t)k D p ((dx1 )/(dt))2 C C ((dx n )/(dt))2 . This problem can be recast as an optimal control problem either as a time optimal problem when the curves are parametrized by arc length, or as the problem of minimizing the energy funcRT 2 tional 12 0 k dx dt (t)k dt over arbitrary curves [15]. In the latter formulation the associated optimal control probRT lems consists of minimizing the integral 12 0 ku(t)k2 dt over the trajectories of dx dt (t) D u(t) that satisfy hx(t); A1 x(t)i D 1. Since there are no abnormal extremals in this case, it follows that the adjoint curve p(t) associated with an optimal trajectory x(t) must maximize Hu D 12 kuk2 C hp(t); ui on the cotangent bundle of the manifold x : hx(t); A1 x(t)i 1 D 0 . The latter is naturally identified with the constrains G1 D G2 D 0, where G1 D hx(t); A1 x(t)i 1 and G2 D hp; A1 xi. According to the Lagrange multiplier rule the correct maximum of Hu subject to these constrains is obtained by maximizing the function Gu D 12 jjujj2 Chp; uiC1 G1 C2 G2 relative to u. The the maximal Hamiltonian is given by H D 12 jjpjj2 C 1 G1 C 2 G2 obtained by substituting u D p. The correct multipliers 1 and 2 are determined by requiring that the integral curves ! of the associated Hamiltonian vector field H respect the hA1 p;pi constraints G1 D G2 D 0. It follows that 1 D 2jjA1 xjj and 2 D 0. Hence the solutions are the projections of the integral curves of hA1 p;
pi 1 H D kpk2 C G1 restricted to G1 D G2 D 0 : 2 2kA1 xk2 (10) The corresponding equations are dx @H D D p; dt @p
and
dp @H hA1 p; pi : D D dt @x kA1 xk2
(11)
The projections of these equations on the ellipsoid that reside on the energy level H D 12 are called geodesics. It is
well known in differential geometry that geodesics are only locally optimal (up to the first conjugate point). The relatively simple case, when the ellipsoid degenerates to the sphere occurs when A D I. Then the above hp;pi Hamiltonian reduces to H D 12 (kpk2 ) C 2kxk2 (kxk2 1) and the corresponding equations are given by dx D p; dt
dp D kpk2 x : dt
(12)
It follows by an easy calculation that the projections x(t) of Eqs. (12) evolve along the great circles because x(t) ^ dx dx dt (t) D x(0) ^ dt (0). The solutions in the general case can be obtained either by the method of separation of variables inspired by the work of C.G.J. Jacobi (see also [2]), or by the isospectral deformation methods discovered by J. Moser [22], which are somewhat mysteriously linked to the following problem. Example 5 (Newmann–Moser problem) This problem concerns the motion of a point mass on the unit sphere hx; xi D 1 that moves in a force field with quadratic potential V D 12 hx; Axi with A an arbitrary symmetric matrix. Then the Principle of Least Action applied to the 1 2 Lagrangian L D 12 k dx defines an optimal dt k 2 hx; Axi RT control problem by maximizing 0 L(x(t); u(t))dt over the trajectories of dx dt D u(t), subject to the constraint G1 D jjxjj2 1 D 0, whose extremal equations etc. In fact, H D 12 (kpk2 C hx; Axi) C 1 G1 C 2 G2 , where hp;xi
G2 D hx; pi; 1 D kxk2 and 2 D
kpk2 2kxk2
hAx; xi.
dx dp D p; D Ax C (hAx; x kpk2 ix dt dt
(13)
Equations (13) can be recast in matrix form dP (t) D [K(t); P(t)] ; dt
dK (t) D [P(t); A] dt
(14)
with P(t) D x(t) ˝ x(t) I and K(t) D x(t) ^ p(t) D x(t) ˝ p(t) p(t) ˝ x(t), where [M; N] denotes the matrix commutator N M M N. Equations (14) admit a Lax pair representation in terms of scalar parameter .
1 dL D P(t); L (t) ; dt where L (t) D P(t) K(t) 2 A ;
(15)
that provides a basis for Moser’s method of proving integrability of Eqs. (13). This method exploits the fact that
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the spectrum of L (t) is constant, and hence the functions k; D Trace(Lk ) are constants of motion for each and k > 0. Moreover, these functions are in involution with each other (to be explained in the next section). Remarkably, Eqs. (11) can be transformed to Eqs. (13) from which then can be inferred that the geodesic ellipsoidal problem is also integrable [22]. It will be shown later that this example is a particular case of a more general situation in which the same integration methods are available.
1. x(t) is the projection of (t) in [0; T] and d dt (t) D E u(t) ((t)), a. e. in [0; T], where Hu(t) () D (F(x; H ¯ u(t))), 2 Tx M. 2. (t) ¤ 0 for all t 2 [0; T]. 3. Hu(t) ((t)) D (t)(F(x(t); u(t))) (t)(F(x(t); v)) for all v 2 U a. e. in [0; T]. 4. If the extremal curve is extremal relative to the fixed terminal time then Hu(t) ((t)) is constant a. e. in [0; T], otherwise, Hu(t) ((t)) D 0 a. e. in [0; T].
Maximum Principle on Manifolds
An absolutely continuous curve (t) that satisfies the conditions of the Maximum Principle is called an extremal. R T Problems of optimization in which a cost functional 0 f (x(t); u(t))dt is to be minimized over the trajectories of a control system in M subject to the prescribed boundary conditions with terminal time either fixed or variable are reduced to boundary problems defined above in the same manner as described in the introductory part. Then each optimal trajectory x(t) is equal to the projection of an ˜ D R M relaextremal for the extended system (1) on M ˜ tive to the extended initial conditions x˜0 D (0; x0 ). If T M is identified with R T M and its points ˜ are written as (0 ; ) the Hamiltonian lifts of the extended control system are of the form
The formulation of the maximum principle for control systems on arbitrary manifolds requires additional geometric concepts and terminology [1,12]. Let M denote an n-dimensional smooth manifold with Tx M and Tx M denoting the tangent and the cotangent space at a point x 2 M. The tangent bundle TM is equal to the union [fTx M : x 2 Mg, and similarly the cotangent bundle T M is equal to fTx M : x 2 Mg. In each of these cases there is a natural bundle projection on the base manifold. In particular, x 2 M is the projection of a point 2 T M if and only if 2 Tx M. The cotangent bundle T M has a canonical symplectic form ! that turns T M into a symplectic manifold. This implies that for each function H on T M there is a vector E V ) for all vecE on T M defined by V (H) D !(H; field H tor fields V on T M. In the symplectic context H is called E is called the Hamiltonian vector field Hamiltonian and H corresponding to H. Each vector field X on M defines a function H X () D (X(x)) on T M. The corresponding Hamiltonian vector E X is called the Hamiltonian lift of X. In particular, field H control systems dx dt (t) D F(x(t); u(t)) lift to Hamiltonians Hu parametrized by controls with Hu () D (X u (x)) D (F(x; u)), 2 Tx M. With these notations in place consider a control system dx dt (t) D F(x(t); u(t)) on M with control functions u(t) taking values in an arbitrary set U in Rm . Suppose that the system satisfies the same assumptions as in Proposition 1. Let Ax 0 (T) denote the reachable set from x0 at time T and let Ax 0 D [fAx 0 (T) : T 0g. The control u that generates trajectory x(t) from x0 to the boundary of either Ax 0 (T) or Ax 0 is called extremal. For extremal trajectories the following version of the Maximum Principle is available. Proposition 6 (Geometric maximum principle (GMP)) Suppose that a trajectory x(t) corresponds to an extremal control u(t) on an interval [0; T]. Then there exists an absolutely continuous curve (t) in T M in the interval [0; T] that satisfies the following conditions:
¯ Hu (0 ; ) D 0 f (x; u) C (F(x; u(t))); 0 2 R ; 2 Tx M :
(16)
For each extremal curve (0 (t); (t)) associated with the extended system 0 (t) is constant because the Hamiltonian Hu (0 ; ) is constant along the first factor of the extended state space. In particular, 0 0 along the extremals that correspond to optimal trajectories. As before, such extremals are classified as normal or abnormal depending whether 0 is equal to zero or not, and in the normal case the variable 0 is traditionally reduced to 1. The Maximum principle is then stated in terms of the reduced Hamiltonian on T M in which 0 appear as parameters. In this context Proposition 6 can be rephrased as follows: Proposition 7 Let x(t) denote an optimal trajectory on an interval [0; T] generated by a control u(t). Then there exist a number 0 2 f0; 1g and an absolutely continuous curve (t) in T M defined on the interval [0; T] that projects onto x(t) and satisfies: 1. (t) ¤ 0 whenever 0 D 0. E 2. d dt (t) D Hu(t) (0 ; (t)) a. e. on [0; T]. 3. Hu(t) (0 ; (t)) Hv (0 ; (t)) for any v 2 U, a. e. in [0; T]. When the initial and the terminal points are replaced
Maximum Principle in Optimal Control
by the submanifolds S0 and S1 there are transversality conditions: 4. (0)(v) D 0, v 2 Tx(0) S0 and (T)(v) D 0, v 2 Tx(T) S1 . The above version of (MP) coincides with the Euclidean version when the variables are expressed in terms of the canonical coordinates. Canonical coordinates are defined as follows. Any choice of coordinates (x1 ; x2 ; : : : ; x n ) on M induces coordinates (v1 ; : : : ; v n ) of vectors in Tx M relative to the basis @x@ ; : : : ; @x@ and it also induces co1 n ordinates (p1 ; p2 ; : : : ; p n ) of covectors in Tx M relative to the dual basis dx1 ; dx2 ; : : : ; dx n . Then (x1 ; x2 ; : : : ; x n ; p1 ; p2 ; : : : ; p n ) serves as a system of coordinates for points in T M. These coordinates in turn define coordinates for tangent vectors in T (T M) relative to the basis @x@ 1 ; : : : ; @x@n ; @p@ 1 ; : : : ; @p@ n . The symplectic form ! can then be expressed in terms of vector fields X D Pn Pn @ @ @ @ iD1 Vi @x i C Pi @p i and Y D iD1 Wi @x i C Q i @p i as !(x;p) (X; Y) D
n X
Q i Vi Pi Wi :
(17)
iD1
The correspondence between functions H and their E is given by Hamiltonian fields H E H(x; p) D
n X @H @ @H @ ; @p i @x i @x i @p i
(18)
iD1
E are given by the and the integral curves (x(t); p(t)) of H usual differential equations dx i @H ; D dt @p i
dp i @H ; D dt @x i
i D 1; : : : ; n :
(19)
Any choice of coordinates on T M that preserves Eq. (17) is called canonical. Canonical coordinates could be defined alternatively as the coordinates that preserve the Hamiltonian equations (19). In terms of the canonical coordinates the Maximum Principle of Proposition 7 coincides with the original version in Proposition 1. Optimal Control Problems on Lie Groups Canonical coordinates are not suitable for all variational problems. For instance, variational problems in geometry and mechanics often have symmetries that govern the solutions; to take advantage of these symmetries it may be necessary to use coordinates that are compatible with the symmetries and which may not necessarily be canonical. That is particularly true for optimal problems on Lie groups that are either right or left invariant.
To elaborate further, assume that G denotes a Lie group (matrix group for simplicity) and that g denotes its Lie algebra. A vector field X on G is called left-invariant if for every g 2 G, X(g) D gA for some matrix A in g, i. e., X is determined by its tangent vector at the group identity. Similarly, right-invariant vector fields are defined as the right translations of matrices in g. Both the left and the right invariant vector fields form a frame on G, that is, Tg G D fgA: A 2 gg D fAg : A 2 gg for all g 2 G. Therefore, the tangent bundle TG can be realized as the product G g either via the left translations (g; A) ! gA, or via the right translations (g; A) ! Ag. Similarly, the cotangent bundle T G can be realized in two ways as G g with g equal to the dual of g. In the left-invariant realization 2 Tg G is identified with (g; l) 2 G g via the formula l(A) D (gA) for all A 2 g. For optimal control problems which are left-invariant it is natural to identify T G as G g via the left translations, and likewise identify T G as G g via the right translations for right-invariant problems. In both cases the realization T G D G g rules out canonical coordinates (assuming that G is non-abelian) and hence the Hamiltonian equations (19) take on a different form. For concreteness sake assume that T G D G g is realized via the left translations. Then it is natural to realize (T G) as (G g ) (g g ) where ((g; l); (A; f )) 2 (G g ) (g g ) denotes tangent vector (A; f ) at the point (g; l). In this representation of T(T G) the symplectic form ! is given by the following expression: !(g;l ) ((A1 ; f1 ); (A2 ; f2 )) D f2 (A1 ) f1 (A2 ) l([A1 ; A2 ]) :
(20)
Functions on Gg that are constant over the first factor, i. e., functions on g , are called left-invariant Hamiltonians. If H is left invariant then integral curves (g(t); l(t)) E are given of the corresponding Hamiltonian vector field H by dg (t) D g(t) dH(l(t)) ; dt
(21)
dl (t) D ad (dH(l(t))(l(t))) dt where dH denotes the differential of H, and where ad (A) : g ! g is given by (ad (A)(l))(X) D l([A; X]), l 2 g , X 2 g, [12,13]. On semi-simple Lie groups linear functions l in g can be identified with matrices L in g via an invariant
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quadratic form h ; i so that Eqs. (21) become dg (t) D g(t) dH(l(t)) ; dt dL (t) D [dH(l(t); L(t))] : dt 1 2
(22)
For instance, the problem of minimizing the integral RT 2 0 ku(t)k dt over the trajectories of ! m X dg u i (t)A i ; u 2 Rm (23) (t) D g(t) A0 C dt iD1
with A0 ; A1 ; : : : ; A m matrices in g leads to the following Hamiltonians: P H 2i C H0 . 1. (Normal extrema) H D 12 m iD1 P 2. (Abnormal extrema) H D H0 C m iD1 u i (t)H i , subject to H i D 0, i D 1; : : : ; m, with each H i equal to the Hamiltonian lift of the left invariant vector field g ! gA i . In the left invariant representation of T G each H i is a linear function on g , i. e., H i (l) D l(A i ) and consequently both Hamiltonians above are left-invariant. In the abnormal case dH D P A0 C m iD1 u i (t)A i , and ! m X dg u i (t)A i ; (t) D g(t) A0 C dt iD1 # (24) " m X dL u i (t)A i ; L(t) ; (t) D A0 C dt iD1
H1 (t) D H2 (t) D D H m (t) D 0 ;
(25)
are the corresponding extremal equations. In the normal case the extremal controls are given by u i D H i , i D 1; : : : ; m, and the corresponding Hamiltonian system is given by Eqs. (22) with dH D A0 C Pm iD1 H i (t)A i . Left invariant Hamiltonian systems [Eqs. (21) and (22)] always admit certain functions, called integrals of motion, which are constant along their solutions. Hamiltonians which admit a “maximal” number of functionally independent integrals of motion are called integrable. For left invariant Hamiltonians on Lie groups there is a deep and beautiful theory directed to characterizing integrable systems [10,13,23]. This topic is discussed in more detail below. Poisson Bracket, Involution and Integrability Integrals of motion are most conveniently discussed in terms of the Poisson bracket. For that reason it becomes
necessary to introduce the notion of a Poisson manifold. A manifold M that admits a bilinear and skew symmetric form f ; g : C 1 (M) C 1 (M) ! C 1 (M) that satisfies the Jacobi identity f f ; fg; hgg C fh; f f ; ggg C fg; fh; f gg D 0 and is a derivation f f g; hg D f fg; hg C gf f ; hg is called Poisson. It is known that every symplectic manifold is Poisson, and it is also known that every Poisson manifold admits a foliation in which each leaf is symplectic. In particular, the cotangent bundle T M is a Poisson manifold E with f f ; hg() D ! ( fE(); h()) for all functions f and h. It is easy to show that F is an integral of motion for H if and only if fF; Hg D 0 from which it follows that F is an integral of motion for H if and only if H is an integral of motion for F. Functions F and H for which fF; Hg D 0 are also said to be in involution. A function H on a 2n dimensional symplectic manifold S is said to be integrable or completely integrable if there exist n functions '1 ; : : : ; 'n with '1 D H which are functionally independent and further satisfy f' i ; ' j g D 0 for each i and j. It is known that such a system of functions is dimensionally maximal. On Lie groups the dual g of the Lie algebra g inherits a Poisson structure from the symplectic form ! [Eq. (17)], with f f ; hg(l) D l([d f ; dh]) for any functions f and h on g . In the literature on Hamiltonian systems this structure is often called Lie–Poisson. The symplectic leaves induced by the Poisson–Lie structure coincide with the coadjoint orbits of G and the solutions of the equation dl dt (t) D ad (dH(l(t))(l(t))) associated with Eqs. (21) evolve on coadjoint orbits of G. Most of the literature on integrable systems is devoted to integrability properties of the above equation considered as a Hamiltonian equation on a coadjoint orbit, or to its semi-simple counterpart dL dt (t) D [dH(L(t)); L(t)]. In this setting integrability is relative to the Poisson–Lie structure on each orbit, which may be of different dimensions. However, integrability can also be defined relative to the entire cotangent structure in which case the system is integrable whenever the number of independent integrals in involution is equal to the dimension of G. Leaving these subtleties aside, left-invariant Hamiltonian systems on Lie groups (22) always admit certain integrals of motion. They fall into two classes: 1. Hamiltonian lifts of right-invariant vector fields on G Poisson commute with any left-invariant Hamiltonian because right-invariant vector fields commute with leftinvariant vector fields. If X(g) D Ag denote a right invariant vector field then its Hamiltonian lift F A is equal to FA (L; g) D hL; g 1 Agi. In view of the formula fFA ; FB g D F[A;B] , the maximal number of functionally independent Hamiltonian lifts of right-invariant vector fields is equal to the rank of g. The rank of a semi-sim-
Maximum Principle in Optimal Control
ple Lie algebra g is equal to the dimension of a maximal abelian subalgebra of g. Such maximal abelian subalgebras are called Cartan subalgebras. 2. Each eigenvalue of L(t) is a constant of motion for dL dt (t) D [dH(L(t); L(t))]. If (L) and (L) denote eigenvalues of L then f; g D 0. Equivalently, the spectral functions ' k (L) D Trace(L k ) are in involution and Poisson commute with H. For three-dimensional Lie groups the above integrals are sufficient for complete integrability. For instance, every left-invariant Hamiltonian is completely integrable on SO3 [12]. In general, it is difficult to determine when the above integrals of motion can be extended to a completely integrable system of functions for a given Hamiltonian H [10,13,25]. Affirmative answers are known only in the exceptional cases in which there are additional symmetries. For instance, integrable system (15) is a particular case of the following more general situation. Example 8 Suppose that a semi-simple Lie group G admits an involutive automorphism ¤ I that splits the Lie algebra g of G into a direct sum g D p C k with k D fA: (A) D Ag and p D fA: (A) D Ag. Such a decomposition is known as a Cartan decomposition and the following Lie algebraic conditions hold [p; p] D k;
[p; k] D p;
[k; k] k :
(26)
Then L 2 g can be written as L D K C P with P 2 p and K 2 k. Assume that H(L) D 12 hK; Ki C hA; Pi, for some A 2 p where h ; i denotes a scalar multiple of the Cartan– Killing form that is positive definite on k. This Hamiltonian describes normal R Textrema for the problem of minimizing the integral 12 0 kU(t)k2 dt over the trajectories of dg (t) D g(t)(A C U(t)); dt
U(t) 2 k :
(27)
With the aid of the above decomposition the HamiltoE are given by nian equations (22) associated with H dg D g(A C K) ; dt dK D [A; P] ; dt dP D [A; K] C [K; P] : dt
(28)
dP Equations dK dt D [A; P]; dt D [A; K] C [K; P] admit two distinct types of integrals of motion. The first type is a con-
sequence of the spectral parameter representation dM D [N ; M ]; with M D P K C (2 1)A dt 1 and N D (P A) ; (29) from which it follows that ;k D Trace(Mk ) are constants of motion for each 2 R and k 2 Z C . The second type follows from the observation that [A; P] is orthogonal (relative to the Cartan–Killing form) to the subalgebra k0 D fX 2 k : [A; X] D 0g. Hence the projection of K(t) on k0 is constant. In many situations these two types of integrals of motion are sufficient for complete integrability [23]. Abnormal Extrema and Singular Problems For simplicity of exposition the discussion will be confined to control affine systems written explicitly as m
X dx u i (t)X i (x) ; D X0 (x) C dt iD1
u D (u1 ; : : : ; u m ) 2 U ;
(30)
with X 0 ; X1 ; : : : ; X m smooth vector fields on a smooth manifold M. Recall that abnormal extrema are absolutely continuous curves (t) ¤ 0 in T M satisfying Pm E E 1. d iD1 u i (t)H i ((t)) a. e. for some dt (t) D H0 ((t)) C E m denote the E 0; : : : ; H admissible control u(t) where H Hamiltonian vector fields associated with Hamiltonian lifts H i () D (X i (x)), 2 Tx M, i D 0; : : : ; m, and P Pm 2. H0 ((t)) C m iD1 u i (t)H i ((t)) H0 ((t)) C iD1 v i H i ((t)) a. e. for any v D (v1 ; : : : ; v m ) 2 U. Abnormal extremals satisfy the conditions of the Maximum Principle independently of any cost functional and can be studied in their own right. However, in practice, they are usually linked to some fixed optimization problem, such as for instance the problem of minimizing R 1 T 2 2 0 ku(t)k dt. In such a case there are several situations that may arise: 1. An optimal trajectory is only the projection of a normal extremal curve. 2. An optimal trajectory is the projection of both a normal and an abnormal extremal curve. 3. An optimal trajectory is only the projection of an abnormal curve (strictly abnormal case). When U is an open subset of R the maximality condition (2) implies that H i ((t)) D 0, i D 1; : : : ; m.
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Then extremal curves which project onto optimal trajectories satisfy another set of constraints fH i ; H j g((t)) D (t)([X i ; X j ](x(t))) D 0, 1 i; j m, known as the Goh condition in the literature on control theory [1]. Case (1) occurs when X1 (x(t)); : : : ; X m (x(t)); [X i ; X j ](x(t)), 1 i; j m span Tx(t) M. The remaining cases occur in the situations where higher order Lie brackets among X0 ; : : : ; X m are required to generate the entire tangent space Tx(t) M along an optimal trajectory x(t). In the second case abnormal extrema can be ignored since every optimal trajectory is the projection of a normal extremal curve. However, that is no longer true in Case (3) as the following example shows. Example 9 (Montgomery [21]) In this example M D R3 with its points parametrized by cylindrical coordinates r; ; z.R The optimal control problem consists of minimizT ing 12 0 (u12 C u22 )dt over the trajectories of dx (t) D u1 (t)X 1 (x(t)) C u2 (t)X 2 (x(t)); dt @ 1 @ @ ; X2 D A(r) ; X1 D @r r @ @z 1 and A(r) D r2 2
where (31)
1 4 r ; 4
(32)
or more explicitly over the solutions of the following system of equations: dr D u1 ; dt
d u2 D ; dt r
dz u2 D A(r) : dt r
(33)
Then normal extremal curves are integral curves of the Hamiltonian vector field associated to H D 12 (H12 C H22 ), with H1 D p r ; H2 D 1r (p A(r)p z ), where p r ; p ; p z denote dual coordinates of co-vectors p defined by p D p r dr C p d C p z dz. @ An easy calculation shows that [X1 ; X2 ] D dA dr @z 2
@ and [X1 ; [X 1 ; X2 ]] D ddrA2 @z . Hence, X1 (x); X 2 (x); [X 1 ; X2 ](x) spans R3 except at x D (r; ; z) where dA dr D 0, that is, on the cylinder r D 1. Since [X 1 ; [X 1 ; X2 ]] ¤ 0 on r D 1, it follows that X1 ; X2 ; [X 1 ; X2 ]; [X 1 ; [X 1 ; X2 ]] span R3 at all points x 2 R3 . The helix r D 1; z( ) D A(1) ; 2 R, generated by u1 D 0; u2 D 1, is a locally optimal trajectory (shown in [21]). It is the projection of an abnormal extremal curve and not the projection of a normal extremal curve.
Trajectories of a control system that are the projections of a constrained Hamiltonian system are called singular [3]. For instance, the helix in the above example is singular. The terminology derives from the singularity
theory of mappings, and in the control theoretic context it is associated to the end point mapping E : u[0;T] ! x(x0 ; u; T), where x(x0 ; u; t) denotes the trajectory of dx dt D F(x(t); u(t)), x(x 0 ; u; 0) D x0 , with the controls u(t) in the class of locally bounded measurable with values in Rm . It is known that the singular trajectories are the projections of the integral curves (t) of the constrained Hamiltonian system, obtained by the Maximum Principle: d E D H((t); u(t)) ; dt
dH ((t); u(t)) D 0 ; du
(34)
where H(; u) D (F(x; u)), 2 Tx M. For an extensive theory of singular trajectories see [3]. Future Directions Since the original publications there has been a considerable effort to obtain the maximum principle under more general conditions and under different technical assumptions. This effort seems to be motivated by two distinct objectives: the first motivation is a quest for a high order maximum principle [4,16,17], while the second motivation is an extension of the maximum principle to differential inclusions and non-smooth problems [9,20,28]. Although there is some indication that the corresponding theoretical approaches do not lead to common theory [5], there still remains an open question how to incorporate these diverse points of view into a universal maximum principle. Bibliography Primary Literature 1. Agrachev AA, Sachkov YL (2005) Control theory from the geometric viewpoint. Encycl Math Sci 87. Springer, Heidelberg 2. Arnold VI (1989) Mathematical methods of classical mechanics. Graduate texts in Mathematics, vol 60. Springer, Heidelberg 3. Bonnard B, Chyba M (2003) Singular trajectories and their role in control. Springer, Heidelberg 4. Bianchini RM, Stefani G (1993) Controllability along a trajectory; a variational approach. SIAM J Control Optim 31:900–927 5. Bressan A (2007) On the intersection of a Clarke cone with a Boltyanski cone (to appear) 6. Berkovitz LD (1974) Optimal control theory. Springer, New York 7. Caratheodory C (1935) Calculus of variations. Teubner, Berlin (reprinted 1982, Chelsea, New York) 8. Clarke FH (1983) Optimization and nonsmooth analysis. Wiley Interscience, New York 9. Clarke FH (2005) Necessary conditions in dynamic optimization. Mem Amer Math Soc 816(173) 10. Fomenko AT, Trofimov VV (1988) Integrable systems on Lie algebras and symmetric spaces. Gordon and Breach 11. Gamkrelidze RV (1978) Principles of optimal control theory. Plenum Press, New York
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12. Jurdjevic V (1997) Geometric control theory. Cambridge Studies in Advanced Mathematics vol 51. Cambridge University Press, Cambridge 13. Jurdjevic V (2005) Hamiltonian systems on complex Lie groups and their homogeneous spaces. Mem Amer Math Soc 178(838) 14. Lee EB, Markus L (1967) Foundations of optimal control theory. Wiley, New York 15. Liu WS, Sussmann HJ (1995) Shortest paths for SR metrics of rank 2 distributions. Mem Amer Math Soc 118(564) 16. Knobloch H (1975) High order necessary conditions in optimal control. Springer, Berlin 17. Krener AJ (1977) The high order maximum principle and its application to singular extremals. SIAM J Control Optim 17:256– 293 18. Kupka IK (1990) The ubiquity of Fuller’s phenomenon. In: Sussmann HJ (ed) Non-linear controllability and Optimal control. Marcel Dekker, New York, pp 313–350 19. Marchal C (1973) Chattering arcs and chattering controls. J Optim Theory App 11:441–468 20. Morduchovich B (2006) Variational analysis and generalized differentiation: I. Basic analysis, II. Applications. Grundlehren Series (Fundamental Principle of Mathematical Sciences). Springer, Berlin 21. Montgomery R (1994) Abnormal minimizers. SIAM J Control Optim 32(6):1605–1620 22. Moser J (1980) Geometry of quadrics and spectral theory. In: The Chern Symposium 1979. Proceedings of the International symposium on Differential Geometry held in honor of S.S. Chern, Berkeley, California. Springer, pp 147–188 23. Perelomov AM (1990) Integrable systems of classical mechanics and Lie algebras. Birkhauser, Basel
24. Pontryagin LS, Boltyanski VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes. Wiley, New York 25. Reiman AG, Semenov Tian-Shansky MA (1994) Group-theoretic methods in the theory of finite dimensional integrable systems. In: Arnold VI, Novikov SP (eds) Encyclopedia of Mathematical Sciences. Springer, Heidelberg 26. Sussmann HJ, Willems J (2002) The brachistochrone problem and modern control theory. In: Anzaldo-Meneses A, Bonnard B, Gauthier J-P, Monroy-Perez F (eds) Contemporary trends in non-linear geometric control theory and its applications. Proceedings of the conference on Geometric Control Theory and Applications held in Mexico City on September 4–6, 2000, to celebrate the 60th anniversary of Velimir Jurdjevic. World Scientific Publishers, Singapore, pp 113–165 27. Sussmann HJ () Set separation, approximating multicones and the Lipschitz maximum principle. J Diff Equations (to appear) 28. Vinter RB (2000) Optimal control. Birkhauser, Boston 29. Young LC (1969) Lectures in the calculus of variations and optimal control. Saunders, Philadelphia 30. Zelikin MI, Borisov VF (1994) Theory of chattering control with applications to aeronautics, Robotics, Economics, and Engineering. Birkhauser, Basel
Books and Reviews Bressan A (1985) A high-order test for optimality of bang-bang controls. SIAM J Control Optim 23:38–48 Griffiths P (1983) Exterior differential systems and the calculus of variations. Birkhauser, Boston
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Measure Preserving Systems
KARL PETERSEN Department of Mathematics, University of North Carolina, Chapel Hill, USA
tems theory. These systems arise from science and technology as well as from mathematics itself, so applications are found in a wide range of areas, such as statistical physics, information theory, celestial mechanics, number theory, population dynamics, economics, and biology.
Article Outline
Introduction: The Dynamical Viewpoint
Glossary Definition of the Subject Introduction: The Dynamical Viewpoint Where do Measure-Preserving Systems Come from? Construction of Measures Invariant Measures on Topological Dynamical Systems Finding Finite Invariant Measures Equivalent to a Quasi-Invariant Measure Finding -finite Invariant Measures Equivalent to a Quasi-Invariant Measure Some Mathematical Background Future Directions Bibliography
Sometimes introducing a dynamical viewpoint into an apparently static situation can help to make progress on apparently difficult problems. For example, equations can be solved and functions optimized by reformulating a given situation as a fixed point problem, which is then addressed by iterating an appropriate mapping. Besides practical applications, this strategy also appears in theoretical settings, for example modern proofs of the Implicit Function Theorem. Moreover, the introduction of the ideas of change and motion leads to new concepts, new methods, and even new kinds of questions. One looks at actions and orbits and instead of always seeking exact solutions begins perhaps to ask questions of a qualitative or probabilistic nature: what is the general behavior of the system, what happens for most initial conditions, what properties of systems are typical within a given class of systems, and so on. Much of the credit for introducing this viewpoint should go to Henri Poincaré [29].
Measure Preserving Systems
Glossary Dynamical system A set acted upon by an algebraic object. Elements of the set represent all possible states or configurations, and the action represents all possible changes. Ergodic A measure-preserving system is ergodic if it is essentially indecomposable, in the sense that given any invariant measurable set, either the set or its complement has measure 0. Lebesgue space A measure space that is isomorphic with the usual Lebesgue measure space of a subinterval of the set of real numbers, possibly together with countably or finitely many point masses. Measure An assignment of sizes to sets. A measure that takes values only between 0 and 1 assigns probabilities to events. Stochastic process A family of random variables (measurable functions). Such an object represents a family of measurements whose outcomes may be subject to chance. Subshift, shift space A closed shift-invariant subset of the set of infinite sequences with entries from a finite alphabet. Definition of the Subject Measure-preserving systems model processes in equilibrium by transformations on probability spaces or, more generally, measure spaces. They are the basic objects of study in ergodic theory, a central part of dynamical sys-
Two Examples Consider two particular examples, one simple and the other not so simple. Decimal or base 2 expansions of numbers in the unit interval raise many natural questions about frequencies of digits and blocks. Instead of regarding the base 2 expansion x D :x0 x1 : : : of a fixed x 2 [0; 1] as being given, we can regard it as arising from a dynamical process. Define T : [0; 1] ! [0; 1] by Tx D 2x mod 1 (the fractional part of 2x) and let P D fP0 D [0; 1/2); P1 D [1/2; 1]g be a partition of [0; 1] into two subintervals. We code the orbit of any point x 2 [0; 1] by 0’s and 1’s by letting x k D i if T k x 2 Pi ; k D 0; 1; 2; : : : . Then reading the expansion of x amounts to applying to the coding the shift transformation and projection onto the first coordinate. This is equivalent to following the orbit of x under T and noting which element of the partition P is entered at each time. Reappearances of blocks amount to recurrence to cylinder sets as x is moved by T, frequencies of blocks correspond to ergodic averages, and Borel’s theorem on normal numbers is seen as a special case of the Ergodic Theorem. Another example concerns Szemerédi’s Theorem [34], which states that every subset A N of the natural numbers which has positive upper density contains arbi-
Measure Preserving Systems
trarily long arithmetic progressions: given L 2 N there are s; m 2 N such that s; s C m; : : : ; s C (L 1)m 2 A. Szemerédi’s proof was ingenious, direct, and long. Furstenberg [15] saw how to obtain this result as a corollary of a strengthening of Poincaré’s Recurrence Theorem in ergodic theory, which he then proved. Again we have an apparently static situation: a set A N of positive density in which we seek arbitrarily long regularly spaced subsets. Furstenberg proposed to consider the characteristic function 1 A of A as a point in the space f0; 1gN of 0’s and 1’s and to form the orbit closure X of this point under the shift transformation . Because A has positive density, it is possible to find a shift-invariant measure on X which gives positive measure to the cylinder set B D [1] D fx 2 X : x1 D 1g. Furstenberg’s Multiple Recurrence Theorem says that given L 2 N there is m 2 N such that (B \ T m B \ \ T (L1)m B) > 0. If y is a point in this intersection, then y contains a block of L 1’s, each at distance m from the next. And since y is in the orbit closure of 1 A , this block also appears in the sequence 1 A 2 f0; 1gN , yielding the result. Aspects of the dynamical argument remain in new combinatorial and harmonic-analytic proofs of the Szemerédi Theorem by T. Gowers [16,17] and T. Tao [35], as well as the extension to the (density zero) set of prime numbers by B. Green and T. Tao [18,36]. A Range of Actions Here is a sample of dynamical systems of various kinds: 1. A semigroup or group G acts on a set X. There is given a map G X ! X, (g; x) ! gx, and it is assumed that g1 (g2 x) D (g1 g2 )x for all g1 ; g2 2 G; x 2 X (1) ex D x
2. 3.
4.
5.
for all x 2 X; if G has an identity element e. (2) A continuous linear operator T acts on a Banach or Hilbert space V. B is a Boolean -algebra (a set together with a zero element 0 and operations _; ^;0 which satisfy the same rules as ;; [; \;c (complementation) do for -algebras of sets); N is a -ideal in B (N 2 N ; B 2 B; B ^ N D B implies B 2 N ; and N1 ; N2 ; 2 N implies _1 nD1 N n 2 N ); and S : B ! B preserves the Boolean -algebra operations and S N N . B is a Boolean -algebra, is a countably additive positive (nonzero except on the zero element of B) function on B, and S : B ! B is as above. Then (B; ) is a measure algebra and S is a measure algebra endomorphism. (X; B; ) is a measure space (X is a set, B is a -algebra of subsets of X, and : B ! [0; 1] is countably additive: If B1 ; B2 ; 2 B are pairwise disjoint, then
P1 ([1 nD1 (B n )); T : X ! X is measurnD1 B n ) D able (T 1 B B) and nonsingular ((B) D 0 implies (T 1 B) D 0 – or, more stringently, and T 1 are equivalent in the sense of absolute continuity). 6. (X; B; ) is a measure space, T : X ! X is a one-toone onto map such that T and T 1 are both measurable (so that T 1 B D B D T B), and (T 1 B) D (B) for all B 2 B. (In practice often T is not one-to-one, or onto, or even well-defined on all of X, but only after a set of measure zero is deleted.) This is the case of most interest for us, and then we call (X; B; ; T) a measurepreserving system. We also allow for the possibility that T is not invertible, or that some other group (such as R or Zd ) or semigroup acts on X, but the case of Z actions will be the main focus of this article. 7. X is a compact metric space and T : X ! X is a homeomorphism. Then (X; T) is a topological dynamical system. 8. M is a compact manifold (C k for some k 2 [1; 1]) and T : M ! M is a diffeomorphism (one-to-one and onto, with T and T 1 both C k ). Then (M; T) is a smooth dynamical system. Such examples can arise from solutions of an autonomous differential equation given by a vector field on M. Recall that in Rn , an ordinary differential equation initial-value problem x 0 D f (x); x(0) D x0 has a unique solution x(t) as long as f satisfies appropriate smoothness conditions. The existence and uniqueness theorem for differential equations then produces a flow according to Tt x0 D x(t), satisfying TsCt x0 D Ts (Tt x0 ). Restricting to a compact invariant set (if there is one) and taking T D T1 (the time 1 map) gives us a smooth system (M; f ). Naturally there are relations and inclusions among these examples of actions. Often problems can be clarified by forgetting about some of the structure that is present or by adding desirable structure (such as topology) if it is not. There remain open problems about representation and realization; for example, taking into account necessary restrictions, which measure-preserving systems can be realized as smooth systems preserving a smooth measure? Sometimes interesting aspects of the dynamics of a smooth system can be due to the presence of a highly nonsmooth subsystem, for example a compact lowerdimensional invariant set. Thus one should be ready to deal with many kinds of dynamical systems. Where do Measure-Preserving Systems Come from? Systems in Equilibrium Besides physical systems, abstract dynamical systems can also represent aspects of biological, economic, or other
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real-world systems. Equilibrium does not mean stasis, but rather that the changes in the system are governed by laws which are not themselves changing. The presence of an invariant measure means that the probabilities of observable events do not change with time. (But of course what happens at time 2 can still depend on what happens at time 1, or, for that matter, at time 3.) We consider first the example of the wide and important class of Hamiltonian systems. Many systems that model physical situations, for example a large number of ideal charged particles in a container, can be studied by means of Hamilton’s equations. The state of the entire system at any time is supposed to be specified by a vector (q; p) 2 R2n , the phase space, with q listing the coordinates of the positions of all of the particles, and p listing the coordinates of their momenta. We assume that there is a time-independent Hamiltonian function H(q; p) such that the time development of the system satisfies Hamilton’s equations: dq i @H ; D dt @p i
dp i @H ; D dt @q i
i D 1; : : : ; n :
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Often the Hamiltonian function is the sum of kinetic and potential energy: H(q; p) D K(p) C U(q) :
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The potential energy U(q) may depend on interactions among the particles or with an external field, while the kinetic energy K(p) depends on the velocities and masses of the particles. As discussed above, solving these equations with initial state (q; p) for the system produces a flow (q; p) ! Tt (q; p) in phase space. According to Liouville’s Theorem, this flow preserves Lebesgue measure on R2n . Calculating dH/dt by means of the Chain Rule and using Hamilton’s equations shows that H is constant on orbits of the flow, and thus each set of constant energy X(H0 ) D f(q; p) : H(q; p) D H0 g is an invariant set. Thus one should consider the flow restricted to the appropriate invariant set. It turns out that there are also natural invariant measures on the sets X(H0 ), namely the ones given by rescaling the volume element dS on X(H0 ) by the factor 1/jjOHjj. For details, see [25]. Systems in equilibrium can also be hiding inside systems not in equilibrium, for example if there is an attractor supporting an SRB measure (for Sinai, Ruelle, and Bowen) (for definitions of the terms used here and more explanations, see the article in this collection by A. Wilkinson). Suppose that T : M ! M is a diffeomorphism on a compact manifold as above, and that m is a version of Lebesgue
measure on M, say given by a smooth volume form. We consider m to be a “physical measure”, corresponding to laboratory measurements of observable quantities, whose values can be determined to lie in certain intervals in R. Quite possibly m is not itself invariant under T, and an experimenter might observe strange or chaotic behavior whenever the state of the system gets close to some compact invariant set X. The dynamics of T restricted to X can in fact be quite complicated – maybe a full shift, which represents completely undeterministic behavior (for example if there is a horseshoe present), or a shift of finite type, or some other complicated topological dynamical system. Possibly m(X) D 0, so that X is effectively invisible to the observer except through its effects. It can happen that there is a T-invariant measure supported on X such that 1 n
n1 X
mT k ! weak ;
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kD0
and then the long-term equilibrium dynamics of the system is described by (X; T; ). For a recent survey on SRB measures, see [39]. Stationary Stochastic Processes A stationary process is a family f f t : t 2 Tg of random variables (measurable functions) on a probability space (˝; F ; P). Usually T is Z; N, or R. For the remainder of this section let us fix T D Z (although the following definition could make sense for T any semigroup). We say that the process f f n : n 2 Zg is stationary if its finitedimensional distributions are translation invariant, in the sense that for each r D 1; 2; : : : , each n1 ; : : : ; nr 2 Z, each choice of Borel sets B1 ; : : : ; Br R, and each s 2 Z, we have Pf! : f n 1 (!) 2 B1 ; : : : ; f n r (!) 2 Br g D Pf! : f n 1 Cs (!) 2 B1 ; : : : ; f n r Cs (!) 2 B r g :
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The f n represent measurements made at times n of some random phenomenon, and the probability that a particular finite set of measurements yield values in certain ranges is supposed to be independent of time. Each stationary process f f n : n 2 Zg on (˝; F ; P) corresponds to a shift-invariant probability measure on the set RZ (with its Borel -algebra) and a single observable, namely the projection 0 onto the 0’th coordinate, as follows. Define : ˝ ! RZ
by (!) D ( f n (!))1 1 ;
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and for each Borel set E RZ , define (E) D P( 1 E). Then examining the values of on cylinder sets – for Borel
Measure Preserving Systems
B1 ; : : : ; Br R, fx 2 RZ : x n i 2 B i ; i D 1; : : : ; rg D Pf! 2 ˝ : f n i (!) 2 B i ; i D 1; : : : ; rg
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– and using stationarity of ( f n ) shows that is invariant under . Moreover, the processes ( f n ) on ˝ and 0 ı n on RZ have the same finite-dimensional distributions, so they are equivalent for the purposes of probability theory. Construction of Measures We review briefly (following [33]) the construction of measures largely due to C. Carathéodory [8], with input from M. Fréchet [13], H. Hahn [19], A. N. Kolmogorov [26], and others, then discuss the application to construction of measures on shift spaces and of stochastic processes in general. The Carathéodory Construction A semialgebra is a family S of subsets of a set X which is closed under finite intersections and such that the complement of any member of S is a finite disjoint union of members of S. Key examples are 1. the family H of half-open subintervals [a; b) of [0; 1); 2. in the space X D AZ of doubly infinite sequences on a finite alphabet A, the family C of cylinder sets (determined by fixing finitely many entries) fx 2 AZ : x n 1 D a1 ; : : : ; x n r D ar g ;
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3. the family C1 of anchored cylinder sets fx 2 AN : x1 D a1 ; : : : ; x r D ar g
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in the space X D AN of one-sided infinite sequences on a finite alphabet A. An algebra is a family of subsets of a set X which is closed under finite unions, finite intersections, and complements. A -algebra is a family of subsets of a set X which is closed under countable unions, countable intersections, and complements. If S is a semialgebra of subsets of X, the algebra A(S) generated by S is the smallest algebra of subsets of X which contains S. A(S) is the intersection of all the subalgebras of the set 2 X of all subsets of X and consists exactly of all finite disjoint unions of elements of S. Given an algebra A, the -algebra B(A) generated by A is the smallest -algebra of subsets of X which contains A. A nonnegative set function on S is a function : S ! [0; 1] such that (;) D 0 if ; 2 S. We say that such a is
finitely additive if whenever S1 ; : : : ; S n 2 S are pairwise disjoint and S D [niD1 S i 2 S, we have (S) D Pn iD1 (S i ); countably additive if whenever S1 ; S2 2 S are pairwise disjoint and S D [1 iD1 S i 2 S, we have (S) D P1 (S ); and i iD1 countably subadditive if whenever S1 ; S2 2 S and P1 S D [1 iD1 (S i ). iD1 S i 2 S, we have (S) A measure is a countably additive nonnegative set function defined on a -algebra. Proposition 1 Let S be a semialgebra and a nonnegative set function on S. In order that have an extension to a finitely additive set function on the algebra A(S) generated by S, it is necessary and sufficient that be finitely additive on S. Proof 1 The stated condition is obviously necessary. Conversely, given which is finitely additive on S, it is natural to define (
n [
Si ) D
iD1
n X
(S i )
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iD1
whenever A D [niD1 S i (with the S i pairwise disjoint) is in the algebra A(S) generated by S. It is necessary to verify that is then well defined on A(S), since each element of A(S) may have more than one representation as a finite disjoint union of members of S. But, given two such representations of a single set A, forming the common refinement and applying finite additivity on S shows that so defined assigns the same value to A both times. Then finite additivity on A(S) of the extended is clear. Proposition 2 Let S be a semialgebra and a nonnegative set function on S. In order that have an extension to a countably additive set function on the algebra A(S) generated by S, it is necessary and sufficient that be (i) finitely additive and (ii) countably subadditive on S. Proof 2 Conditions (i) and (ii) are clearly necessary. If is finitely additive on S, then by Proposition 1 has an extension to a finitely additive nonnegative set function, which we will still denote by , on A(S). Let us see that this extension is countably subadditive on A(S). Suppose that A1 ; A2 ; 2 A(S) are pairwise disjoint and their union A 2 A(S). Then A is a finite disjoint union of sets in S, as is each A i : AD
AD
1 [ iD1 m [ jD1
A i ; each A i D
ni [
Si k ;
kD1
R j ; each A i 2 A(S) ; each S i k ; R j 2 S :
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Measure Preserving Systems
Since each R j 2 S, by countable subadditivity of on S, and using R j D R j \ A, (R j ) D (
ni 1 [ [
S i k \R j )
iD1 kD1
ni 1 X X
(S i k \R j ); (13)
iD1 kD1
and hence, by finite additivity of on A(S), (A) D
m X
(R j )
jD1
D
ni X 1 X m X
(S i k \ R j )
iD1 kD1 jD1
ni 1 X X
(S i k ) D
iD1 kD1
1 X
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(A i ) :
Theorem 1 In order that a nonnegative set function on an algebra A of subsets of a set X have an extension to a (countably additive) measure on the -algebra B(A) generated by A, it is necessary and sufficient that be countably additive on A. Here is a sketch of how the extension can be constructed. Given a countably additive nonnegative set function on an algebra A of subsets of a set X, one defines the outer measure that it determines on the family 2 X of all subsets of X by 1 X
In this way, beginning with the semialgebra H of halfopen subintervals of [0; 1) and [a; b) D ba, one arrives at Lebesgue measure on the -algebra M of Lebesgue measurable sets and on its sub- -algebra B(H ) of Borel sets.
iD1
Now finite additivity of on an algebra A implies that is monotonic on the algebra: if A; B 2 A and A B, then (A) (B). Thus if A1 ; A2 ; 2 A(S) are pairwise disjoint and their union A 2 A(S), then for each n P we have niD1 (A i ) D ([niD1 A i ) (A), and hence P1 iD1 (A i ) (A).
(E) D inf f
of all subsets of X as above. Then the family M of measurable subsets of X is a -algebra containing A (and hence B(A)) and all subsets of X which have measure 0. The restriction jM is a (countably additive) measure which agrees on A with . If is -finite on A (so that there are X1 ; X2 ; 2 A with (X i ) < 1 for all i and X D [1 iD1 X i ), then on B(A) is the only extension of on A to B(A).
(A i ) : A i 2 A; E [1 iD1 A i g : (15)
Measures on Shift Spaces The measures that determine stochastic processes are also frequently constructed by specifying data on a semialgebra of cylinder sets. Given a finite alphabet A, denote by ˝(A) D AZ and ˝ C (A) D AN the sets of two and onesided sequences, respectively, with entries from A. These are compact metric spaces, with d(x; y) D 2n when n D inffjkj : x k ¤ y k g. In both cases, the shift transformation defined by ( x)n D x nC1 for all n is a homeomorphism. Suppose (cf. [3]) that for every k D 1; 2; : : : we are given a function g k : Ak ! [0; 1], and that these functions satisfy, for all k, 1. g k (B) 0 for all B 2 Ak ; P g (Bi) D g k (B) for all B 2 Ak ; 2. P i2A kC1 3. i2A g 1 (i) D 1. Then Theorems 1 and 2 imply that there is a unique measure on the Borel subsets of ˝ C (A) such that for all k D 1; 2; : : : and B 2 Ak
iD1
fx 2 ˝ C (A) : x1 : : : x k D Bg D g k (B) :
is a nonnegative, countably subadditive, monoThen tonic set function on 2 X . Define a set E to be -measurable if for all T X, (T) D (T \ E) C (T \ E c ) :
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This ingenious definition can be partly motivated by noting that if is to be finitely additive on the family M of -measurable sets, which should contain X, then at least this condition must hold when T D X. It is amazing that then this definition readily, with just a little set theory and a few "’s, yields the following theorem. Theorem 2 Let be a countably additive nonnegative set function on an algebra A of subsets of a set X, and let be the outer measure that it determines on the family 2 X
4.
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If in addition the g k also satisfy P i2A g kC1 (iB) D g k (B) for all k D 1; 2; : : : and all B 2 Ak , then there is a unique shift-invariant measure on the Borel subsets of ˝ C (A) (also ˝(A)) such that for all n, all k D 1; 2; : : : and B 2 Ak fx 2 ˝ C (A) : x n : : : x nCk1 D Bg D g k (B) : (18) This follows from the Carathéodory theorem by beginning with the semialgebra C1 of anchored cylinder sets or the semialgebra C of cylinder sets determined by finitely many consecutive coordinates, respectively.
There are two particularly important examples of this construction. First, let our finite alphabet be A D f0; : : : ;
Measure Preserving Systems
d 1g, and let p D (p0 ; : : : ; p d1 ) be a probability vecP tor: all p i 0 and d1 iD0 p i D 1. For any block B D b1 : : : b k 2 Ak , define g k (B) D p b 1 : : : p b k :
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The resulting measure p is the product measure on ˝(A) D AZ of infinitely many copies of the probability measure determined by p on the finite sample space A. The measure-preserving system (˝; B; ; ) (with B the -algebra of Borel subsets of ˝(A), or its completion), is denoted by B(p) and is called the Bernoulli system determined by p. This system models an infinite number of independent repetitions of an experiment with finitely many outcomes, the ith of which has probability p i on each trial. This construction can be generalized to model stochastic processes which have some memory. Again let A D f0; : : : ; d 1g, and let p D (p0 ; : : : ; p d1 ) be a probability vector. Let P be a d d stochastic matrix with rows and columns indexed by A. This means that all entries of P are nonnegative, and the sum of the entries in each row is 1. We regard P as giving the transition probabilities between pairs of elements of A. Now we define for any block B D b 1 : : : b k 2 Ak
frequently T D Z; N; R; Zd , or Rd ). We give a brief description, following [4]. Let T be an arbitrary index set. We aim to produce a R-valued stochastic process indexed by T, that is to say, a Borel probability measure P on ˝ D RT , which has prespecified finite-dimensional distributions. Suppose that for every ordered k-tuple t1 ; : : : ; t k of distinct elements of T we are given a Borel probability measure t1 :::t k on R k . Denoting f 2 RT also by ( f t : t 2 T), we want it to be the case that, for each k, each choice of distinct t1 ; : : : t k 2 T, and each Borel set B R k , Pf( f t : t 2 T) : ( f t1 ; : : : ; f t k ) 2 Bg D t1 :::t k (B) : (22) For consistency, we will need, for example, that t1 t2 (B1 B2 ) D t2 t1 (B2 B1 ) ; since
Pf( f t1 ; f t2 ) 2 A1 A2 g D Pf( f t2 ; f t1 ) 2 A2 A1 g : (24) Thus we assume: 1. For any k D 1; 2; : : : and permutation of 1; : : : ; k, if : R k ! R k is defined by (x1 ; : : : ; x k ) D (x1 ; : : : ; x k ) ;
g k (B) D p b 1 Pb 1 b 2 Pb 2 b 3 : : : Pb k1 b k :
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Using the g k to define a nonnegative set function p;P on the semialgebra C1 of anchored cylinder subsets of ˝ C (A), one can verify that p;P is (vacuously) finitely additive and countably subadditive on C1 and therefore extends to a measure on the Borel -algebra of ˝ C (A), and its completion. The resulting stochastic process is a (one-step, finite-state) Markov process. If p and P also satisfy pP D p ;
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then condition 4. above is satisfied, and the Markov process is stationary. In this case we call the (one or two-sided) measure-preserving system the Markov shift determined by p and P. Points in the space are conveniently pictured as infinite paths in a directed graph with vertices A and edges corresponding to the nonzero entries of P. A process with a longer memory, say of length m, can be produced by repeating the foregoing construction after recoding with a sliding block code to the new alphabet Am : for each ! 2 ˝(A), let ((!))n D !n !nC1 : : : !nCm1 2 Am . The Kolmogorov Consistency Theorem There is a generalization of this method to the construction of stochastic processes indexed by any set T. (Most
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then for all k and all Borel B R k t1 :::t k (B) D t1 :::t k (1 B) :
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Further, since leaving the value of one of the f t j free does not change the probability in (22), we also should have 2. For any k D 1; 2; : : : , distinct t1 ; : : : ; t k ; t kC1 2 T, and Borel set B R k , t1 :::t k (B) D t1 :::t k t kC1 (B R) :
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Theorem 3 (Kolmogorov Consistency Theorem [26]) Given a system of probability measures t1 :::t k as above indexed by finite ordered subsets of a set T, in order that there exist a probability measure P on RT satisfying (22) it is necessary and sufficient that the system satisfy 1. and 2. above. When T D Z; R, or N, as in the example with the g k above, the problem of consistency with regard to permutations of indices does not arise, since we tacitly use the order in T in specifying the finite-dimensional distributions. In case T is a semigroup, by adding conditions on the given data t1 :::t k it is possible to extend this construction also to produce stationary processes indexed by T, in parallel with the above constructions for T D Z or N.
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Measure Preserving Systems
Invariant Measures on Topological Dynamical Systems
Ergodicity and Unique Ergodicity
Existence of Invariant Measures Let X be a compact metric space and T : X ! X a homeomorphism (although usually it is enough just that T be a continuous map). Denote by C (X) the Banach space of continuous real-valued functions on X with the supremum norm and by M(X) the set of Borel probability measures on X. Given the weak topology, according to which Z Z f n d ! f d n ! if and only if X
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M(X) is a convex subset of the dual space C (X) of all continuous linear functionals from C (X) to R. With the
weak topology it is metrizable and (by Alaoglu’s Theorem) compact. Denote by MT (X) the set of T-invariant Borel probability measures on X. A Borel probability measure on X is in M(X) if and only if (T 1 B) D (B) for all Borel sets B X ;
X
Proposition 3 For every compact topological dynamical system (X; T) (with X not empty) there is always at least one T-invariant Borel probability measure on X. Proof 3 Let m be any Borel probability measure on X. For example, we could pick a point x0 2 X and let m be the point mass ıx 0 at x0 defined by ıx 0 ( f ) D f (x0 ) for all f 2 C (X) :
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Form the averages 1 n
n1 X
mT i ;
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iD0
which are also in M(X). By compactness, fA n mg has a weak cluster point , so that there is a subsequence A n k m ! weak :
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Then 2 M(X); and is T-invariant, because for each f 2 C (X) 1 j( f k!1 n k
j( f T) ( f )j D lim
Equivalently (using the Ergodic Theorem), (X; B; ; T) is ergodic if and only if for each f 2 L1 (X; B; ) 1 n
n1 X kD1
f (T k x) !
Z f d almost everywhere : (36) X
It can be shown that the ergodic measures on (X; T) are exactly the extreme points of the compact convex set MT (X), namely those 2 MT (X) for which there do not exist 1 ; 2 2 MT (x) with 1 ¤ 2 and s 2 (0; 1) such that D s1 C (1 s) 2 :
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(29)
equivalently, Z Z f ıT d D f d for all f 2 C (X): (30) ( f T) D
An m D
B 2 B ; (T 1 B4B) D 0 implies (B) D 0 or 1 : (35)
X
for all f 2 C (X) ;
X
Among the T-invariant measures on X are the ergodic ones, those for which (X; B; ; T) (with B the -algebra of Borel subsets of X) forms an ergodic measure-preserving system. This means that there are no proper T-invariant measurable sets:
T n k ) ( f )j D 0 ; (34)
both terms inside the absolute value signs being bounded.
The Krein-Milman Theorem states that in a locally convex topological vector space such as C (X) every compact convex set is the closed convex hull of its extreme points. Thus every nonempty such set has extreme points, and so there always exist ergodic measures for (X; T). A topological dynamical system (X; T) is called uniquely ergodic if there is only one T-invariant Borel probability measure on X, in which case, by the foregoing discussion, that measure must be ergodic. There are many examples of topological dynamical systems which are uniquely ergodic and of others which are not. For now, we just remark that translation by a generator on a compact monothetic group is always uniquely ergodic, while group endomorphisms and automorphisms tend to be not uniquely ergodic. Bernoulli and (nonatomic) Markov shifts are not uniquely ergodic, because they have many periodic orbits, each of which supports an ergodic measure. Finding Finite Invariant Measures Equivalent to a Quasi-Invariant Measure Let (X; B; m) be a -finite measure space, and suppose that T : X ! X is an invertible nonsingular transformation. Thus we assume that T is one-to-one and onto (maybe after a set of measure 0 has been deleted), that T and T 1 are both measurable, so that T B D B D T 1 B ;
(38)
Measure Preserving Systems
and that T and T 1 preserve the -ideal of sets of measure 0: m(B) D 0 if and only if m(T
1
B) D 0
if and only if m(TB) D 0 :
(B) D (39)
In this situation we say that m is quasi-invariant for T. A nonsingular system (X; B; m; T) as above may model a nonequilibrium situation in which events that are impossible (measure 0) at any time are also impossible at any other time. When dealing with such a system, it can be useful to know whether there is a T-invariant measure that is equivalent to m (in the sense of absolute continuity – they have the same sets of measure 0–in which case we write m), for then one would have available machinery of the measure-preserving situation, such as the Ergodic Theorem and entropy in their simplest forms. Also, it is most useful if the measures are -finite, so that tools such as the Radon-Nikodym and Tonelli-Fubini theorems will be available. We may assume that m(X) D 1. For if X D [1 iD1 X i with each X i 2 B and m(X i ) < 1, disjointifying (replace X i by X i n X i1 for i 2) and deleting any X i that have measure 0, we may replace m by 1 X iD1
mj X i : 2 i m(X i )
(40)
Definition 1 Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. We say that A; B 2 B are T-equivalent, and write A T B, if there are two sequences of pairwise disjoint sets, A1 ; A2 ; : : : and B1 ; B2 ; : : : and integers n1 ; n2 ; : : : such that
AD
1 [ iD1
Ai ; B D
1 [
1 X D [1 iD1 X i ; B D [ iD1 B i , then
ni
B i ; and T A i D B i for all i :
iD1
(41)
1 X
(B i ) D
iD1
D
1 X
(T n i X i )
iD1
1 X
(42)
(X i ) D (X) ;
iD1
so that (X n B) D 0 and hence m(X n B) D 0. For the converse, one tries to show that if X is T-nonshrinkable, then for each A 2 B the following limit exists: 1 n!1 n
lim
n1 X
m(T k A) :
(43)
kD0
The condition of T-nonshrinkability not being easy to check, subsequent authors gave various necessary and sufficient conditions for the existence of a finite equivalent invariant measure: 1. Dowker [11]. Whenever A 2 B and m(A) > 0, lim inf n!1 m(T n A) > 0. 2. Calderón [6]. Whenever A 2 B and m(A) > 0; P k lim in f n!1 n1 1 kD0 m(T A) > 0. 3. Dowker [12]. Whenever A 2 B and m(A) > 0, P k lim supn!1 n1 1 kD0 m(T A) > 0. Hajian and Kakutani [20] showed that the condition m(A) > 0 implies lim sup m(T n A) > 0
(44)
n!1
is not sufficient for existence of a finite equivalent invariant measure. They also gave another necessary and sufficient condition. Definition 3 A measurable set W X is called wandering if the sets T i W; i 2 Z, are pairwise disjoint. W is called weakly wandering if there are infinitely many integers n i such that T n i W and T n j W are disjoint whenever ni ¤ n j .
Definition 2 Let (X; B; m; T) be as above. A measurable set A X is called T-nonshrinkable if A is not T-equivalent to any proper subset: whenever B A and B T A we have m(A n B) D 0.
Theorem 5 (Hajian-Kakutani [20]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a finite invariant measure m if and only if there are no weakly wandering sets of positive measure.
Theorem 4 (Hopf [23]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a finite invariant measure m if and only if X is T-nonshrinkable.
Finding -finite Invariant Measures Equivalent to a Quasi-Invariant Measure
Proof 4 We present just the easy half. If m is T-invariant and X T B, with corresponding decompositions
While being able to replace a quasi-invariant measure by an equivalent finite invariant measure would be great, it
First Necessary and Sufficient Conditions
971
972
Measure Preserving Systems
may be impossible, and then finding a -finite equivalent measure would still be pretty good. Hopf’s nonshrinkability condition was extended to the -finite case by Halmos: Theorem 6 (Halmos [21]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a -finite invariant measure m if and only if X is a countable union of T-nonshrinkable sets. Another necessary and sufficient condition is given easily in terms of solvability of a cohomological functional equation involving the Radon-Nikodym derivative w of mT with respect to m, defined by Z
w dm for all B 2 B :
m(TB) D
(45)
B
Proposition 4 ([21]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a -finite invariant measure m if and only if there is a measurable function f : X ! (0; 1) such that f (Tx) D w(x) f (x) a.e.
(46)
Proof 5 If m is -finite and T-invariant, let f D dm/d be the Radon-Nikodym derivative of m with respect to , so that Z
f d for all B 2 B :
m(B) D
(47)
B
Then for all B 2 B, since T D , f d D f T d ; while also ZB ZT B w dm D w f dm ; m(TB) D m(TB) D
B
Definition 4 A nonsingular system (X; B; m; T) (with m(X) D 1) is called conservative if there are no wandering sets of positive measure. It is called completely dissipative if there is a wandering set W such that ! 1 [ i T W D m(X) : (51) m iD1
Note that if (X; B; m; T) is completely dissipative, it is easy to construct a -finite equivalent invariant measure. With W as above, define D m on W and push along the orbit of W, letting D mT n on each T n W. We want to claim that this allows us to restrict attention to the conservative case, which follows once we know that the system splits into a conservative and a completely dissipative part. Theorem 7 (Hopf Decomposition [24]) Given a nonsingular map T on a probability space (X; B; m), there are disjoint measurable sets C and D such that 1. X D C [ D; 2. C and D are invariant: TC D C D T 1 C, T D D D D T 1 D; 3. TjC is conservative; 4. If D ¤ ;, then Tj D is completely dissipative. Proof 6 Assume that the family W of wandering sets with positive measure is nonempty, since otherwise we can take C D X and D D ;. Partially order W by W1 W2 if m(W1 n W2 ) D 0 :
Z
Z
Conservativity and Recurrence
(48)
B
so that f T D w f a.e. Conversely, given such an f , let
We want to apply Zorn’s Lemma to find a maximal element in W . Let fW : 2 g be a chain (linearly ordered subset) in W . Just forming [2 W may result in a nonmeasurable set, so we have to use the measure to form a measure-theoretic essential supremum of the chain. So let s D supfm(W ) : 2 g ;
Z (B) D B
1 f
dm for all B 2 B :
Then for all B 2 B Z Z 1 1 (TB) D dm D f f T dmT TB B Z Z 1 1 D w dm D fT f dm D (B) : B
(49)
(54)
and let
B
(53)
so that s 2 (0; 1]. If there is a such that m(W ) D s, let W be that W . Otherwise, for each k choose k 2 so that s k D m(Wk ) " s ;
(50)
(52)
WD
1 [ kD1
Wk :
(55)
Measure Preserving Systems
We claim that in either case W is an upper bound for the chain fW : 2 g. In both cases we have m(W) D s. Note that if ; 2 are such that m(W ) m(W ), then W W . For if W W , then m(W n W ) D 0, and thus m(W ) D m(W \ W ) C m(W n W ) D m(W \ W ) D m(W ) m(W ) ; (56) so that m(W nW ) D 0; W W , and hence W D W . Thus in the first case W 2 W is an upper bound for the chain. In the second case, by discarding the measure 0 set 1 [
(Wk n WkC1 ) ;
B D
1 [
T i B
(58)
iD1
is wandering. In fact much more is true.
m(W \ W ) C m(W n W )
ZD
nonsingular system is recurrent: almost every point of each set of positive measure returns at some future time to that set. This is easy to see, because for each B 2 B, the set
(57)
kD1
we may assume that W is the increasing union of the Wk . Then W Wk for all k, and W is wandering: if some T n W \ W ¤ ;, then there must be a k such that T n Wk \ Wk ¤ ;. Moreover, W W for all 2 . For let 2 be given. Choose k with s k D m(Wk ) > m(W ). By the above, we have Wk W . Since W is the increasing union of the Wk , we have W Wk for all k. Therefore W W , and W is an upper bound in W for the given chain. By Zorn’s Lemma, there is a maximal element W i in W . Then D D [1 iD1 T W is T-invariant, Tj D is completely dissipative, and C D X n D cannot contain any wandering set of positive measure, by maximality of W , so TjC is conservative. Because of this decomposition, when looking for a -finite equivalent invariant measure we may assume that the nonsingular system (X; B; m; T) is conservative, for if not we can always construct one on the dissipative part. Remark 1 If (X; B; m) is nonatomic and T : X ! X is nonsingular, invertible, and ergodic, in the sense that if A 2 B satisfies T 1 A D A D TA then either m(A) D 0 or m(Ac ) D 0, then T is conservative. For if W is a wandering set of positive measure, taking any A W with i 0 < m(A) < m(W) and forming [1 iD1 T A will produce an invariant set of positive measure whose complement also has positive measure. We want to reduce the problem of existence of a -finite equivalent invariant measure to that of a finite one by using first-return maps to sets of finite measure. For this purpose it will be necessary to know that every conservative
Theorem 8 ([21]) For any nonsingular system (X; B; m; T) the following properties are equivalent: 1. The system is incompressible: for each B 2 B such that T 1 B B, we have m(B n T 1 B) D 0. 2. The system is recurrent; for each B 2 B, with B defined as above, m(B n B ) D 0. 3. The system is conservative: there are no wandering sets of positive measure. 4. The system is infinitely recurrent: for each B 2 B, almost every point of B returns to B infinitely many times, equivalently, m Bn
1 [ 1 \
! T
i
B Dm Bn
nD0 iDn
1 \
! T
n
B
D 0:
nD0
(59) There is a very slick proof by F. B. Wright [38] of this result in the even more general situation of a Boolean -algebra homomorphism (reproduced in [28]). Using First-Return Maps, and Counterexamples to Existence Now given a nonsingular conservative system (X; B; m; T) and a set B 2 B, for each x 2 B there is a smallest n B (x) 1 such that T n B (x) 2 B :
(60)
We define the first-return map TB : B ! B by TB (x) D T n B (x) (x) for all x 2 B :
(61)
Using derivative maps, it is easy to reduce the problem of existence of a -finite equivalent invariant measure to that of existence of finite equivalent invariant measures, in a way. Theorem 9 (see [14]) Let T be a conservative nonsingular transformation on a probability space (X; B; m). Then there is a -finite T-invariant measure m if and only if there is an increasing sequence of sets B n 2 B with [1 nD1 B n D X such that for each n the first-return map TB n
973
974
Measure Preserving Systems
has a finite invariant measure equivalent to m restricted to B n . Proof 7 Given a -finite equivalent invariant measure , let the B n be sets of finite -measure that increase to X. Conversely, given such a sequence B n with finite invariant measures n for the first-return maps TB n , extend 1 in the obvious way to an (at least -finite) invariant measure i on the full orbit A1 D [1 iD1 T B1 . Then replace B2 by B2 n A1 , and continue. There are many more checkable conditions for existence of a -finite equivalent invariant measure in the literature. There are also examples of invertible ergodic nonsingular systems for which there does not exist any -finite equivalent invariant measure due to Ornstein [27] and subsequently Chacon [9], Brunel [5], L. Arnold [2], and others. Invariant Measures for Maps of the Interval or Circle Finally we mention sample theorems from a huge array of such results about existence of finite invariant measures for maps of an interval or of the circle. Theorem 10 (“Folklore Theorem” [1]) Let X D (0; 1) and denote by m Lebesgue measure on X. Let T : X ! X be a map for which there is a finite or countable partition ˛ D fA i g of X into half-open intervals [a i ; b i ) satisfying the following conditions. Denote by A0i the interior of each interval A i . Suppose that 1. for each i, T : A0i ! X is one-to-one and onto; 2. T is C 2 on each A0i ; 3. there is an n such that inf inf j(T n )0 (x)j > 1 ; i x2A0 i
(62)
4. for each i, sup j x;y;z2A0i
T 00 (x) j 1 everywhere. Then there is a unique finite invariant measure equivalent to Lebesgue measure m, and in fact is ergodic and the Radon-Nikodym derivative d/dm has a continuous version. Some Mathematical Background Lebesgue Spaces Definition 5 Two measure spaces (X; B; ) and (Y; C ; ) are isomorphic (sometimes also called isomorphic mod 0) if there are subsets X 0 X and Y0 Y such that (X0 ) D 0 D (Y0 ) and a one-to-one onto map : X n X0 ! Y n Y0 such that and 1 are measurable and ( 1 C) D (C) for all measurable C Y n Y0 . Definition 6 A Lebesgue space is a finite measure space that is isomorphic to a measure space consisting of a (possibly empty) finite subinterval of R with the -algebra of Lebesgue measurable sets and Lebesgue measure, possibly together with countably many atoms (point masses). The measure algebra of a measure space (X; B; ) conˆ with Bˆ the Boolean -algebra (see sists of the pair (Bˆ ; ), Sect. “A Range of Actions”, 3.) of B modulo the -ideal of sets of measure 0, together with the operations induced by ˆ is induced on Bˆ by on B. Evset operations in B, and ˆ is a metric space with the metery measure algebra (Bˆ ; ) ˆ ric d(A; B) D (A4B) for all A; B 2 Bˆ . It is nonatomic if whenever A; B 2 Bˆ and A < B (which means A ^ B D A), either A D 0 or A D B. A homomorphism of measure alˆ is a Boolean -algebra homogebras : (Cˆ; ˆ ) ! (Bˆ ; ) ˆ D ˆ (C) for all Cˆ 2 Cˆ. The inˆ C) morphism such that ( verse of any factor map : X ! Y from a measure space (X; B; ) to a measure space (Y; C ; ) induces a homoˆ We say morphism of measure algebras (Cˆ; ˆ ) ! (Bˆ ; ). that a measure algebra is normalized if the measure of the ˆ 0 ) D 1. maximal element is 1: (0 We work within the class of Lebesgue spaces because (1) they are the ones commonly encountered in the wide range of naturally arising examples; (2) they allow us to assume if we wish that we are dealing with a familiar space such as [0; 1] or f0; 1gN ; and (3) they have the following useful properties.
Measure Preserving Systems
(Carathéodory [7]) Every normalized and nonatomic measure algebra whose associated metric space is separable (has a countable dense set) is measure-algebra isomorphic with the measure algebra of the unit interval with Lebesgue measure. (von Neumann [37]) Every complete separable metric space with a Borel probability measure on the completion of the Borel sets is a Lebesgue space. (von Neumann [37]) Every homomorphism : (Cˆ; ˆ of the measure algebras of two Lebesgue ˆ ) ! (Bˆ ; ) spaces (Y; C ; ) and (X; B; ) comes from a factor map: there are a set X0 X with (X0 ) D 0 and a measurable map : X n X 0 ! Y such that coincides with the map induced by 1 from Cˆ to Bˆ .
Every countable or finite measurable partition of a complete measure space is an R-partition. The orbit partition of a measure-preserving transformation is often not an R-partition. (For example, if the transformation is ergodic, the corresponding factor space will be trivial, consisting of just one cell, rather than corresponding to the partition into orbits as required.) For any set B X, let B0 D B and B1 D B c D X n B. Definition 9 A basis C D fC1 ; C2 ; : : : g for a complete measure space (X; B; ) is called complete, and the space is called complete with respect to the basis, if for every 0,1-sequence e 2 f0; 1gN , 1 \
C ie i ¤ ; :
(66)
iD1
Rokhlin Theory V. A. Rokhlin [31] provided an axiomatic, intrinsic characterization of Lebesgue spaces. The key ideas are the concept of a basis and the correspondence of factors with complete sub--algebras and (not necessarily finite or countable) measurable partitions of a special kind. Definition 7 A basis for a complete measure space (X; B; ) is a countable family C D fC1 ; C2 ; : : : g of measurable sets which generates B: For each B 2 B there is C 2 B(C ) (the smallest -algebra of subsets of X that contains C ) such that B C and (C n B) D 0; and separates the points of X: For each x; y 2 X with x ¤ y, there is C i 2 C such that either x 2 C i ; y … C i or else y 2 Ci ; x … Ci . Coarse sub--algebras of B may not separate points of X and thus may lead to equivalence relations, partitions, and factor maps. Partitions of the following kind deserve careful attention. Definition 8 Let (X; B; ) be a complete measure space and a partition of X, meaning that up to a set of measure 0, X is the union of the elements of , which are pairwise disjoint up to sets of measure 0. We call an Rpartition if there is a countable family D D fD1 ; D2 ; : : : g of -saturated sets (that is, each D i is a union of elements of ) such that for all distinct E; F 2 ; there is D i such that either E D i ; F ª D i (so F D ci ) or F D i ; E ª D i (so E
D ci )
(64)
:
Any such family D is called a basis for . Note that each element of an R-partition is necessarily measurable: if C 2 with basis fD i g, then \ (65) CD fD i : C D i g :
C is called complete mod 0 (and (X; B; ) is called complete mod 0 with respect to C , if there is a complete measure space (X 0 ; B0 ; 0 ) with a complete basis C 0 such that X is a full-measure subset of X 0 , and C i D C 0i \ X for all i D 1; 2; : : : .
From the definition of basis, each intersection in (66) contains at most one point. The space f0; 1gN with Bernoulli 1/2; 1/2 measure on the completion of the Borel sets has the complete basis C i D f! : ! i D 0g. Proposition 5 If a measure space is complete mod 0 with respect to one basis, then it is complete mod 0 with respect to every basis. Theorem 12 ([31]) A measure space is a Lebesgue space (that is, isomorphic mod 0 with the usual Lebesgue measure space of a possibly empty subinterval of R possibly together with countably many atoms) if and only if it has a complete basis. In a Lebesgue space (X; B; ) there is a one-to-one onto correspondence between complete sub--algebras of B (that is, those for which the restriction of the measure yields a complete measure space) and R-partitions of X: Given an R-partition , let B() denote the -algebra generated by , which consists of all sets in B that are -saturated – unions of members of – and let B() denote the completion of B() with respect to . Conversely, given a complete sub- -algebra C B, define an equivalence relation on X by x y if for all A 2 C , either x; y 2 A or else x; y 2 Ac . The measure ˆ has a countable dense set Cˆ0 (take a countalgebra (Cˆ; ) ˆ and, for each i; j for which able dense set fBˆ i g for (Bˆ ; ) it is possible, choose Cˆ i j within distance 1/2 j of Bˆ i ). Then representatives C i 2 C of the Cˆi j will be a basis for the partition corresponding to the equivalence relation .
975
976
Measure Preserving Systems
Given any family fB g, of complete sub--algebras of B, their join is the intersection of all the sub--algebras that contain their union: [ _ B D B( B ) ; (67)
and their infimum is just their intersection: \ ^ B D B( B ) :
(68)
These -algebra operations correspond to the supremum and infimum of the corresponding families of R-partitions. We say that a partition 1 is finer than a partition 2 , and write 1 2 , if every element of 2 is a union of elements of 1 . Given any family f g of R-partitions, there W is a coarsest R-partition which refines all of them, V and a finest R-partition which is coarser than all of them. We have _ ^ ^ _ B( ) D B( ) ; B( ) D B( ) : (69)
Now we discuss the relationship among factor maps : X ! Y from a Lebesgue space (X; B; ) to a complete measure space (Y; C ; ), complete sub--algebras of B, and R-partitions of X. Given such a factor map , BY D 1 C is a complete sub--algebra of B, and the equivalence relation x1 x2 if (x1 ) D (x2 ) determines an R-partition Y . (A basis for can be formed from a countable dense set in Bˆ Y as above.) Conversely, given a complete sub--algebra C
B, the identity map (X; B; ) ! (X; C ; ) is a factor map. Alternatively, given an R-partition of X, we can form a measure space (X/; B(); ) and a factor map : X ! X/ as follows. The space X/ is just itself; that is, the points of X/ are the members (cells, or atoms) of the partition . B() consists of the -saturated sets in B considered as subsets of , and is the restriction of to B(). Completeness of (X; B; ) forces completeness of (X/; B(); ). The map : X ! X/ is defined by letting (x) D (x) D the element of to which x belongs. Thus for a Lebesgue space (X; B; ), there is a perfect correspondence among images under factor maps, complete sub--algebras of B, and R-partitions of X. Theorem 13 If (X; B; ) is a Lebesgue space and (Y; C ; ) is a complete measure space that is the image of (X; B; ) under a factor map, then (Y; C ; ) is also a Lebesgue space. Theorem 14 Let (X; B; ) be a Lebesgue space, (Y; C ; ) a separable measure space (that is, one with a countable basis as above, equivalently one with a countable dense set in
its measure algebra), and : X ! Y a measurable map ( 1 C B). Then ' is also forward measurable: if A X is measurable, then (A) Y is measurable. Theorem 15 Let (X; B; ) be a Lebesgue space. 1. Every measurable subset of X, with the restriction of B and , is a Lebesgue space. Conversely, if a subset A of X with the restrictions of B and is a Lebesgue space, then A is measurable (A 2 B). 2. The product of countably many Lebesgue spaces is a Lebesgue space. ˆ (defined 3. Every measure algebra isomorphism of (Bˆ ; ) as above) is induced by a point isomorphism mod 0. Disintegration of Measures Every R-partition of a Lebesgue space (X; B; ) has associated with it a canonical system of conditional measures: Using the notation of the preceding section, for -almost every C 2 X/, there are a -algebra BC of subsets of C and a measure mC on BC such that: 1. (C; BC ; mC ) is a Lebesgue space; 2. for every A 2 B, A \ C 2 BC for -almost every C 2 ; 3. for every A 2 B, the map C ! mC (A \ C) is B()measurable on X/; 4. for every A 2 B, Z (A) D mC (A \ C) d (C) : (70) X/
It follows that for f 2 L1 (X), (a version of) its conditional expectation (see the next section) with respect to the factor algebra corresponding to is given by Z E( f jB()) D f dmC on - a.e. C 2 ; (71) C
since the right-hand side is B()-measurable and for each A 2 B(), its integral over any B 2 B() is, as required, (A \ B) (use the formula on B/(jB)). It can be shown that a canonical system of conditional measures for an R-partition of a Lebesgue space is essentially unique, in the sense that any two measures mC and m0C will be equal for -almost all C 2 . Also, any partition of a Lebesgue space that has a canonical system of conditional measures must be an R-partition. These conditional systems of measures can be used to prove the ergodic decomposition theorem and to show that every factor situation is essentially projection of a skew product onto the base (see [32]).
Measure Preserving Systems
Theorem 16 Let (X; B; ) be a Lebesgue space. If is an R-partition of X, f(C; BC ; mC )g is a canonical system of conditional measures for , and A 2 B, define (AjC) D mC (A \ C). Then: 1. for every A 2 B, (Aj(x)) is a measurable function of x 2 X; 2. if (n ) is an increasing sequence of R-partitions of X, then for each A 2 B _ (Ajn (x)) ! (Aj n (X)) a.e. d ; (72) n
3. if ( n ) is a decreasing sequence of R-partitions of X, then for each A 2 B ^ (Ajn (x)) ! (Aj n (X)) a.e. d : (73) n
This is a consequence of the Martingale and Reverse Martingale Convergence Theorems. The statements hold just as well for f 2 L1 (X) as for f D 1 A for some A 2 B.
each F 2 F we know whether or not x 2 F. When F is the -algebra generated by a finite measurable partition ˛ of X and f is the characteristic function of a set A 2 B, the conditional expectation gives the conditional probabilities of A with respect to all the sets in ˛: E(11 A jF )(x) D (Aj˛(x))
(79) D (A \ F)/(F) if x 2 F 2 ˛ : R We write E( f ) D E( f jf;; Xg) D X f d for the expectation of any integrable function f . A measurable function f on X is independent of a sub- -algebra F B if for each (a; b) R and F 2 F we have ( f 1 (a; b)) \ F) D ( f 1 (a; b)) (F) :
(80)
A function : R ! R is convex if whenever t1 ; : : : ; t n 0 P and niD1 t i D 1, n n X X ti xi ) t i (x i ) for all x1 ; : : : ; x n 2 R : (81) ( iD1
iD1
Theorem 17 Let (X; B; ) be a probability space and F
B a sub- -algebra.
Conditional Expectation Let (X; B; ) be a -finite measure space, f 2 L1 (X), and F B a sub--algebra of B. Then Z (F) D f d (74) F
defines a finite signed measure on F which is absolutely continuous with respect to restricted to F . So by the Radon-Nikodym Theorem there is a function g 2 L1 (X; F ; ) such that Z (F) D g d for all F 2 F : (75)
1. E(jF ) is a positive contraction on L p (X) for each p 1. 2. If f 2 L1 (X) is F -measurable, then E( f jF ) D f a.e. If f 2 L1 (X) is F -measurable, then E( f gjF ) D f E(gjF ) for all g 2 L1 (X). 3. If F1 F2 are sub- -algebras of B, then E(E ( f jF2)jF1 ) D E( f jF1 ) a.e. for each f 2 L1 (X). 4. If f 2 L1 (X) is independent of the sub- -algebra F
B, then E( f jF ) D E( f ) a.e. 5. If : R ! R is convex, f and ı f 2 L1 (X), and F B is a sub- -algebra, then (E( f jF )) E( ı f jF ) a.e.
F
Any such function g, which is unique as an element of L1 (X; F ; ) (and determined only up to sets of -measure 0) is called a version of the conditional expectation of f with respect to F , and denoted by g D E( f jF ) :
(76)
As an element of L1 (X; B; ), E( f jF ) is characterized by the following two properties: E( f jF ) is F -measurable ; Z Z E( f jF ) d D f d for all F 2 F : F
(77) (78)
F
We think of E( f jF )(x) as our expected value for f if we are given the information in F , in the sense that for
The Spectral Theorem A separable Hilbert space is one with a countable dense set, equivalently a countable orthonormal basis. A normal operator is a continuous linear operator S on a Hilbert space H such that SS D S S, S being the adjoint operator defined by (S f ; g) D ( f ; S g) for all f ; g 2 H . A continuous linear operator S is unitary if it is invertible and S D S 1 . Two operators S1 and S2 on Hilbert spaces H1 and H2 , respectively, are called unitarily equivalent if there is a unitary operator U : H1 ! H2 which carries S1 to S2 : S2 U D U S1 . The following brief account follows [10,30]. Theorem 18 Let S : H ! H be a normal operator on a separable Hilbert space H . Then there are mutually singular Borel probability measures 1 ; 1 ; 2 ; : : : such that
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S is unitarily equivalent to the operator M on the direct sum Hilbert space 2
2
L (C; 1 ) ˚ L (C; 1 ) ˚
2 M kD1
˚ ˚
m M
! 2
L (C; 2 ) !
L2 (C; m ) ˚ : : :
(82)
kD1
defined by
The various L2 spaces and multiplication operators involved in the above theorem can be amalgamated into a coherent whole, resulting in the following convenient form of the Spectral Theorem for normal operators Theorem 19 Let S : H ! H be a normal operator on a separable Hilbert space H . There are a finite measure space (X; B; ) and a bounded measurable function h : X ! C such that S is unitarily equivalent to the operator of multiplication by h on L2 (X; B; ). The form of the Spectral Theorem given in Theorem 18 is useful for discussing absolute continuity and multiplicity properties of the spectrum of a normal operator. Another form, involving spectral measures, has useful consequences such as the functional calculus.
M(( f1;1 (z1;1 ); f1;2 (z1;2 ); : : : ); ( f1;1 (z1;1 )); ( f2;1 (z2;1 ); f2;2 (z2;2 )); : : : ) D (z1;1 f1;1 (z1;1 ); z1;2 f1;2 (z1;2 ); : : : ); (z1;1 f1;1 (z1;1 )); (z2;1 f2;1 (z2;1 ); z2;2 f2;2 (z2;2 )); : : : ) : (83) The measures i are supported on the spectrum (S) of S, the (compact) set of all 2 C such that S I does not have a continuous inverse. Some of the i may be 0. They are uniquely determined up to absolute continuity equivalence. The smallest absolute continuity class with respect to which all the i are absolutely continuous is called the maximum spectral type of S. A measure representing this P type is i i /2 i . We have in mind the example for which H D L2 (X; B; ) and S f D f ı T (the “Koopman operator”) for a measure-preserving system (X; B; ; T) on a Lebesgue space (X; B; ), which is unitary: it is linear, continuous, invertible, preserves scalar products, and has spectrum equal to the unit circle. The proof of Theorem 18 can be accomplished by first decomposing H (in a careful way) into the direct sum of pairwise orthogonal cyclic subspaces H n : each H n is the closed linear span of fS i (S ) j f n : i; j 0g for some f n 2 H . This means that for each n the set fp(S; S ) f n : p is a polynomial in two variablesg is dense in H n . Similarly, by the Stone-Weierstrass Theorem the set Pn of all polynomials p(z; z) is dense in the set C ((SjH n )) of continuous complex-valued functions on (SjH n ). We define a bounded linear functional n on Pn by (p) D (p(S; S ) f n ; f n )
(84)
and extend it by continuity to a bounded linear functional on C ((SjH n )). It can be proved that this functional is positive, and therefore, by the Riesz Representation Theorem, it corresponds to a positive Borel measure on (SjH n ).
Theorem 20 Let S : H ! H be a normal operator on a separable Hilbert space H . There is a unique projectionvalued measure E defined on the Borel subsets of the spectrum (S) of S such that E( (S)) D I (D the identity on H ); 1 [
E
!! f D
Ai
iD1
1 X (E(A i )) f
(85)
iD1
whenever A1 ; A2 ; : : : are pairwise disjoint Borel subsets of (S) and f 2 H , with the series converging in norm; and Z SD
(S)
dE() :
(86)
Spectral integrals such as the one in (86) can be defined by reducing to complex measures f ;g (A) D (E(A) f ; g), for f ; g 2 H and A (S) a Borel set. Given a bounded Borel measurable function on (S), the operator Z V D (S) D
(S)
() dE()
(87)
is determined by specifying that Z (V f ; g) D
(S)
() d f ;g for all f ; g 2 H :
(88)
Then Sk D
Z (S)
k dE() for all k D 0; 1; : : : :
(89)
Measure Preserving Systems
These spectral integrals sometimes behave a bit strangely: Z If V1 D 1 () dE() (S) Z and V2 D 2 () dE() ; (90) (S) Z then V1 V2 D 1 () 2 () dE() : (S)
Finally, if f 2 H and is a finite positive Borel measure that is absolutely continuous with respect to f ; f , then there is g in the closed linear span of fS i (S ) j f : i; j 0g such that D g;g . Theorem 20 can be proved by applying Theorem 19, which allows us to assume that H D L2 (X; B; ) and S is multiplication by h 2 L1 (X; B; ). For any Borel set A (S), let E(A) be the projection operator given by multiplication by the 0; 1-valued function 1 A ı h. Future Directions The mathematical study of dynamical systems arose in the late nineteenth and early twentieth century, along with measure theory and probability theory, so it is a young field with many interesting open problems. New questions arise continually from applications and from interactions with other parts of mathematics. Basic aspects of the problems of classification, topological or smooth realization, and systematic construction of measure-preserving systems remain open. There is much work to be done to understand the relations among systems and different types of systems (factors and relative properties, joinings and disjointness, various notions of equivalence with associated invariants). There is a continual need to determine properties of classes of systems and of particular systems arising from applications or other parts of mathematics such as probability, number theory, geometry, algebra, and harmonic analysis. Some of these questions are mentioned in more detail in the other articles in this collection. Bibliography Primary Literature 1. Adler RL (1973) F-expansions revisited. Lecture Notes in Math, vol 318. Springer, Berlin 2. Arnold LK (1968) On -finite invariant measures. Z Wahrscheinlichkeitstheorie Verw Geb 9:85–97 3. Billingsley P (1978) Ergodic theory and information. Robert E Krieger Publishing Co, Huntington NY, pp xiii,194; Reprint of the 1965 original 4. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York, pp xiv,593
5. Brunel A (1966) Sur les mesures invariantes. Z Wahrscheinlichkeitstheorie Verw Geb 5:300–303 6. Calderón AP (1955) Sur les mesures invariantes. C R Acad Sci Paris 240:1960–1962 7. Carathéodory C (1939) Die Homomorphieen von Somen und die Multiplikation von Inhaltsfunktionen. Annali della R Scuola Normale Superiore di Pisa 8(2):105–130 8. Carathéodory C (1968) Vorlesungen über Reelle Funktionen, 3rd edn. Chelsea Publishing Co, New York, pp x,718 9. Chacon RV (1964) A class of linear transformations. Proc Amer Math Soc 15:560–564 10. Conway JB (1990) A Course in Functional Analysis, vol 96, 2nd edn. Springer, New York, pp xvi,399 11. Dowker YN (1955) On measurable transformations in finite measure spaces. Ann Math 62(2):504–516 12. Dowker YN (1956) Sur les applications mesurables. C R Acad Sci Paris 242:329–331 13. Frechet M (1924) Des familles et fonctions additives d’ensembles abstraits. Fund Math 5:206–251 14. Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Co., New York, pp v,143 15. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Anal Math 31:204–256 16. Gowers WT (2001) A new proof of Szemerédi’s theorem. Geom Funct Anal 11(3):465–588 17. Gowers WT (2001) Erratum: A new proof of Szemerédi’s theorem. Geom Funct Anal 11(4):869 18. Green B, Tao T (2007) The primes contain arbitrarily long arithmetic progressions. arXiv:math.NT/0404188 19. Hahn H (1933) Über die Multiplikation total-additiver Mengenfunktionen. Annali Scuola Norm Sup Pisa 2:429–452 20. Hajian AB, Kakutani S (1964) Weakly wandering sets and invariant measures. Trans Amer Math Soc 110:136–151 21. Halmos PR (1947) Invariant measures. Ann Math 48(2):735–754 22. Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, pp xviii,802 23. Hopf E (1932) Theory of measure and invariant integrals. Trans Amer Math Soc 34:373–393 24. Hopf E (1937) Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1st edn. Springer, Berlin, pp iv,83 25. Khinchin AI (1949) Mathematical Foundations of Statistical Mechanics. Dover Publications Inc, New York, pp viii,179 26. Kolmogorov AN (1956) Foundations of the Theory of Probability. Chelsea Publishing Co, New York, pp viii,84 27. Ornstein DS (1960) On invariant measures. Bull Amer Math Soc 66:297–300 28. Petersen K (1989) Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge, pp xii,329 29. Poincaré H (1987) Les Méthodes Nouvelles de la Mécanique Céleste. Tomes I, II,III. Les Grands Classiques Gauthier-Villars. Librairie Scientifique et Technique Albert Blanchard, Paris 30. Radjavi H, Rosenthal P (1973) Invariant subspaces, 2nd edn. Springer, Mineola, pp xii,248 31. Rohlin VA (1952) On the fundamental ideas of measure theory. Amer Math Soc Transl 1952:55 32. Rohlin VA (1960) New progress in the theory of transformations with invariant measure. Russ Math Surv 15:1–22
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33. Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York, pp xx,444 34. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:199–245 35. Tao T (2006) Szemerédi’s regularity lemma revisited. Contrib Discret Math 1:8–28 36. Tao T (2006) Arithmetic progressions and the primes. Collect Math Extra:37–88 37. von Neumann J (1932) Einige Sätze über messbare Abbildungen. Ann Math 33:574–586 38. Wright FB (1961) The recurrence theorem. Amer Math Mon 68:247–248 39. Young L-S (2002) What are SRB measures, and which dynamical systems have them? J Stat Phys 108:733–754
Books and Reviews Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, pp xiii,194 Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic Theory. Fundamental Principles of Mathematical Sciences, vol 245. Springer, New York, pp x,486 Denker M, Grillenberger C, Sigmund K (1976) Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol 527. Springer, Berlin, pp iv,360
Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Co, New York, pp v,143 Glasner E (2003) Ergodic Theory via Joinings. Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence, pp xii,384 Halmos PR (1960) Lectures on Ergodic Theory. Chelsea Publishing Co, New York, pp vii,101 Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge, pp xviii,802 Hopf E (1937) Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1st edn. Springer, Berlin, pp iv,83 Jacobs K (1960) Neue Methoden und Ereignisse der Ergodentheorie. Jber Dtsch Math Ver 67:143–182 Petersen K (1989) Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge, pp xii,329 Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York, pp xx,444 Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, pp x,168 Walters P (1982) An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol 79. Springer, New York, pp ix,250
Mechanical Systems: Symmetries and Reduction
Mechanical Systems: Symmetries and Reduction JERROLD E. MARSDEN1 , TUDOR S. RATIU2 1 Control and Dynamical Systems, California Institute of Technology, Pasadena, USA 2 Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Article Outline Glossary Definition of the Subject Introduction Symplectic Reduction Symplectic Reduction – Further Discussion Reduction Theory: Historical Overview Cotangent Bundle Reduction Future Directions Acknowledgments Appendix: Principal Connections Bibliography Glossary Lie group action A process by which a Lie group, acting as a symmetry, moves points in a space. When points in the space that are related by a group element are identified, one obtains the quotient space. Free action An action that moves every point under any nontrivial group element. Proper action An action that obeys a compactness condition. Momentum mapping A dynamically conserved quantity that is associated with the symmetry of a mechanical system. An example is angular momentum, which is associated with rotational symmetry. Symplectic reduction A process of reducing the dimension of the phase space of a mechanical system by restricting to the level set of a momentum map and also identifying phase space points that are related by a symmetry. Poisson reduction A process of reducing the dimension of the phase space of a mechanical system by identifying phase space points that are related by a symmetry. Equivariance Equivariance of a momentum map is a property that reflects the consistency of the mapping with a group action on its domain and range. Momentum cocycle A measure of the lack of equivariance of a momentum mapping.
Singular reduction A reduction process that leads to non-smooth reduced spaces. Often associated with non-free group actions. Coadjoint orbit The orbit of an element of the dual of the Lie algebra under the natural action of the group. KKS (Kostant-Kirillov-Souriau) form The natural symplectic form on coadjoint orbits. Cotangent bundle A mechanical phase space that has a structure that distinguishes configurations and momenta. The momenta lie in the dual to the space of velocity vectors of configurations. Shape space The space obtained by taking the quotient of the configuration space of a mechanical system by the symmetry group. Principal connection A mathematical object that describes the geometry of how a configuration space is related to its shape space. Related to geometric phases through the subject of holonomy. In turn related to locomotion in mechanical systems. Mechanical connection A special (principal) connection that is built out of the kinetic energy and momentum map of a mechanical system with symmetry. Magnetic terms These are expressions that are built out of the curvature of a connection. They are so named because terms of this form occur in the equations of a particle moving in a magnetic field. Definition of the Subject Reduction theory is concerned with mechanical systems with symmetries. It constructs a lower dimensional reduced space in which associated conservation laws are enforced and symmetries are “factored out” and studies the relation between the dynamics of the given system with the dynamics on the reduced space. This subject is important in many areas, such as stability of relative equilibria, geometric phases and integrable systems. Introduction Geometric mechanics has developed in the last 30 years or so into a mature subject in its own right, and its applications to problems in Engineering, Physics and other physical sciences has been impressive. One of the important aspects of this subject has to do with symmetry; even things as apparently simple as the symmetry of a system such as the n-body problem under the group of translations and rotations in space (the Euclidean group) or a wheeled vehicle under the planar Euclidean group turns out to have profound consequences. Symmetry often gives conservation laws through Noether’s theorem and these conservation laws can be used to reduce the dimension of a system.
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In fact, reduction theory is an old and time-honored subject, going back to the early roots of mechanics through the works of Euler, Lagrange, Poisson, Liouville, Jacobi, Hamilton, Riemann, Routh, Noether, Poincaré, and others. These founding masters regarded reduction theory as a useful tool for simplifying and studying concrete mechanical systems, such as the use of Jacobi’s elimination of the node in the study of the n-body problem to deal with the overall rotational symmetry of the problem. Likewise, Liouville and Routh used the elimination of cyclic variables (what we would call today an Abelian symmetry group) to simplify problems and it was in this setting that the Routh stability method was developed. The modern form of symplectic reduction theory begins with the works of Arnold [7], Smale [168], Meyer [117], and Marsden and Weinstein [110]. A more detailed survey of the history of reduction theory can be found in the first sections of this article. As was the case with Routh, this theory has close connections with the stability theory of relative equilibria, as in Arnold [8] and Simo, Lewis, and Marsden [166]. The symplectic reduction method is, in fact, by now so well known that it is used as a standard tool, often without much mention. It has also entered many textbooks on geometric mechanics and symplectic geometry, such as Abraham and Marsden [1], Arnold [9], Guillemin and Sternberg [57], Libermann and Marle [89] and Mcduff and Salamon [116]. Despite its relatively old age, research in reduction theory continues vigorously today. It will be assumed that the reader is familiar with the basic concepts in [104]. For the statements of the bulk of the theorems, it is assumed that the manifolds involved are finite dimensional and are smooth unless otherwise stated. While many interesting examples are infinite-dimensional, the general theory in the infinite dimensional case is still not in ideal shape; see, for example, Chernoff and Marsden [39], Marsden and Hughes [96] and Mielke [118] and examples and discussion in [92]. Notation To keep things reasonably systematic, we have adopted the following universal conventions for some common maps and other objects: Configuration space of a mechanical system: Q Phase space: P Cotangent bundle projection: Q : T Q ! Q Tangent bundle projection: Q : TQ ! Q Quotient projection: P;G : P ! P/G Tangent map: T' : TM ! T N for the tangent of a map ': M ! N
Thus, for example, the symbol T Q;G denotes the quotient projection from T Q to (T Q)/G. The Lie algebra of a Lie group G is denoted g. Actions of G on a space is denoted by concatenation. For example, the action of a group element g on a point q 2 Q is written as gq or g q The infinitesimal generator of a Lie algebra element 2 g for an action of G on P is denoted P , a vector field on P Momentum maps are denoted J : P ! g . Pairings between vector spaces and their duals are denoted by simple angular brackets: for example, the pairing between g and g is denoted h; i for 2 g and 2g Inner products are denoted with double angular brackets: hhu; vii. Symplectic Reduction Roughly speaking, here is how symplectic reduction goes: given the symplectic action of a Lie group on a symplectic manifold that has a momentum map, one divides a level set of the momentum map by the action of a suitable subgroup to form a new symplectic manifold. Before the division step, one has a manifold (that can be singular if the points in the level set have symmetries) carrying a degenerate closed 2-form. Removing such a degeneracy by passing to a quotient space is a differential-geometric operation that was promoted by Cartan [26]. The “suitable subgroup” related to a momentum mapping was identified by Smale [168] in the special context of cotangent bundles. It was Smale’s work that inspired the general symplectic construction by Meyer [117] and the version we shall use, which makes explicit use of the properties of momentum maps, by Marsden and Weinstein [110]. Momentum Maps Let G be a Lie group, g its Lie algebra, and g be its dual. Suppose that G acts symplectically on a symplectic manifold P with symplectic form denoted by ˝. We shall denote the infinitesimal generator associated with the Lie algebra element by P and we shall let the Hamiltonian vector field associated to a function f : P ! R be denoted X f . A momentum map is a map J : P ! g , which is defined by the condition P D XhJ;i
(1)
for all 2 g, and where hJ; i : P ! R is defined by the natural pointwise pairing. We call such a momentum
Mechanical Systems: Symmetries and Reduction
map equivariant when it is equivariant with respect to the given action on P and the coadjoint action of G on g . That is, J(g z) D Adg 1 J(z)
(2)
for every g 2 G, z 2 P, where g z denotes the action of g on the point z, Ad denotes the adjoint action, and Ad the coadjoint action. Note that when we write Adg 1 , we literally mean the adjoint of the linear map Ad g 1 : g ! g. The inverse of g is necessary for this to be a left action on g . Some authors let that inverse be understood in the notation. However, such a convention would be a notational disaster since we need to deal with both left and right actions, a distinction that is essential in mechanics. A quadruple (P; ˝; G; J), where (P; ˝) is a given symplectic manifold and J : P ! g is an equivariant momentum map for the symplectic action of a Lie group G, is sometimes called a Hamiltonian G-space. Taking the derivative of the equivariance identity (2) with respect to g at the identity yields the condition of infinitesimal equivariance: Tz J( P (z)) D ad J(z)
(3)
for any 2 g and z 2 P. Here, ad : g ! g; 7! [; ] is the adjoint map and ad : g ! g is its dual. A computation shows that (3) is equivalent to hJ; [; ]i D fhJ; i; hJ; ig
(4)
for any ; 2 g, that is, hJ; i : g ! F (P) is a Lie algebra homomorphism, where F (P) denotes the Poisson algebra of smooth functions on P. The converse is also true if the Lie group is connected, that is, if G is connected then an infinitesimally equivariant action is equivariant (see §12.3 in [104]). The idea that an action of a Lie group G with Lie algebra g on a symplectic manifold P should be accompanied by such an equivariant momentum map J : P ! g and the fact that the orbits of this action are themselves symplectic manifolds both occur already in Lie [90]; the links with mechanics also rely on the work of Lagrange, Poisson, Jacobi and Noether. In modern form, the momentum map and its equivariance were rediscovered by Kostant [78] and Souriau [169,170] in the general symplectic case and by Smale [168] for the case of the lifted action from a manifold Q to its cotangent bundle P D T Q. Recall that the equivariant momentum map in this case is given explicitly by ˝ ˛ ˝ ˛ J(˛q ); D ˛q ; Q (q) ; (5) where ˛q 2 Tq Q; 2 g, and where the angular brackets denote the natural pairing on the appropriate spaces.
Smale referred to J as the “angular momentum” by generalization from the special case G D SO(3), while Souriau used the French word “moment”. Marsden and Weinstein [110], followed usage emerging at that time and used the word “moment” for J, but they were soon corrected by Richard Cushman and Hans Duistermaat, who suggested that the proper English translation of Souriau’s French word was “momentum,” which fit better with Smale’s designation as well as standard usage in mechanics. Since 1976 or so, most people who have contact with mechanics use the term momentum map (or mapping). On the other hand, Guillemin and Sternberg popularized the continuing use of “moment” in English, and both words coexist today. It is a curious twist, as comes out in work on collective nuclear motion Guillemin and Sternberg [56] and plasma physics (Marsden and Weinstein [111] and Marsden, Weinstein, Ratiu, Schmid, and Spencer [114]), that moments of inertia and moments of probability distributions can actually be the values of momentum maps! Mikami and Weinstein [119] attempted a linguistic reconciliation between the usage of “moment” and “momentum” in the context of groupoids. See [104] for more information on the history of the momentum map and Sect. “Reduction Theory: Historical Overview” for a more systematic review of general reduction theory. Momentum Cocycles and Nonequivariant Momentum Maps Consider a momentum map J : P ! g that need not be equivariant, where P is a symplectic manifold on which a Lie group G acts symplectically. The map : G ! g that is defined by (g) :D J(g z) Adg 1 J(z) ;
(6)
where g 2 G and z 2 P is called a nonequivariance or momentum one-cocycle. Clearly, is a measure of the lack of equivariance of the momentum map. We shall now prove a number of facts about . The first claim is that does not depend on the point z 2 P provided that the symplectic manifold P is connected (otherwise it is constant on connected components). To prove this, we first recall the following equivariance identity for infinitesimal generators: Tq ˚ g P (q) D (Ad g )P (g q) ; (7) i. e.; ˚ g P D Ad g 1 P : This is an easy Lie group identity that is proved, for example, in [104]; see Lemma 9.3.7. One shows that (g) is constant by showing that its Hamiltonian vector field vanishes. Using the fact that (g)
983
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Mechanical Systems: Symmetries and Reduction
is independent of z along with the basic identity Ad g h D Ad g Ad h and its consequence Ad(g h)1 D Adg 1 Adh 1 , shows that satisfies the cocycle identity (gh) D (g) C Adg 1 (h)
(8)
for any g; h 2 G. This identity shows that produces a new action : G g ! g defined by (g; ) :D Adg 1 C (g)
(9)
with respect to which the momentum map J is obviously equivariant. This action is not linear anymore – it is an affine action. For 2 g, let (g) D h(g); i. Differentiating the definition of , namely ˝ ˛ (g) D hJ(g z); i J(z); Ad g 1 with respect to g at the identity in the direction 2 g shows that Te () D ˙ (; ) ;
(10)
where ˙ (; ), which is called the infinitesimal nonequivariance two-cocycle, is defined by ˙ (; ) D hJ; [; ]i fhJ; i; hJ; ig :
(11)
Since does not depend on the point z 2 P, neither does ˙ . Also, it is clear from this definition that ˙ measures the lack of infinitesimal equivariance of J. Another way to look at this is to notice that from the derivation of Eq. (10), for z 2 P and 2 g, we have Tz J( P (z)) D ad J(z) C ˙ (; ) :
(12)
Comparison of this relation with Eq. (3) also shows the relation between ˙ and the infinitesimal equivariance of J. The map ˙ : g g ! R is bilinear, skew-symmetric, and, as can be readily verified, satisfies the two-cocycle identity ˙ ([; ]; ) C ˙ ([; ]; ) C ˙ ([ ; ]; ) D 0 ;
(13)
for all ; ; 2 g. The Symplectic Reduction Theorem There are many precursors of symplectic reduction theory. When G is Abelian, the components of the momentum map form a system of functions in involution (i. e. the Poisson bracket of any two is zero). The use of k such functions to reduce a phase space to one having 2k fewer dimensions may be found already in the work of Lagrange,
Poisson, Jacobi, and Routh; it is well described in, for example, Whittaker [179]. In the nonabelian case, Smale [168] noted that Jacobi’s elimination of the node in SO(3) symmetric problems can be understood as division of a nonzero angular momentum level by the SO(2) subgroup which fixes the momentum value. In his setting of cotangent bundles, Smale clearly stated that the coadjoint isotropy group G of 2 g (defined to be the group of those g 2 G such that g D , where the dot indicates the coadjoint action), leaves the level set J1 () invariant (Smale [168], Corollary 4.5). However, he only divided by G after fixing the total energy as well, in order to obtain the “minimal” manifold on which to analyze the reduced dynamics. The goal of his “topology and mechanics” program was to use topology, and specifically Morse theory, to study relative equilibria, which he did with great effectiveness. Marsden and Weinstein [110] combined Souriau’s momentum map for general symplectic actions, Smale’s idea of dividing the momentum level by the coadjoint isotropy group, and Cartan’s idea of removing the degeneracy of a 2-form by passing to the leaf space of the form’s null foliation. The key observation was that the leaves of the null foliation are precisely the (connected components of the) orbits of the coadjoint isotropy group (a fact we shall prove in the next section as the reduction lemma). An analogous observation was made in Meyer [117], except that Meyer worked in terms of a basis for the Lie algebra g and identified the subgroup G as the group which left the momentum level set J1 () invariant. In this way, he did not need to deal with the equivariance properties of the coadjoint representation. In the more general setting of symplectic manifolds with an equivariant momentum map for a symplectic group action, the fact that G acts on J1 () follows directly from equivariance of J. Thus, it makes sense to form the symplectic reduced space which is defined to be the quotient space P D J1 ()/G :
(14)
Roughly speaking, the symplectic reduction theorem states that, under suitable hypotheses, P is itself a symplectic manifold. To state this precisely, we need a short excursion on level sets of the momentum map and some facts about quotients. Free and Proper Actions The action of a Lie group G on a manifold M is called a free action if g m D m for some g 2 G and m 2 M implies that g D e, the identity element.
Mechanical Systems: Symmetries and Reduction
An action of G on M is called proper when the map G M ! M M; (g; m) 7! (g m; m) is a proper map – that is, inverse images of compact sets are compact. This is equivalent to the statement that if mk is a convergent sequence in M and if g k m k converges in M, then g k has a convergent subsequence in G. As is shown in, for example, [2] and Duistermaat and Kolk [48], freeness, together with properness implies that the quotient space M/G is a smooth manifold and that the projection map : M ! M/G is a smooth surjective submersion. Locally Free Actions An action of G on M is called infinitesimally free at a point m 2 M if M (m) D 0 implies that D 0. An action of G on M is called locally free at a point m 2 M if there is a neighborhood U of the identity in G such that g 2 U and g m D m implies g D e. Proposition 1 An action of a Lie group G on a manifold M is locally free at m 2 M if and only if it is infinitesimally free at m. A free action is obviously locally free. The converse is not true because the action of any discrete group is locally free, but need not be globally free. When one has an action that is locally free but not globally free, one is lead to the theory of orbifolds, as in Satake [164]. In fact, quotients of manifolds by locally free and proper group actions are orbifolds, which follows by the use of the Palais slice theorem (see Palais [144]). Orbifolds come up in a variety of interesting examples involving, for example, resonances; see, for instance, Cushman and Bates [44] and Alber, Luther, Marsden, and Robbins [3] for some specific examples. Symmetry and Singularities If is a regular value of J then we claim that the action is automatically locally free at the elements of the corresponding level set J1 (). In this context it is convenient to introduce the notion of the symmetry algebra at z 2 P defined by gz D f 2 g j P (z) D 0g : The symmetry algebra gz is the Lie algebra of the isotropy subgroup Gz of z 2 P defined by Gz D fg 2 G j g z D zg : The following result (due to Smale [168] in the special case of cotangent bundles and in general to Arms, Marsden,
and Moncrief [5]), is important for the recognition of regular as well as singular points in the reduction process. Proposition 2 An element 2 g is a regular value of J if and only if gz D 0 for all z 2 J1 (). In other words, points are regular points precisely when they have trivial symmetry algebra. In examples, this gives an easy way to recognize regular points. For example, for the double spherical pendulum (see, for example, Marsden and Scheurle [108] or [95]), one can say right away that the only singular points are those with both pendula pointing vertically (either straight down or straight up). This result holds whether or not J is equivariant. This result, connecting the symmetry of z with the regularity of , suggests that points with symmetry are bifurcation points of J. This observation turns out to have many important consequences, including some related key convexity theorems. Now we are ready to state the symplectic reduction theorem. We will be making two sets of hypotheses; other variants are discussed in the next section. The following notation will be convenient in the statement of the results. SR (P; ˝) is a symplectic manifold, G is a Lie group that acts symplectically on P and has an equivariant momentum map J : P ! g . SRFree G acts freely and properly on P. SRRegular Assume that 2 g is a regular value of J and that the action of G on J1 () is free and proper From the previous discussion, note that condition SRFree implies condition SRRegular. The real difference is that SRRegular assumes local freeness of the action of G (which is equivalent to being a regular value, as we have seen), while SRFree assumes global freeness (on all of P). Theorem 3 (Symplectic reduction theorem) Assume that condition SR and that either the condition SRFree or the condition SRRegular holds. Then P is a symplectic manifold, and is equipped with the reduced symplectic form ˝ that is uniquely characterized by the condition ˝; ˝ D i
(15)
where : J1 () ! P is the projection to the quotient space and where i : J1 () ! P is the inclusion. The above procedure is often called point reduction because one is fixing the value of the momentum map at a point 2 g . An equivalent reduction method called orbit reduction will be discussed shortly.
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Coadjoint Orbits
Mathematical Physics Links
A standard example (due to Marsden and Weinstein [110]) that we shall derive in detail in the next section, is the construction of the coadjoint orbits in g of a group G by reduction of the cotangent bundle T G with its canonical symplectic structure and with G acting on T G by the cotangent lift of left (resp. right) group multiplication. In this case, one finds that (T G) D O , the coadjoint orbit through 2 g . The reduced symplectic form is given by the Kostant, Kirillov, Souriau coadjoint form, also referred to as the KKS form:
Another example in Marsden and Weinstein [110] came from general relativity, namely the reduction of the cotangent bundle of the space of Riemannian metrics on a manifold M by the action of the group of diffeomorphisms of M. In this case, restriction to the zero momentum level is the divergence constraint of general relativity, and so one is led to a construction of a symplectic structure on a space closely related to the space of solutions of the Einstein equations, a question revisited in Fischer, Marsden, and Moncrief [51] and Arms, Marsden, and Moncrief [6]. Here one sees a precursor of an idea of Atiyahf and Bott [11], which has led to some of the most spectacular applications of reduction in mathematical physics and related areas of pure mathematics, especially low-dimensional topology.
!O ()(ad ; ad ) D h; [; ]i ;
(16)
where ; 2 g; 2 O , ad : g ! g is the adjoint operator defined by ad :D [; ] and ad : g ! g is its dual. In this formula, one uses the minus sign for the left action and the plus sign for the right action. We recall that coadjoint orbits, like any group orbit is always an immersed manifold. Thus, one arrives at the following result (see also Theorem 7): Corollary 4 Given a Lie group G with Lie algebra g and any point 2 g , the reduced space (T G) is the coadjoint orbit O through the point ; it is a symplectic manifold with symplectic form given by (16). This example, which “explains” Kostant, Kirillov and Souriau’s formula for this structure, is typical of many of the ensuing applications, in which the reduction procedure is applied to a “trivial” symplectic manifold to produce something interesting. Orbit Reduction An important variant of the symplectic reduction theorem is called orbit reduction and, roughly speaking, it constructs J1 (O)/G, where O is a coadjoint orbit in g . In the next section – see Theorem 8 – we show that orbit reduction is equivalent to the point reduction considered above. Cotangent Bundle Reduction The theory of cotangent bundle reduction is a very important special case of general reduction theory. Notice that the reduction of T G above to give a coadjoint orbit is a special case of the more general procedure in which G is replaced by a configuration manifold Q. The theory of cotangent bundle reduction will be outlined in the historical overview in this chapter, and then treated in some detail in the following chapter.
Singular Reduction In the preceding discussion, we have been making hypotheses that ensure the momentum levels and their quotients are smooth manifolds. Of course, this is not always the case, as was already noted in Smale [168] and analyzed (even in the infinite-dimensional case) in Arms, Marsden, and Moncrief [5]. We give a review of some of the current literature and history on this singular case in Sect. “Reduction Theory: Historical Overview”. For an outline of this subject, see [142] and for a complete account of the technical details, see [138]. Reduction of Dynamics Along with the geometry of reduction, there is also a theory of reduction of dynamics. The main idea is that a G-invariant Hamiltonian H on P induces a Hamiltonian H on each of the reduced spaces, and the corresponding Hamiltonian vector fields X H and X H are -related. The reverse of reduction is reconstruction and this leads one to the theory of classical geometric phases (Hannay–Berry phases); see Marsden, Montgomery, and Ratiu [98]. Reduction theory has many interesting connections with the theory of integrable systems; we just mention some selected references Kazhdan, Kostant, and Sternberg [72]; Ratiu [154,155,156]; Bobenko, Reyman, and Semenov-Tian-Shansky [22]; Pedroni [148]; Marsden and Ratiu [103]; Vanhaecke [174]; Bloch, Crouch, Marsden, and Ratiu [19], which the reader can consult for further information. Symplectic Reduction – Further Discussion The symplectic reduction theorem leans on a few key lemmas that we just state. The first refers to the reflexivity of
Mechanical Systems: Symmetries and Reduction
the operation of taking the symplectic orthogonal complement. Lemma 5 Let (V; ˝) be a finite dimensional symplectic vector space and W V be a subspace. Define the symplectic orthogonal to W by W ˝ D fv 2 V j ˝(v; w) D 0 for all w 2 Wg : Then ˝ W˝ DW:
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In what follows, we denote by G z and G z the G and G -orbits through the point z 2 P; note that if z 2 J1 () then G z J1 (). The key lemma that is central for the symplectic reduction theorem is the following. Lemma 6 (Reduction lemma) Let P be a Poisson manifold and let J : P ! g be an equivariant momentum map of a Lie group action by Poisson maps of G on P. Let G denote the coadjoint orbit through a regular value 2 g of J. Then J1 (G ) D G J1 () D fg z j g 2 G and J(z) D g; (ii) G z D (G z) \ J1 (); (iii) J1 () and G z intersect cleanly, i. e., (i)
Tz (G z) D Tz (G z) \ Tz (J1 ()); (iv) if (P; ˝) is symplectic, then Tz (J1 ()) D (Tz (G z))˝ ; i. e., the sets Tz (J1 ()) and Tz (G z) are ˝-orthogonal complements of each other. Refer to Fig. 1 for one way of visualizing the geometry associated with the reduction lemma. As it suggests, the two manifolds J1 () and G z intersect in the orbit of the isotropy group G z and their tangent spaces Tz J1 () and Tz (G z) are orthogonal and intersect in symplectically the space Tz G z . Notice from the statement (iv) that Tz (J1 ())˝ Tz (J1 ()) provided that G z D G z. Thus, J1 () is coisotropic if G D G; for example, this happens if D 0 or if G is Abelian. Remarks on the Reduction Theorem 1. Even if ˝ is exact; say ˝ D d and the action of G leaves invariant, ˝ need not be exact. Perhaps the simplest example is a nontrivial coadjoint orbit of SO(3), which is a sphere with symplectic form given by the area form (by Stokes’ theorem, it cannot be ex-
Mechanical Systems: Symmetries and Reduction, Figure 1 The geometry of the reduction lemma
act). That this is a symplectic reduced space of T SO(3) (with the canonical symplectic structure, so is exact) is shown in Theorem 7 below. 2. Continuing with the previous remark, assume that ˝ D d and that the G principal bundle J1 () ! P :D J1 ()/G is trivializable; that is, it admits 2 a global section s : P ! J1 (). Let :D s i 1 ˝ (P ). Then the reduced symplectic form ˝ D d is exact. This statement does not imply that the one-form descends to the reduced space, only that the reduced symplectic form is exact and one if its primitives is . In fact, if one changes the global section, another primitive of ˝ is found which differs from by a closed one-form on P . 3. The assumption that is a regular value of J can be relaxed. The only hypothesis needed is that be a clean value of J, i. e., J1 () is a manifold and Tz (J1 ()) D ker Tz J. This generalization applies, for instance, for zero angular momentum in the three dimensional two body problem, as was noted by Marsden and Weinstein [110] and Kazhdan, Kostant, and Sternberg [72]; see also Guillemin and Sternberg [57]. Here are the general definitions: If f : M ! N is a smooth map, a point y 2 N is called a clean value if f 1 (y) is a submanifold and for each x 2 f 1 (y), Tx f 1 (y) D ker Tx f . We say that f intersects a submanifold L N cleanly if f 1 (L) is a submanifold of M and Tx ( f 1 (L)) D (Tx f )1 (T f (x) L). Note that regular values of f are clean values and that if f intersects the submanifold L transversally, then it intersects it cleanly. Also note that the definition of clean intersection of two manifolds is equivalent to the statement that the inclusion map of either one of them intersects the other cleanly. The reduction lemma is an example of this situation.
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4. The freeness and properness of the G action on J1 () are used only to guarantee that P is a manifold; these hypotheses can thus be replaced by the requirement that P is a manifold and : J1 () ! P a submersion; the proof of the symplectic reduction theorem remains unchanged. 5. Even if is a regular value (in the sense of a regular value of the mapping J), it need not be a regular point (also called a generic point) in g ; that is, a point whose coadjoint orbit is of maximal dimension. The reduction theorem does not require that be a regular point. For example, if G acts on itself on the left by group multiplication and if we lift this to an action on T G by the cotangent lift, then the action is free and so all are regular values, but such values (for instance, the zero element in so(3) ) need not be regular. On the other hand, in many important stability considerations, a regularity assumption on the point is required; see, for instance, Patrick [145], Ortega and Ratiu [136] and Patrick, Roberts, and Wulff [146].
Nonequivariant Reduction We now describe how one can carry out reduction for a nonequivariant momentum map. If J : P ! g is a nonequivariant momentum map on the connected symplectic manifold P with nonequivariance group one-cocycle consider the affine action (9) and let e G be the isotropy subgroup of 2 g relative to this action. Then, under the same regularity assumptions (for example, assume that G acts freely and properly on P, or that is a regular value of J and that e G acts freely and properly on J1 ()), the quotient manifold P :D J1 ()/e G is a symplectic manifold whose symplectic form is uniquely determined by the relation ˝ D ˝ . The proof of this statement is identical to i the one given above with the obvious changes in the meaning of the symbols. When using nonequivariant reduction, one has to remember that G acts on g in an affine and not a linear manner. For example, while the coadjoint isotropy subgroup at the origin is equal to G; that is, G0 D G, this is no longer the case for the affine action, where e G 0 in general does not equal G. Coadjoint Orbits as Symplectic Reduced Spaces We now examine Corollary 4 – that is, that coadjoint orbits may be realized as reduced spaces – a little more closely. Realizing them as reduced spaces shows that they are symplectic manifolds See, Chap. 14 in [104] for a “di-
rect” or “bare hands” argument. Historically, a direct argument was found first, by Kirillov, Kostant and Souriau in the early 1960’s and the (minus) coadjoint symplectic structure was found to be ! (ad ; ad ) D h; [; ]i
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Interestingly, this is the symplectic structure on the symplectic leaves of the Lie–Poisson bracket, as is shown in, for example, [104]. (See the historical overview in Sect. “Reduction Theory: Historical Overview” below and specifically, see Eq. (21) for a quick review of the Lie– Poisson bracket). The strategy of the reduction proof, as mentioned in the discussion in the last section, is to show that the coadjoint symplectic form on a coadjoint orbit O of the point , at a point 2 O, may be obtained by symplectically reducing T G at the value . The following theorem (due to Marsden and Weinstein [110]), and which is an elaboration on the result in Corollary 4, formulates the result for left actions; of course there is a similar one for right actions, with the minus sign replaced by a plus sign. Theorem 7 (Reduction to coadjoint orbits) Let G be a Lie group and let G act on G (and hence on T G by cotangent lift) by left multiplication. Let 2 g and let JL : T G ! g be the momentum map for the left action. Then is a regular value of JL , the action of G is free and proper, the symplectic reduced space J1 L ()/G is identified via left translation with O , the coadjoint orbit through , and the reduced symplectic form coincides with ! given in Eq. (18). Remarks 1. Notice that, as in the general Symplectic Reduction Theorem 3, this result does not require to be a regular (or generic) point in g ; that is, arbitrarily nearby coadjoint orbits may have a different dimension. 2. The form ! on the orbit need not be exact even though ˝ is. An example that shows this is SO(3), whose coadjoint orbits are spheres and whose symplectic structure is, as shown in [104], a multiple of the area element, which is not exact by Stokes’ Theorem. Orbit Reduction So far, we have presented what is usually called point reduction. There is another point of view that is called orbit reduction, which we now summarize. We assume the same set up as in the symplectic reduction theorem, with P connected, G acting symplectically, freely, and properly on P with an equivariant momentum map J : P ! g .
Mechanical Systems: Symmetries and Reduction
The connected components of the point reduced spaces P can be regarded as the symplectic leaves of the Poisson manifold (P/G; f; gP/G ) in the following way. Form a map i : P ! P/G defined by selecting an equivalence class [z]G for z 2 J1 () and sending it to the class [z]G . This map is checked to be well-defined and smooth. We then have the commutative diagram
One then checks that i is a Poisson injective immersion. Moreover, the i -images in P/G ofthe con- nected components of the symplectic manifolds P ; ˝ are its symplectic leaves (see [138] and references therein for details). As sets, i P D J1 O /G ; where O g is the coadjoint orbit through 2 g . The set PO :D J1 O /G is called the orbit reduced space associated to the orbit O . The smooth manifold structure (and hence the topology) on PO is the one that makes the map i : P ! PO into a diffeomorphism. For the next theorem, which characterizes the symplectic form and the Hamiltonian dynamics on PO , recall the coadjoint orbit symplectic structure of Kirillov, Kostant and Souriau that was established in the preceding Theorem 7: !O ()(g (); g ()) D h; [; ]i ;
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for ; 2 g and 2 O . We also recall that an injectively immersed submanifold of S of Q is called an initial submanifold of Q when for any smooth manifold P, a map g : P ! S is smooth if and only if ı g : P ! Q is smooth, where : S ,! Q is the inclusion. Theorem 8 (Symplectic orbit reduction theorem) In the setup explained above: (i)
The momentum map J is transverse to the coadjoint orbit O and hence J1 (O ) is an initial submanifold
of P. Moreover, the projection O : J1 O ! PO is a surjective submersion. (ii) PO is a symplectic manifold with the symplectic form ˝O uniquely characterized by the relation O ˝O D JO !O C iO ˝;
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where JO is the restriction of J to J1 O and iO : J1 O ,! P is the inclusion. (iii) The map i : P ! PO is a symplectic diffeomorphism. (iv) (Dynamics). Let H be a G-invariant function on P and e : P/G ! R by H D e define H H ı . Then the Hamiltonian vector field X H is also G-invariant and hence induces a vector field on P/G, which coincides with the Hamiltonian vector field Xe H . Moreover, the flow of Xe H leaves the symplectic leaves PO of P/G invariant. This flow restricted to the symplectic leaves is again Hamiltonian relative to the symplectic form ˝O and the e O given by Hamiltonian function H e H O ı O D H ı i O : Note that if O is an embedded submanifold of g then J is transverse to O and hence J1 (O ) is automatically an embedded submanifold of P. The proof of this theorem when O is an embedded submanifold of g can be found in Marle [91], Kazhdan, Kostant, and Sternberg [72], with useful additions given in Marsden [94] and Blaom [17]. For nonfree actions and when O is not an embedded submanifold of g see [138]. Further comments on the historical context of this result are given in the next section. Remarks 1. A similar result holds for right actions. 2. Freeness and properness of the G -action on J1 () are only needed indirectly. In fact these conditions are sufficient but not necessary for P to be a manifold. All that is needed is for P to be a manifold and to be a submersion and the above result remains unchanged. 3. Note that the description of the symplectic structure on J1 (O)/G is not as simple as it was for J1 ()/G, while the Poisson bracket description is simpler on J1 (O)/G. Of course, the symplectic structure depends only on the orbit O and not on the choice of a point on it. Cotangent Bundle Reduction Perhaps the most important and basic reduction theorem in addition to those already presented is the cotangent bundle reduction theorem. We shall give an exposition of
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the key aspects of this theory in Sect. “Cotangent Bundle Reduction” and give a historical account of its development, along with references in the next section. At this point, to orient the reader, we note that one of the special cases is cotangent bundle reduction at zero (see Theorem 10). This result says that if one has, again for simplicity, a free and proper action of G on Q (which is then lifted to T Q by the cotangent lift), then the reduced space at zero of T Q is given by T (Q/G), with its canonical symplectic structure. On the other hand, reduction at a nonzero value is a bit more complicated and gives rise to modifications of the standard symplectic structure; namely, one adds to the canonical structure, the pull-back of a closed two form on Q to T Q. Because of their physical interpretation (discussed, for example, in [104]), such extra terms are called magnetic terms. In Sect. “Cotangent Bundle Reduction”, we state the basic cotangent bundle reduction theorems along with providing some of the other important notions, such as the mechanical connection and the locked inertia tensor. Other notions that are important in mechanics, such as the amended potential, can be found in [95]. Reduction Theory: Historical Overview We have already given bits an pieces of the history of symplectic reduction and momentum maps. In this section we take a broader view of the subject to put things in historical and topical context. History before 1960 In the preceding sections, reduction theory has been presented as a mathematical construction. Of course, these ideas are rooted in classical work on mechanical systems with symmetry by such masters as Euler, Lagrange, Hamilton, Jacobi, Routh, Riemann, Liouville, Lie, and Poincaré. The aim of their work was, to a large extent, to eliminate variables associated with symmetries in order to simplify calculations in concrete examples. Much of this work was done using coordinates, although the deep connection between mechanics and geometry was already evident. Whittaker [179] gives a good picture of the theory as it existed up to about 1910. A highlight of this early theory was the work of Routh [161,163] who studied reduction of systems with cyclic variables and introduced the amended potential for the reduced system for the purpose of studying, for instance, the stability of a uniformly rotating state – what we would call today a relative equilibrium, terminology introduced later by Poincaré. Smale [168] eventually put the amended potential into a nice geometric setting. Routh’s
work was closely related to the reduction of systems with integrals in involution studied by Jacobi and Liouville around 1870; the Routh method corresponds to the modern theory of Lagrangian reduction for the action of Abelian groups. The rigid body, whose equations were discovered by Euler around 1740, was a key example of reduction – what we would call today either reduction to coadjoint orbits or Lie–Poisson reduction on the Hamiltonian side, or Euler– Poincaré reduction on the Lagrangian side, depending on one’s point of view. Lagrange [81] already understood reduction of the rigid body equations by a method not so far from what one would do today with the symmetry group SO(3). Many later authors, unfortunately, relied so much on coordinates (especially Euler angles) that there is little mention of SO(3) in classical mechanics books written before 1990, which by today’s standards, seems rather surprising! In addition, there seemed to be little appreciation until recently for the role of topological notions; for example, the fact that one cannot globally split off cyclic variables for the S1 action on the configuration space of the heavy top. The Hopf fibration was patiently waiting to be discovered in the reduction theory for the classical rigid body, but it was only explicitly found later on by H. Hopf [64]. Hopf was, apparently, unaware that this example is of great mechanical interest – the gap between workers in mechanics and geometers seems to have been particularly wide at that time. Another noteworthy instance of reduction is Jacobi’s elimination of the node for reducing the gravitational (or electrostatic) n-body problem by means of the group SE(3) of Euclidean motions, around 1860 or so. This example has, of course, been a mainstay of celestial mechanics. It is related to the work done by Riemann, Jacobi, Poincaré and others on rotating fluid masses held together by gravitational forces, such as stars. Hidden in these examples is much of the beauty of modern reduction, stability and bifurcation theory for mechanical systems with symmetry. While both symplectic and Poisson geometry have their roots in the work of Lagrange and Jacobi, it matured considerably with the work of Lie [90], who discovered many remarkably modern concepts such as the Lie– Poisson bracket on the dual of a Lie algebra. See Weinstein [176] and Marsden and Ratiu [104] for more details on the history. How Lie could have viewed his wonderful discoveries so divorced from their roots in mechanics remains a mystery. We can only guess that he was inspired by Jacobi, Lagrange and Riemann and then, as mathematicians often do, he quickly abstracted the ideas, losing valuable scientific and historical connections along the way.
Mechanical Systems: Symmetries and Reduction
As we have already hinted, it was the famous paper Poincaré [153] where we find what we call today the Euler– Poincaré equations – a generalization of the Euler equations for both fluids and the rigid body to general Lie algebras. (The Euler–Poincaré equations are treated in detail in [104]). It is curious that Poincaré did not stress either the symplectic ideas of Lie, nor the variational principles of mechanics of Lagrange and Hamilton – in fact, it is not clear to what extent he understood what we would call today Euler–Poincaré reduction. It was only with the development and physical application of the notion of a manifold, pioneered by Lie, Poincaré, Weyl, Cartan, Reeb, Synge and many others, that a more general and intrinsic view of mechanics was possible. By the late 1950’s, the stage was set for an explosion in the field. 1960–1972 Beginning in the 1960’s, the subject of geometric mechanics indeed did explode with the basic contributions of people such as (alphabetically and nonexhaustively) Abraham, Arnold, Kirillov, Kostant, Mackey, MacLane, Segal, Sternberg, Smale, and Souriau. Kirillov and Kostant found deep connections between mechanics and pure mathematics in their work on the orbit method in group representations, while Arnold, Smale, and Souriau were in closer touch with mechanics. The modern vision of geometric mechanics combines strong links to important questions in mathematics with the traditional classical mechanics of particles, rigid bodies, fields, fluids, plasmas, and elastic solids, as well as quantum and relativistic theories. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to the “hidden” symmetries underlying integrable systems. As we have already mentioned, reduction theory concerns the removal of variables using symmetries and their associated conservation laws. Variational principles, in addition to symplectic and Poisson geometry, provide fundamental tools for this endeavor. In fact, conservation of the momentum map associated with a symmetry group action is a geometric expression of the classical Noether theorem (discovered by variational, not symplectic methods). Arnold and Smale The modern era of reduction theory began with the fundamental papers of Arnold [7] and Smale [168]. Arnold focused on systems whose configuration manifold is a Lie group, while Smale focused on bifurcations of relative equilibria. Both Arnold and Smale linked their theory strongly with examples. For Arnold, they were the same
examples as for Poincaré, namely the rigid body and fluids, for which he went on to develop powerful stability methods, as in Arnold [8]. With hindsight, we can say that Arnold [7] was picking up on the basic work of Poincaré for both rigid body motion and fluids. In the case of fluids, G is the group of (volume preserving) diffeomorphisms of a compact manifold (possibly with boundary). In this setting, one obtains the Euler equations for (incompressible) fluids by reduction from the Lagrangian formulation of the equations of motion, an idea exploited by Arnold [7] and Ebin and Marsden [49]. This sort of description of a fluid goes back to Poincaré (using the Euler–Poincaré equations) and to the thesis of Ehrenfest (as geodesics on the diffeomorphism group), written under the direction of Boltzmann. For Smale, the motivating example was celestial mechanics, especially the study of the number and stability of relative equilibria by a topological study of the energymomentum mapping. He gave an intrinsic geometric account of the amended potential and in doing so, discovered what later became known as the mechanical connection. (Smale appears to not to have recognized that the interesting object he called ˛ is, in fact, a principal connection; this was first observed by Kummer [79]). One of Smale’s key ideas in studying relative equilibria was to link mechanics with topology via the fact that relative equilibria are critical points of the amended potential. Besides giving a beautiful exposition of the momentum map, Smale also emphasized the connection between singularities and symmetry, observing that the symmetry group of a phase space point has positive dimension if and only if that point is not a regular point of the momentum map restricted to a fiber of the cotangent bundle (Smale [168], Proposition 6.2) – a result we have proved in Proposition 2. He went on from here to develop his topology and mechanics program and to apply it to the planar n-body problem. The topology and mechanics program definitely involved reduction ideas, as in Smale’s construction of the quotients of integral manifolds, as in I c;p /S 1 (Smale [168], page 320). He also understood Jacobi’s elimination of the node in this context, although he did not attempt to give any general theory of reduction along these lines. Smale thereby set the stage for symplectic reduction: he realized the importance of the momentum map and of quotient constructions, and he worked out explicit examples like the planar n-body problem with its S1 symmetry group. (Interestingly, he pointed out that one should really use the nonabelian group SE(2); his feeling of unease with fixing the center of mass of an n-body system is remarkably perceptive).
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Synthesis The problem of synthesizing the Lie algebra reduction methods of Arnold [7] with the techniques of Smale [168] on the reduction of cotangent bundles by Abelian groups, led to the development of reduction theory in the general context of symplectic manifolds and equivariant momentum maps in Marsden and Weinstein [110] and Meyer [117], as we described in the last section. Both of these papers were completed by 1972. Poisson Manifolds Meanwhile, things were also gestating from the viewpoint of Poisson brackets and the idea of a Poisson manifold was being initiated and developed, with much duplication and rediscovery (see, Section 10.1 in [104] for additional information). A basic example of a noncanonical Poisson bracket is the Lie–Poisson bracket on g , the dual of a Lie algebra g. This bracket (which comes with a plus or minus sign) is given on two smooth functions on g by
ı f ıg ; ; (21) f f ; gg˙ () D ˙ ; ı ı where ı f /ı is the derivative of f , but thought of as an element of g. These Poisson structures, including the coadjoint orbits as their symplectic leaves, were known to Lie [90], although, as we mentioned previously, Lie does not seem to have recognized their importance in mechanics. It is also not clear whether or not Lie realized that the Lie Poisson bracket is the Poisson reduction of the canonical Poisson bracket on T G by the action of G. (See, Chap. 13 in [104] for an account of this theory). The first place we know of that has this clearly stated (but with no references, and no discussion of the context) is Bourbaki [24], Chapter III, Section 4, Exercise 6. Remarkably, this exercise also contains an interesting proof of the Duflo–Vergne theorem (with no reference to the original paper, which appeared in 1969). Again, any hint of links with mechanics is missing. This takes us up to about 1972. Post 1972 An important contribution was made by Marle [91], who divides the inverse image of an orbit by its characteristic foliation to obtain the product of an orbit and a reduced manifold. In particular, as we saw in Theorem 8, P is symplectically diffeomorphic to an “orbit-reduced” space P Š J 1 (O )/G, where O is a coadjoint orbit of G. From this it follows that the P are symplectic leaves in the
Poisson space P/G. The related paper of Kazhdan, Kostant, and Sternberg [72] was one of the first to notice deep links between reduction and integrable systems. In particular, they found that the Calogero–Moser systems could be obtained by reducing a system that was trivially integrable; in this way, reduction provided a method of producing an interesting integrable system from a simple one. This point of view was used again by, for example, Bobenko, Reyman, and Semenov–Tian–Shansky [22] in their spectacular group theoretic explanation of the integrability of the Kowalewski top. Noncanonical Poisson Brackets The Hamiltonian description of many physical systems, such as rigid bodies and fluids in Eulerian variables, requires noncanonical Poisson brackets and constrained variational principles of the sort studied by Lie and Poincaré. As discussed above, a basic example of a noncanonical Poisson bracket is the Lie–Poisson bracket on the dual of a Lie algebra. From the mechanics perspective, the remarkably modern book (but which was, unfortunately, rather out of touch with the corresponding mathematical developments) by Sudarshan and Mukunda [172] showed via explicit examples how systems such as the rigid body could be written in terms of noncanonical brackets, an idea going back to Pauli [147], Martin [115] and Nambu [129]. Others in the physics community, such as Morrison and Greene [128] also discovered noncanonical bracket formalisms for fluid and magnetohydrodynamic systems. In the 1980’s, many fluid and plasma systems were shown to have a noncanonical Poisson formulation. It was Marsden and Weinstein [111,112] who first applied reduction techniques to these systems. The reduction philosophy concerning noncanonical brackets can be summarized by saying Any mechanical system has its roots somewhere as a cotangent bundle and one can recover noncanonical brackets by the simple process of Poisson reduction. For example, in fluid mechanics, this reduction is implemented by the Lagrange-to-Euler map. This view ran contrary to the point of view, taken by some researchers, that one should proceed by analogy or guesswork to find Poisson structures and then to try to limit the guesses by the constraint of Jacobi’s identity. In the simplest version of the Poisson reduction process, one starts with a Poisson manifold P on which a group G acts by Poisson maps and then forms the quotient space P/G, which, if not singular, inherits a natural Poisson structure itself. Of course, the Lie–Poisson struc-
Mechanical Systems: Symmetries and Reduction
ture on g is inherited in exactly this way from the canonical symplectic structure on T G. One of the attractions of this Poisson bracket formalism was its use in stability theory. This literature is now very large, but Holm, Marsden, Ratiu, and Weinstein [63] is representative. The way in which the Poisson structure on P is related to that on P/G was clarified in a generalization of Poisson reduction due to Marsden and Ratiu [103], a technique that has also proven useful in integrable systems (see, e. g., Pedroni [148] and Vanhaecke [174]). Reduction theory for mechanical systems with symmetry has proven to be a powerful tool that has enabled key advances in stability theory (from the Arnold method to the energy-momentum method for relative equilibria) as well as in bifurcation theory of mechanical systems, geometric phases via reconstruction – the inverse of reduction – as well as uses in control theory from stabilization results to a deeper understanding of locomotion. For a general introduction to some of these ideas and for further references, see Marsden, Montgomery, and Ratiu [98]; Simo, Lewis, and Marsden [166]; Marsden and Ostrowski [99]; Marsden and Ratiu [104]; Montgomery [122,123,124,125,126]; Blaom [16,17] and Kanso, Marsden, Rowley, and Melli-Huber [71]. Tangent and Cotangent Bundle Reduction The simplest case of cotangent bundle reduction is the case of reduction of P D T Q at D 0; the answer is simply P0 D T (Q/G) with the canonical symplectic form. Another basic case is when G is Abelian. Here, (T Q) Š T (Q/G), but the latter has a symplectic structure modified by magnetic terms, that is, by the curvature of the mechanical connection. An Abelian version of cotangent bundle reduction was developed by Smale [168]. Then Satzer [165] studied the relatively simple, but important case of cotangent bundle reduction at the zero value of the momentum map. The full generalization of cotangent bundle reduction for nonabelian groups at arbitrary values of the momentum map appears for the first time in Abraham and Marsden [1]. It was Kummer [79] who first interpreted this result in terms of a connection, now called the mechanical connection. The geometry of this situation was used to great effect in, for example, Guichardet [54,66], Iwai [67], and Montgomery [120,123,124]. We give an account of cotangent bundle reduction theory in the following section. The Gauge Theory Viewpoint Tangent and cotangent bundle reduction evolved into what we now term as the “bundle picture” or the “gauge
theory of mechanics”. This picture was first developed by Montgomery, Marsden, and Ratiu [127] and Montgomery [120,121]. That work was motivated and influenced by the work of Sternberg [171] and Weinstein [175] on a “Yang–Mills construction” which is, in turn, motivated by Wong’s equations, i. e., the equations for a particle moving in a Yang–Mills field. The main result of the bundle picture gives a structure to the quotient spaces (T Q)/G and (TQ)/G when G acts by the cotangent and tangent lifted actions. The symplectic leaves in this pic´ ture were analyzed by Zaalani [182], Cushman and Sniatycki [47] and Marsden and Perlmutter [102]. The work of Perlmutter and Ratiu [149] gives a unified study of the Poisson bracket on (T Q)/G in both the Sternberg and Weinstein realizations of the quotient. As mentioned earlier, we shall review some of the basics of cotangent bundle reduction theory in Sect. “Cotangent Bundle Reduction”. Further information on this theory may be found in [1,95], and [92], as well as a number of the other references mentioned above. Lagrangian Reduction A key ingredient in Lagrangian reduction is the classical work of Poincaré [153] in which the Euler–Poincaré equations were introduced. Poincaré realized that the equations of fluids, free rigid bodies, and heavy tops could all be described in Lie algebraic terms in a beautiful way. The importance of these equations was realized by Hamel [58,59] and Chetayev [40], but to a large extent, the work of Poincaré lay dormant until it was revived in the Russian literature in the 1980’s. The more recent developments of Lagrangian reduction were motivated by attempts to understand the relation between reduction, variational principles and Clebsch variables in Cendra and Marsden [34] and Cendra, Ibort, and Marsden [33]. In Marsden and Scheurle [109] it was shown that, for matrix groups, one could view the Euler–Poincaré equations via the reduction of Hamilton’s variational principle from TG to g. The work of Bloch, Krishnaprasad, Marsden and Ratiu [21] established the Euler–Poincaré variational structure for general Lie groups. The paper of Marsden and Scheurle [109] also considered the case of more general configuration spaces Q on which a group G acts, which was motivated by both the Euler–Poincaré case as well as the work of Cendra and Marsden [34] and Cendra, Ibort, and Marsden [33]. The Euler– Poincaré equations correspond to the case Q D G. Related ideas stressing the groupoid point of view were given in Weinstein [177]. The resulting reduced equations were
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called the reduced Euler–Lagrange equations. This work is the Lagrangian analogue of Poisson reduction, in the sense that no momentum map constraint is imposed. Lagrangian reduction proceeds in a way that is very much in the spirit of the gauge theoretic point of view of mechanical systems with symmetry. It starts with Hamilton’s variational principle for a Lagrangian system on a configuration manifold Q and with a symmetry group G acting on Q. The idea is to drop this variational principle to the quotient Q/G to derive a reduced variational principle. This theory has its origins in specific examples such as fluid mechanics (see, for example, Arnold [7] and Bretherton [25]), while the systematic theory of Lagrangian reduction was begun in Marsden and Scheurle [109] and further developed in Cendra, Marsden, and Ratiu [35]. The latter reference also introduced a connection to realize the space (TQ)/G as the fiber product T(Q/G) g˜ of T(Q/G) with the associated bundle formed using the adjoint action of G on g. The reduced equations associated to this construction are called the Lagrange–Poincaré equations and their geometry has been fairly well developed. Note that a G-invariant Lagrangian L on TQ induces a Lagrangian l on (TQ)/G. Until recently, the Lagrangian side of the reduction story had lacked a general category that is the Lagrangian analogue of Poisson manifolds in which reduction can be repeated. One candidate is the category of Lie algebroids, as explained in Weinstein [177]. Another is that of Lagrange–Poincaré bundles, developed in Cendra, Marsden, and Ratiu [35]. Both have tangent bundles and Lie algebras as basic examples. The latter work also develops the Lagrangian analogue of reduction for central extensions and, as in the case of symplectic reduction by stages, cocycles and curvatures enter in a natural way. This bundle picture and Lagrangian reduction has proven very useful in control and optimal control problems. For example, it was used in Chang, Bloch, Leonard, Marsden and Woolsey [38] to develop a Lagrangian and Hamiltonian reduction theory for controlled mechanical systems and in Koon and Marsden [76] to extend the falling cat theorem of Montgomery [123] to the case of nonholonomic systems as well as to nonzero values of the momentum map. Finally we mention that the paper Cendra, Marsden, Pekarsky, and Ratiu [37] develops the reduction theory for Hamilton’s phase space principle and the equations on the reduced space, along with a reduced variational principle, are developed and called the Hamilton–Poincaré equations. Even in the case Q D G, this collapses to an interesting variational principle for the Lie–Poisson equations on g .
Legendre Transformation Of course the Lagrangian and Hamiltonian sides of the reduction story are linked by the Legendre transformation. This mapping descends at the appropriate points to give relations between the Lagrangian and the Hamiltonian sides of the theory. However, even in standard cases such as the heavy top, one must be careful with this approach, as is already explained in, for example, Holm, Marsden, and Ratiu [61]. For field theories, such as the Maxwell–Vlasov equations, this issues is also important, as explained in Cendra, Holm, Hoyle and Marsden [31] (see also Tulczyjew and Urba´nski [173]). Nonabelian Routh Reduction Routh reduction for Lagrangian systems, which goes back Routh [161,162,163] is classically associated with systems having cyclic variables (this is almost synonymous with having an Abelian symmetry group). Modern expositions of this classical theory can be found in Arnold, Koslov, and Neishtadt [10] and in [104], §8.9. Routh Reduction may be thought of as the Lagrangian analog of symplectic reduction in that a momentum map is set equal to a constant. A key feature of Routh reduction is that when one drops the Euler–Lagrange equations to the quotient space associated with the symmetry, and when the momentum map is constrained to a specified value (i. e., when the cyclic variables and their velocities are eliminated using the given value of the momentum), then the resulting equations are in Euler–Lagrange form not with respect to the Lagrangian itself, but with respect to a modified function called the Routhian. Routh [162] applied his method to stability theory; this was a precursor to the energy-momentum method for stability that synthesizes Arnold’s and Routh’s methods (see Simo, Lewis and Marsden [166]). Routh’s stability method is still widely used in mechanics. The initial work on generalizing Routh reduction to the nonabelian case was that of Marsden and Scheurle [108]. This subject was further developed in Jalnapurkar and Marsden [69] and Marsden, Ratiu and Scheurle [105]. The latter reference used this theory to give some nice formulas for geometric phases from the Lagrangian point of view. Semidirect Product Reduction In the simplest case of a semidirect product, one has a Lie group G that acts on a vector space V (and hence on its dual V ) and then one forms the semidirect product S D GsV , generalizing the semidirect product structure of the Euclidean group SE(3) D SO(3) s R3 .
Mechanical Systems: Symmetries and Reduction
Consider the isotropy group G a 0 for some a0 2 V . The semidirect product reduction theorem states that each of the symplectic reduced spaces for the action of G a 0 on T G is symplectically diffeomorphic to a coadjoint orbit in (g s V ) , the dual of the Lie algebra of the semidirect product. This semidirect product theory was developed by Guillemin and Sternberg [55,56], Ratiu [154,157,158], and Marsden, Ratiu, and Weinstein [106,107]. The Lagrangian reduction analog of semidirect product theory was developed by Holm, Marsden and Ratiu [61,62]. This construction is used in applications where one has advected quantities (such as the direction of gravity in the heavy top, density in compressible fluids and the magnetic field in MHD) as well as to geophysical flows. Cendra, Holm, Hoyle and Marsden [31] applied this idea to the Maxwell–Vlasov equations of plasma physics. Cendra, Holm, Marsden, and Ratiu [32] showed how Lagrangian semidirect product theory fits into the general framework of Lagrangian reduction. The semidirect product reduction theorem has been proved in Landsman [82], Landsman [83], Chap. 4 as an application of a stages theorem for his special symplectic reduction method. Even though special symplectic reduction generalizes Marsden–Weinstein reduction, the special reduction by stages theorem in Landsman [82] studies a setup that, in general, is different to the ones in the reduction by stages theorems of [92]. Singular Reduction Singular reduction starts with the observation of Smale [168] that we have already mentioned: z 2 P is a regular point of a momentum map J if and only if z has no continuous isotropy. Motivated by this, Arms, Marsden, and Moncrief [5,6] showed that (under hypotheses which include the ellipticity of certain operators and which can be interpreted more or less, as playing the role of a properness assumption on the group action in the finite dimensional case) the level sets J1 (0) of an equivariant momentum map J have quadratic singularities at points with continuous symmetry. While such a result is easy to prove for compact group actions on finite dimensional manifolds (using the equivariant Darboux theorem), the main examples of Arms, Marsden, and Moncrief [5] were, in fact, infinite dimensional – both the phase space and the group. Singular points in the level sets of the momentum map are related to convexity properties of the momentum map in that the singular points in phase space map to corresponding singular points in the the image polytope. The paper of Otto [143] showed that if G is a Lie group acting properly on an almost Kähler manifold then the or-
bit space J1 ()/G decomposes into symplectic smooth manifolds constructed out of the orbit types of the G-action on P. In some related work, Huebschmann [65] has made a careful study of the singularities of moduli spaces of flat connections. The detailed structure of J1 (0)/G for compact Lie groups acting on finite dimensional manifolds was determined by Sjamaar and Lerman [167]; their work was extended to proper Lie group actions and to J1 (O )/G by Bates and Lerman [12], with the assumption that O be locally closed in g . Ortega [130] and [138] redid the entire singular reduction theory for proper Lie group actions starting with the point reduced spaces J1 ()/G and also connected it to the more algebraic approach of Arms, Cushman, and Gotay [4]. Specific examples of singular reduction, with further references, may be found in Lerman, Montgomery, and Sjamaar [84] and [44]. One of these, the “canoe” is given in detail in [92]. In fact, this is an example of singular reduction in the case of cotangent bundles, and much more can be said in this case; see Olmos and Dias [150,151]. Another approach to singular reduction based on the technique of blowing up singularities, and which was also designed for the case of singular cotangent bundle reduction, was started in Hernandez and Marsden [60] and Birtea, Puta, Ratiu, and Tudoran [14], a technique which requires further development. Singular reduction has been extensively used in the study of the persistence, bifurcation, and stability of relative dynamical elements; see [41,42,53,85,86,132,134,135, 136,139,146,159,160,180,181]. Symplectic Reduction Without Momentum Maps The reduction theory presented so far needs the existence of a momentum map. However, more primitive versions of this procedure based on foliation theory (see Cartan [26] and Meyer [117]) do not require the existence of this object. Working in this direction, but with a mathematical program that goes beyond the reduction problem, Condevaux, Dazord, and Molino [43] introduced a concept that generalizes the momentum map. This object is defined via a connection that associates an additive holonomy group to each canonical action on a symplectic manifold. The existence of the momentum map is equivalent to the vanishing of this group. Symplectic reduction has been carried out using this generalized momentum map in Ortega and Ratiu [140,141]. Another approach to symplectic reduction that is able to avoid the possible non-existence of the momentum map is based on the optimal momentum map introduced and studied in Ortega and Ratiu [137], Ortega [131], and [92].
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This distribution theoretical approach can also deal with reduction of Poisson manifolds, where the standard momentum map does not exist generically.
Ratiu [36] and Bloch [18] for a more detailed historical review. Multisymplectic Reduction
Reduction of Other Geometric Structures Besides symplectic reduction, there are many other geometric structures on which one can perform similar constructions. For example, one can reduce Kähler, hyperKähler, Poisson, contact, Jacobi, etc. manifolds and this can be done either in the regular or singular cases. We refer to [138] for a survey of the literature for these topics. The Method of Invariants This method seeks to parametrize quotient spaces by group invariant functions. It has a rich history going back to Hilbert’s invariant theory. It has been of great use in bifurcation with symmetry (see Golubitsky, Stewart, and Schaeffer [52] for instance). In mechanics, the method was developed by Kummer, Cushman, Rod and coworkers in the 1980’s; see, for example, Cushman and Rod [45]. We will not attempt to give a literature survey here, other than to refer to Kummer [80], Kirk, Marsden, and Silber [73], Alber, Luther, Marsden, and Robbins [3] and the book of Cushman and Bates [44] for more details and references. Nonholonomic Systems Nonholonomic mechanical systems (such as systems with rolling constraints) provide a very interesting class of systems where the reduction procedure has to be modified. In fact this provides a class of systems that gives rise to an almost Poisson structure, i. e. a bracket which does not necessarily satisfy the Jacobi identity. Reduction theory for nonholonomic systems has made a lot of progress, but many interesting questions still remain. In these types of systems, there is a natural notion of a momentum map, but in general it is not conserved, but rather obeys a momentum equation as was discovered by Bloch, Krishnaprasad, Marsden, and Murray [20]. This means, in particular, that point reduction in such a situation may not be appropriate. Nevertheless, Poisson reduction in the almost Poisson and almost symplectic setting is interesting and from the mathematical point of view, point reduction is also interesting, although, as remarked, one has to be cautious with how it is applied to, for example, nonholonomic systems. A few references are Koiller [75], Bates and Sniatycki [13], Bloch, Krishnaprasad, Marsden, and Murray [20], Koon and Marsden [77], Blankenstein and Van Der Schaft [15], ´ Cushman, Sniatycki [46], Planas-Bielsa [152], and Ortega and Planas-Bielsa [133]. We refer to Cendra, Marsden, and
Reduction theory is by no means completed. For example, for PDE’s, the multisymplectic (as opposed to symplectic) framework seems appropriate, both for relativistic and nonrelativistic systems. In fact, this approach has experienced somewhat of a revival since it has been realized that it is rather useful for numerical computation (see Marsden, Patrick, and Shkoller [100]). Only a few instances and examples of multisymplectic and multi-Poisson reduction are really well understood (see Marsden, Montgomery, Morrison, and Thompson [97]; CastrillónLópez, Ratiu and Shkoller [30], Castrillón-López, Garcia Pérez and Ratiu [27], Castrillón-López and Ratiu [28], Castrillón-López and Marsden [29]), so one can expect to see more activity in this area as well. Discrete Mechanical Systems Another emerging area, also motivated by numerical analysis, is that of discrete mechanics. Here the idea is to replace the velocity phase space TQ by Q Q, with the role of a velocity vector played by a pair of nearby points. This has been a powerful tool for numerical analysis, reproducing standard symplectic integration algorithms and much more. See, for example, Wendlandt and Marsden [178], Kane, Marsden, Ortiz and West [70], Marsden and West [113], Lew, Marsden, Ortiz, and West [87] and references therein. This subject, too, has its own reduction theory. See Marsden, Pekarsky, and Shkoller [101], Bobenko and Suris [23] and Jalnapurkar, Leok, Marsden and West [68]. Discrete mechanics also has some intriguing links with quantization, since Feynman himself first defined path integrals through a limiting process using the sort of discretization used in the discrete action principle (see Feynman and Hibbs [50]). Cotangent Bundle Reduction As mentioned earlier, the cotangent bundle reduction theorems are amongst the most basic and useful of the symplectic reduction theorems. Here we only present the regular versions of the theorems. Cotangent bundle reduction theorems come in two forms – the embedding cotangent bundle reduction theorem and the bundle cotangent bundle reduction theorem. We start with a smooth, free, and proper, left action ˚: GQ ! Q
Mechanical Systems: Symmetries and Reduction
of the Lie group G on the configuration manifold Q and lift it to an action on T Q. This lifted action is symplectic with respect to the canonical symplectic form on T Q, which we denote ˝can , and has an equivariant momentum map J : T Q ! g given by ˝
˛ hJ(˛q ); i D ˛q ; Q (q) ; where 2 g. Letting 2 g , the aim of this section is to determine the structure of the symplectic reduced space ((T Q) ; ˝ ), which, by Theorem 3, is a symplectic manifold. We are interested in particular in the question of to what extent ((T Q) ; ˝ ) is a synthesis of a cotangent bundles and a coadjoint orbit.
For 2 g and q 2 Q, notice that, under the condition CBR1, (iQ ˛ )(q) D hJ(˛ (q)); i D h0 ; i ; and so iQ ˛ is a constant function on Q. Therefore, for 2 g , iQ d˛ D £Q ˛ diQ ˛ D 0 ;
since the Lie derivative is zero by G -invariance of ˛ . It follows that There is a unique two-form ˇ on Q such that ˇ D d˛ : Q;G
Since Q;G is a submersion, ˇ is closed (it need not be exact). Let
Cotangent Bundle Reduction: Embedding Version In this version of the theorem, we first form the quotient manifold Q :D Q/G ; which we call the -shape space. Since the action of G on Q is smooth, free, and proper, so is the action of the isotropy subgroup G and therefore, Q is a smooth manifold and the canonical projection Q;G : Q ! Q is a surjective submersion. Consider the G -action on Q and its lift to T Q. This lifted action is of course also symplectic with respect to the canonical symplectic form ˝can and has an equivariant momentum map J : T Q ! g obtained by restricting J; that is, for ˛q 2 Tq Q, J (˛q ) D J(˛q )jg : Let 0 :D jg 2 g be the restriction of to g . Notice that there is a natural inclusion of submanifolds J1 () (J )1 (0 ) :
(23)
(22)
Since the actions are free and proper, and 0 are regular values, so these sets are indeed smooth manifolds. Note that, by construction, 0 is G -invariant. There will be two key assumptions relevant to the embedding version of cotangent bundle reduction. Namely, CBR1 In the above setting, assume there is a G -invariant one-form ˛ on Q with values in (J )1 (0 ); and the condition (which by (22), is a stronger condition) CBR2 Assume that ˛ in CBR1 takes values in J1 ().
B D Q ˇ ; where Q : T Q ! Q is (following our general conventions for maps) the cotangent bundle projection. Also, to avoid confusion with the canonical symplectic form ˝can on T Q, we shall denote the canonical symplectic form on T Q , the cotangent bundle of -shape space, by !can . Theorem 9 (Cotangent bundle reduction – embedding version) (i)
If condition CBR1 holds, then there is a symplectic embedding ' : ((T Q) ; ˝ ) ! (T Q ; !can B ) ;
onto a submanifold of T Q covering the base Q/G . (ii) The map ' in (i) gives a symplectic diffeomorphism of ((T Q) ; ˝ ) onto (T Q ; !can B ) if and only if g D g . (iii) If CBR2 holds, then the image of ' equals the vector subbundle [TQ;G (V )]ı of T Q , where V TQ is the vector subbundle consisting of vectors tangent to the G-orbits, that is, its fiber at q 2 Q equals Vq D f Q (q) j 2 gg, and ı denotes the annihilator relative to the natural duality pairing between TQ and T Q . Remarks 1. A history of this result can be found in Sect. “Reduction Theory: Historical Overview”. 2. As shown in the appendix on Principal Connections (see Proposition A2) the required one form ˛ may be constructed satisfying condition CBR1 from a connec-
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tion on the -shape space bundle Q;G : Q ! Q/G and an ˛ satisfying CBR2 can be constructed using a connection on the shape space bundle Q;G : Q ! Q/G. 3. Note that in the case of Abelian reduction, or, more generally, the case in which G D G , the reduced space is symplectically diffeomorphic to T (Q/G) with the symplectic structure given by ˝can B . In particular, if D 0, then the symplectic form on T (Q/G) is the canonical one, since in this case one can choose ˛ D 0 which yields B D 0. 4. The term B on T Q is usually called a magnetic term, a gyroscopic term, or a Coriolis term. The terminology “magnetic” comes from the Hamiltonian description of a particle of charge e moving according to the Lorentz force law in R3 under the influence of a magnetic field B. This motion takes place in T R3 but with the nonstandard symplectic structure dq i ^ dp i ec B; i D 1; 2; 3, where c is the speed of light and B is regarded as a closed two-form: B D B x dy ^ dz B y dx ^ dz C Bz dx ^ dy (see §6.7 in [104] for details) The strategy for proving this theorem is to first deal with the case of reduction at zero and then to treat the general case using a momentum shift. Reduction at Zero The reduced space at D 0 is, as a set, (T Q)0 D J1 (0)/G since, for D 0; G D G. Notice that in this case, there is no distinction between orbit reduction and symplectic reduction. Theorem 10 (Reduction at zero) Assume that the action of G on Q is free and proper, so that the quotient Q/G is a smooth manifold. Then 0 is a regular value of J and there is a symplectic diffeomorphism between (T Q)0 and T (Q/G) with its canonical symplectic structure. The Case G D G If one is reducing at zero, then clearly G D G . However, this is an important special case of the general cotangent bundle reduction theorem that, for example, includes the case of Abelian reduction. The key assumption here is that G D G , which indeed is always the case if G is Abelian. Theorem 11 Assume that the action of G on Q is free and proper, so that the quotient Q/G is a smooth manifold. Let 2 g , assume that G D G , and assume that CBR2 holds. Then is a regular value of J and there is a symplectic
diffeomorphism between (T Q) and T (Q/G), the latter with the symplectic form !can B ; here, !can is the canoni ˇ , where cal symplectic form on T (Q/G) and B D Q/G the two form ˇ on Q/G is defined by ˇ D d˛ : Q;G
Example Consider the reduction of a general cotangent bundle T Q by G D SO(3). Here G Š S 1 , if ¤ 0, and so the reduced space is embedded into the cotangent bundle T (Q/S 1 ). A specific example is the case of Q D SO(3). 2 , the sphere Then the reduced space (T SO(3)) is Skk of radius kk which is a coadjoint orbit in so(3) . In this 2 and the embedding of case, Q/G D SO(3)/S 1 Š Skk 2 2 Skk into T Skk is the zero section. Magnetic Terms and Curvature Using the results of the preceding section, we will now show how one can interpret the magnetic term B as the curvature of a connection on a principal bundle. We saw in the preamble to the Cotangent Bundle Reduction Theorem 9 that iQ d˛ D 0 for any 2 g , which was used to drop d˛ to the quotient. In the language of principal bundles, this may be rephrased by saying that d˛ is horizontal and thus, once a connection is introduced, the covariant exterior derivative of ˛ coincides with d˛ . There are two methods to construct a form ˛ with the properties in Theorem 9. We continue to work under the general assumption that G acts on Q freely and properly. First Method Construction of ˛ from a connection A 2 ˝ 1 (Q; g ) on the principal bundle Q;G : Q ! Q/G . To carry this out, one shows that the choice ˛ :D h0 ; A i 2 ˝ 1 (Q) satisfies the condition CBR1 in Theorem 9, where, as above, 0 D jg . The two-form d˛ may be interpreted in terms of curvature. In fact, one shows that d˛ is the 0 -component of the curvature two-form. We summarize these results in the following statement. Proposition 12 If the principal bundle Q;G : Q ! Q/G with structure group G has a connection A , then ˛ (q) can be taken to equal A (q) 0 and B is induced on T Q by d˛ (a two-form on Q), which equals the 0 -component of the curvature B of A .
Mechanical Systems: Symmetries and Reduction
Second Method Construction of ˛ from a connection A 2 ˝ 1 (Q; g) on the principal bundle Q;G : Q ! Q/G. One can show that the choice (A1), that is, ˛ :D h; Ai 2 ˝ 1 (Q) satisfies the condition CBR2 in Theorem 9. As with the first method, there is an interpretation of the two-form d˛ in terms of curvature as follows. Proposition 13 If the principal bundle Q;G : Q ! Q/G with structure group G has a connection A, then ˛ (q) can be taken to equal A(q) and B is the pull back to T Q of d˛ 2 ˝ 2 (Q), which equals the -component of the two form B C [A; A] 2 ˝ 2 (Q; g), where B is the curvature of A.
(T Q) embeds as a vector subbundle of T (Q/G ). The bundle version of this theorem says, roughly speaking, that (T Q) is a coadjoint orbit bundle over T (Q/G) with fiber the coadjoint orbit O through . Again we utilize a choice of connection A on the shape space bundle Q;G : Q ! Q/G. A key step in the argument is to utilize orbit reduction and the identification (T Q) Š (T Q)O . Theorem 15 (Cotangent bundle reduction – bundle version) The reduced space (T Q) is a locally trivial fiber bundle over T (Q/G) with typical fiber O. This point of view is explored further and the exact nature of the coadjoint orbit bundle is identified and its symplectic structure is elaborated in [92]. Poisson Version
Coadjoint Orbits We now apply the Cotangent Bundle Reduction Theorem 9 to the case Q D G and with the G-action given by left translation. The right Maurer–Cartan form R is a flat connection associated to this action (see Theorem A13) and hence ˝ ˛ d˛ (g)(u g ; v g ) D ; [ R ; R ](g)(u g ; v g ) ˛ ˝ D ; [Tg R g 1 u g ; Tg R g 1 v g ] : Recall from Theorem 7 that the reduced space (T G) is the coadjoint orbit O endowed with the negative orbit and, according to the Cotangent Bunsymplectic form ! dle Reduction Theorem, it symplectically embeds as the zero section into (T O ; !can B ), where B D O ˇ , O : T O ! O is the cotangent bundle projection, G;G ˇ D d˛ , and G;G : G ! O is given by G;G (g) D Adg . The derivative of G;G is given by ˇ d ˇˇ Ad D ad Adg Tg G;G (Te L g ) D dt ˇtD0 g exp(t) . for any 2 g. Then a computation shows that ˇ D ! Thus, the embedding version of the cotangent bundle reduction theorem produces the following statement which, of course, can be easily checked directly. ) symplectically Corollary 14 The coadjoint orbit (O ; ! embeds as the zero section into the symplectic manifold ). (T O ; !can C O !
Cotangent Bundle Reduction: Bundle Version The embedding version of the cotangent bundle reduction theorem presented in the preceding section states that
This same type of argument as above shows the following, which we state slightly informally. Theorem The Poisson reduced space (T Q)/G is diffeomorphic to the coadjoint bundle of Q;G : Q ! Q/G. This diffeomorphism is implemented by a connection A 2 ˝ 1 (Q; g). Thus the fiber of (T Q)/G ! T (Q/G) is isomorphic to the Lie–Poisson space g . There is an interesting formula for the Poisson structure on (T Q)/G that was originally computed in Montgomery, Marsden, and Ratiu [127], Montgomery [121]. Further developments in Cendra, Marsden, Pekarsky, and Ratiu [37] and Perlmutter and Ratiu [149] gives a unified study of the Poisson bracket on (T Q)/G in both the Sternberg and Weinstein realizations of the quotient. Finally, we refer to, for instance, Lewis, Marsden, Montgomery and Ratiu [88] for an application of this result; in this case, the dynamics of fluid systems with free boundaries is studied. Coadjoint Orbit Bundles The details of the nature of the bundle and its associated symplectic structure that was sketched in Theorem 15 is due to Marsden and Perlmutter [102]; see also Za´ alani [182] Cushman and Sniatycki [47], and [149]. An exposition may be found in [92]. Future Directions One of the goals of reduction theory and geometric mechanics is to take the analysis of mechanical systems with symmetries to a deeper level of understanding. But much more needs to be done. As has already been explained,
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there is still a need to put many classical concepts, such as quasivelocities, into this context, with a resultant strengthening of the theory and its applications. In addition, links with Dirac structures, groupoids and algebroids is under development and should lead to further advances. Finally we mention that while much of this type of work has been applied to field theories (such as electromagnetism and gravity), greater insight is needed for many topics, stressenergy-momentum tensors being one example. Acknowledgments This work summarizes the contributions of many people. We are especially grateful to Alan Weinstein, Victor Guillemin and Shlomo Sternberg for their incredible insights and work over the last few decades. We also thank Hernán Cendra and Darryl Holm, our collaborators on the Lagrangian context and Juan-Pablo Ortega, a longtime collaborator on Hamiltonian reduction and other projects; he along with Gerard Misiolek and Matt Perlmutter were our collaborators on [92], a key recent project that helped us pull many things together. We also thank many other colleagues for their input and invaluable support over the years; this includes Larry Bates, Tony Bloch, Marco Castrillón-López, Richard Cushman, Laszlo Fehér, Mark Gotay, John Harnad, Eva Kanso, Thomas Kappeler, P.S. Krishnaprasad, Naomi Leonard, Debra Lewis, James Montaldi, George Patrick, Mark Roberts, Miguel Rodríguez´ Olmos, Steve Shkoller, Je¸drzej Sniatycki, Leon Takhtajan, Karen Vogtmann, and Claudia Wulff. Appendix: Principal Connections
One can alternatively use right actions, which is common in the principal bundle literature, but we shall stick with the case of left actions for the main exposition. Vectors that are infinitesimal generators, namely those of the form Q (q) are called vertical since they are sent to zero by the tangent of the projection map Q;G . Definition A1 A connection, also called a principal connection on the bundle Q;G : Q ! Q/G is a Lie algebra valued 1-form A : TQ ! g
where g denotes the Lie algebra of G, with the following properties: (i) the identity A( Q (q)) D holds for all 2 g; that is, A takes infinitesimal generators of a given Lie algebra element to that same element, and (ii) we have equivariance: A(Tq ˚ g (v)) D Ad g (A(v)) for all v 2 Tq Q, where ˚ g : Q ! Q denotes the given action for g 2 G and where Ad g denotes the adjoint action of G on g. A remark is noteworthy at this point. The equivariance identity for infinitesimal generators noted previously (see (7)), namely, Tq ˚ g Q (q) D (Ad g )Q (g q) ; shows that if the first condition for a connection holds, then the second condition holds automatically on vertical vectors. If the G-action on Q is a right action, the equivariance condition (ii) in Definition A1 needs to be changed to A(Tq ˚ g (v)) D Ad g 1 (A(v)) for all g 2 G and v 2 Tq Q.
In preparation for the next section which gives a brief exposition of the cotangent bundle reduction theorem, we now give a review and summary of facts that we shall need about principal connections. An important thing to keep in mind is that the magnetic terms in the cotangent bundle reduction theorem will appear as the curvature of a connection.
Since A is a Lie algebra valued 1-form, for each q 2 Q, we get a linear map A(q) : Tq Q ! g and so we can form its dual A(q) : g ! Tq Q. Evaluating this on produces an ordinary 1-form:
Principal Connections Defined
This 1-form satisfies two important properties given in the next Proposition.
We consider the following basic set up. Let Q be a manifold and let G be a Lie group acting freely and properly on the left on Q. Let
Proposition A2 For any connection A and 2 g , the corresponding 1-form ˛ defined by (A1) takes values in J1 () and satisfies the following G-equivariance property:
Q;G : Q ! Q/G denote the bundle projection from the configuration manifold Q to shape space S D Q/G. We refer to Q;G : Q ! Q/G as a principal bundle.
Associated One-Forms
˛ (q) D A(q) () :
(A1)
˚ g ˛ D ˛Adg : Notice in particular, if the group is Abelian or if is G-invariant, (for example, if D 0), then ˛ is an invariant 1-form.
Mechanical Systems: Symmetries and Reduction
Horizontal and Vertical Spaces Associated with any connection are vertical and horizontal spaces defined as follows. Definition A3 Given the connection A, its horizontal space at q 2 Q is defined by H q D fv q 2 Tq Q j A(v q ) D 0g and the vertical space at q 2 Q is, as above, Vq D fQ (q) j 2 gg : The map v q 7! verq (v q ) :D [A(q)(v q )]Q (q) is called the vertical projection, while the map v q 7! horq (v q ) :D v q verq (v q ) is called the horizontal projection. Because connections map infinitesimal generators of a Lie algebra elements to that same Lie algebra element, the vertical projection is indeed a projection for each fixed q onto the vertical space and likewise with the horizontal projection. By construction, we have v q D verq (v q ) C hor q (v q ) and so Tq Q D H q ˚ Vq and the maps hor q and verq are projections onto these subspaces. It is sometimes convenient to define a connection by the specification of a space H q declared to be the horizontal space that is complementary to V q at each point, varies smoothly with q and respects the group action in the sense that H gq D Tq ˚ g (H q ). Clearly this alternative definition of a principal connection is equivalent to the definition given above. Given a point q 2 Q, the tangent of the projection map Q;G restricted to the horizontal space H q gives an isomorphism between H q and T[q] (Q/G). Its inverse 1 Tq Q;G j H q : TQ;G (q) (Q/G) ! H q is called the horizontal lift to q 2 Q.
manifold Q is a Riemannian manifold. Of course, the Riemannian structure will often be that defined by the kinetic energy of a given mechanical system. Thus, assume that Q is a Riemannian manifold, with metric denoted hh; ii and that G acts freely and properly on Q by isometries, so Q;G : Q ! Q/G is a principal G-bundle. In this context we may define the horizontal space at a point simply to be the metric orthogonal to the vertical space. This therefore defines a connection called the mechanical connection. Recall from the historical survey in the introduction that this connection was first introduced by Kummer [79] following motivation from Smale [168] and [1]. See also Guichardet [54], who applied these ideas in an interesting way to molecular dynamics. The number of references since then making use of the mechanical connection is too large to survey here. In Proposition A5 we develop an explicit formula for the associated Lie algebra valued 1-form in terms of an inertia tensor and the momentum map. As a prelude to this formula, we show the following basic link with mechanics. In this context we write the momentum map on TQ simply as J : TQ ! g . Proposition A4 The horizontal space of the mechanical connection at a point q 2 Q consists of the set of vectors v q 2 Tq Q such that J(v q ) D 0. For each q 2 Q, define the locked inertia tensor I(q) to be the linear map I(q) : g ! g defined by hI(q); i D hhQ (q); Q (q)ii
(A2)
for any ; 2 g. Since the action is free, I(q) is nondegenerate, so (A2) defines an inner product. The terminology “locked inertia tensor” comes from the fact that for coupled rigid or elastic systems, I(q) is the classical moment of inertia tensor of the rigid body obtained by locking all the joints of the system. In coordinates, j
I ab D g i j K ai K b ;
(A3)
where [Q (q)] i D K ai (q) a define the action functions K ai . Define the map A : TQ ! g which assigns to each v q 2 Tq Q the corresponding angular velocity of the locked system: A(q)(v q ) D I(q)1 (J(v q )) ;
(A4)
where L is the kinetic energy Lagrangian. In coordinates, The Mechanical Connection As an example of defining a connection by the specification of a horizontal space, suppose that the configuration
A a D I ab g i j K bi v j
since J a (q; p) D p i K ai (q).
(A5)
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We defined the mechanical connection by declaring its horizontal space to be the metric orthogonal to the vertical space. The next proposition shows that A is the associated connection one-form. Proposition A5 The g-valued one-form defined by (A4) is the mechanical connection on the principal G-bundle Q;G : Q ! Q/G. Given a general connection A and an element 2 g , we can define the -component of A to be the ordinary oneform ˛ given by ˛ (q) D A(q) 2 Tq Q; ˛ ˝ D ; A(q)(v q )
i. e.;
˛ ˝ ˛ (q); v q
Curvature The curvature B of a connection A is defined as follows.
for all v q 2 Tq Q. Note that ˛ is a G -invariant one-form. It takes values in J1 () since for any 2 g, we have ˝ ˛ ˝ ˛ J(˛ (q)); D ˛ (q); Q D h; A(q)(Q (q))i D h; i : In the Riemannian context, Smale [168] constructed ˛ by a minimization process. Let ˛q] 2 Tq Q be the tangent vector that corresponds to ˛q 2 Tq Q via the metric hh; ii on Q. A(q)
Tq Q
Proposition A6 The 1-form ˛ (q) D 2 associated with the mechanical connection A given by (A4) is characterized by K(˛ (q)) D inffK(ˇq ) j ˇq 2 J1 () \ Tq Qg ; (A6) where K(ˇ q ) D T Q. See Fig. 2.
] 2 1 2 kˇ q k
proof in Smale [168] or of Proposition 4.4.5 in [1]. Thus, one of the merits of the previous proposition is to show easily that this variational definition of ˛ does indeed yield a smooth one-form on Q with the desired properties. Note also that ˛ (q) lies in the orthogonal space to Tq Q \ J1 () in the fiber Tq Q relative to the bundle metric on T Q defined by the Riemannian metric on Q. It also follows that ˛ (q) is the unique critical point of the kinetic energy of the bundle metric on T Q restricted to the fiber Tq Q \ J1 ().
is the kinetic energy function on
The proof is a direct verification. We do not give here it since this proposition will not be used later in this book. The original approach of Smale [168] was to take (A6) as the definition of ˛ . To prove from here that ˛ is a smooth one-form is a nontrivial fact; see the
Mechanical Systems: Symmetries and Reduction, Figure 2 The extremal characterization of the mechanical connection
Definition A7 The curvature of a connection A is the Lie algebra valued two-form on Q defined by B(q)(u q ; v q ) D dA(horq (u q ); hor q (v q )) ;
(A7)
where d is the exterior derivative. When one replaces vectors in the exterior derivative with their horizontal projections, then the result is called the exterior covariant derivative and one writes the preceding formula for B as B D dA A :
For a general Lie algebra valued k-form ˛ on Q, the exterior covariant derivative is the k C 1-form dA ˛ defined on tangent vectors v0 ; v1 ; : : : ; v k 2 Tq Q by dA ˛(v0 ; v1 ; : : : ; v k ) D d˛ horq (v0 ); hor q (v1 ); : : : ; hor q (v k ) :
(A8)
Here, the symbol dA reminds us that it is like the exterior derivative but that it depends on the connection A. Curvature measures the lack of integrability of the horizontal distribution in the following sense. Proposition A8 On two vector fields u, v on Q one has B(u; v) D A [hor(u); hor(v)] : Given a general distribution D TQ on a manifold Q one can also define its curvature in an analogous way directly in terms of its lack of integrability. Define vertical vectors at q 2 Q to be the quotient space Tq Q/D q and define the curvature acting on two horizontal vector fields u, v (that is, two vector fields that take their values in the distribution) to be the projection onto the quotient of their Jacobi–Lie bracket. One can check that this operation depends only on the point values of the vector fields, so indeed defines a two-form on horizontal vectors.
Mechanical Systems: Symmetries and Reduction
Cartan Structure Equations We now derive an important formula for the curvature of a principal connection. Theorem A9 (Cartan structure equations) For any vector fields u, v on Q we have B(u; v) D dA(u; v) [A(u); A(v)] ;
(A9)
where the bracket on the right hand side is the Lie bracket in g. We write this equation for short as B D dA [A; A] :
If the G-action on Q is a right action, then the Cartan Structure Equations read B D dA C [A; A]. The following Corollary shows how the Cartan Structure Equations yield a fundamental equivariance property of the curvature. Corollary A10 For all g 2 G we have ˚ g B D Ad g ı B. If the G-action on Q is on the right, equivariance means ˚ g B D Ad g 1 ı B. Bianchi Identity The Bianchi Identity, which states that the exterior covariant derivative of the curvature is zero, is another important consequence of the Cartan Structure Equations. Corollary A11 If B D dA A 2 ˝ 2 (Q; g) is the curvature two-form of the connection A, then the Bianchi Identity holds: dA B D 0 : This form of the Bianchi identity is implied by another version, namely dB D [B; A]^ ; where the bracket on the right hand side is that of Lie algebra valued differential forms, a notion that we do not develop here; see the brief discussion at the end of §9.1 in [104]. The proof of the above form of the Bianchi identity can be found in, for example, Kobayashi and Nomizu [74]. Curvature as a Two-Form on the Base We now show how the curvature two-form drops to a twoform on the base with values in the adjoint bundle. The associated bundle to the given left principal bundle Q;G : Q ! Q/G via the adjoint action is called the
adjoint bundle. It is defined in the following way. Consider the free proper action (g; (q; )) 2 G (Q g) 7! (gq; Ad g ) 2 Qg and form the quotient g˜ :D QG g :D (Q g)/G which is easily verified to be a vector bundle g˜ : g˜ ! Q/G, where g˜ (g; ) :D Q;G (q). This vector bundle has an additional structure: it is a Lie algebra bundle; that is, a vector bundle whose fibers are Lie algebras. In this case the bracket is defined pointwise: [g˜ (g; ); g˜ (g; )] :D g˜ (g; [; ]) for all g 2 G and ; 2 g. It is easy to check that this defines a Lie bracket on every fiber and that this operation is smooth as a function of Q;G (q). The curvature two-form B 2 ˝ 2 (Q; g) (the vector space of g-valued two-forms on Q) naturally induces a two-form B on the base Q/G with values in g˜ by B(Q;G (q)) Tq Q;G (u); Tq Q;G (v) :D g˜ q; B(u; v) (A10) for all q 2 Q and u; v 2 Tq Q. One can check that B is well defined. B D Since (A10) can be equivalently written as Q;G g˜ ı id Q B and Q;G is a surjective submersion, it follows that B is indeed a smooth two-form on Q/G with values in g˜ . Associated Two-Forms Since B is a g-valued two-form, in analogy with (A1), for every 2 g we can define the -component of B, an ordinary two-form B 2 ˝ 2 (Q) on Q, by ˝ ˛ B (q)(u q ; v q ) :D ; B(q)(u q ; v q ) (A11) for all q 2 Q and u q ; v q 2 Tq Q. The adjoint bundle valued curvature two-form B induces an ordinary two-form on the base Q/G. To obtain it, we consider the dual g˜ of the adjoint bundle. This is a vector bundle over Q/G which is the associated bundle relative to the coadjoint action of the structure group G of the principal (left) bundle Q;G : Q ! Q/G on g . This vector bundle has additional structure: each of its fibers is a Lie–Poisson space and the associated Poisson tensors on each fiber depend smoothly on the base, that is, g˜ : g˜ ! Q/G is a Lie–Poisson bundle over Q/G. Given 2 g , define the ordinary two-form B on Q/G by B Q;G (q) Tq Q;G (u q ); Tq Q;G (v q ) ˛ ˝ :D g˜ (q; ); B (Q;G (q)) Tq Q;G (u q ); Tq Q;G (v q ) ˛ ˝ D ; B(q)(u q ; v q ) D B (q)(u q ; v q ) ; (A12)
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where q 2 Q, u q ; v q 2 Tq Q, and in the second equality h; i : g˜ g˜ ! R is the duality pairing between the coadjoint and adjoint bundles. Since B is well defined and smooth, so is B . Proposition A12 Let A 2 ˝ 1 (Q; g) be a connection oneform on the (left) principal bundle Q;G : Q ! Q/G and B 2 ˝ 2 (Q; g) its curvature two-form on Q. If 2 g , the corresponding two-forms B 2 ˝ 2 (Q) and B 2 ˝ 2 (Q/G) defined by (A11) and (A12), respectively, are re B D B . In addition, B satisfies the follated by Q;G lowing G-equivariance property: ˚ g B D BAdg : B , where Thus, if G D G then d˛ D B D Q;G ˛ (q) D A(q) ().
Further relations between ˛ and the -component of the curvature will be studied in the next section when discussing the magnetic terms appearing in cotangent bundle reduction. The Maurer–Cartan Equations A consequence of the structure equations relates curvature to the process of left and right trivialization and hence to momentum maps. Theorem A13 (Maurer–cartan equations) Let G be a Lie group and let R : TG ! g be the map (called the right Maurer–Cartan form) that right translates vectors to the identity: R (v g ) D Tg R g 1 (v g ) : Then d R [ R ; R ] D 0 : There is a similar result for the left trivialization L , namely the identity d L C [ L ; L ] D 0 : Of course there is much more to this subject, such as the link with classical connection theory, Riemannian geometry, etc. We refer to [92] for further basic information and references and to Bloch [18] for applications to nonholonomic systems, and to Cendra, Marsden, and Ratiu [35] for applications to Lagrangian reduction.
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n-Body Problem and Choreographies
n-Body Problem and Choreographies SUSANNA TERRACINI Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, Italia Article Outline Glossary Definition of the Subject Introduction Simple Choreographies and Relative Equilibria Symmetry Groups and Equivariant Orbits The 3-Body Problem Minimizing Properties of Simple Choreographies Generalized Orbits and Singularities Asymptotic Estimates at Collisions Absence of Collision for Locally Minimal Paths Future Directions Bibliography Glossary Central configurations Are the critical points of the potential constrained on the unitary moment of inertia ellipsoid. Central configurations are associated to particular solutions to the n-body problem: the relative equilibrium and the homographic motions defined in Definition 1. Choreographical solution A choreographical solution of the n-body problem is a solution such that the particles move on the same curve, exchanging their positions after a fixed time. This property can be regarded as a symmetry of the trajectory. This notion finds a natural generalization in that of G-equivariant trajectory defined in Definition 2 for a given group of symmetries-G. The G-equivariant minimization technique consists in seeking action minimizing trajectories among all G-equivariant paths. Collision and singularities When a trajectory can not be extended beyond a certain time b we say that a singularity occurs. Singularities can be collisions if the solution admits a limit configuration as t ! b. In such a case we term b a collision instant. n-Body problem The n-body problem is the system of differential equations (1) associated with suitable initial or boundary value data. A solution or trajectory is a doubly differentiable path q(t) D (q1 (t); : : : ; q n (t)) satisfying (1) for all t. The weaker notion of generalized solution is defined in Definition 5 applies to trajectories found by variational methods.
Variational approach The variational approach to the n-body problem consists in looking at trajectories as critical points of the action functional defined in (4). Such critical points can be (local) minimizers, or constrained minimizers or mountain pass, or other type. Definition of the Subject The motion of n-point particles of positions x i (t) 2 R3 and masses m i > 0, interacting in accordance with Newton’s law of gravitation, satisfies the system of differential equations: n X xi x j m i x¨ i (t) D G mi m j ; jx i x j j3 j¤i; jD1
i D 1; : : : ; n; ;
t 2R:
(1)
The n-body problem consists in solving Eq. (1) associated with initial or boundary conditions. A simple choreography is a periodic solution to the n-body problem Eq. (1) where the bodies lie on the same curve and exchange their mutual positions after a fixed time, namely, there exists a function x : R ! R such that x i (t) D x(t C (i 1)) ; i D 1; : : : ; n ;
t 2 R;
(2)
where D 2/n. Introduction The two-body problem can be reduced, by the conservation of the linear momentum, to the one center Kepler problem and can be completely solved either by exploiting the conservation laws (angular momentum, energy and the Lenz vector), or by performing the Levi–Civita change of coordinates reducing the problem to that of an harmonic oscillator [58]. The three-body problem is much more complicated than the two-body and can not be solved in a simple way. A major study of the Earth–Moon–Sun system was undertaken by Delaunay in his La Théorie du mouvement de la lune. In the restricted three-body problem, the mass of one of the bodies is negligible; the circular restricted threebody problem is the special case in which two of the bodies are in circular orbits and was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century, Poincaré at the end of the 19th century and Moser in the 20th century. Poincaré’s work on the restricted three-body problem was the foundation of deterministic chaos theory.
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A very basic – still fundamental – question concerns the real number of degrees of freedom of the n-body problem. As the motion of each point particle is represented by 3-dimensional vectors, the n-body problem has 3n degrees of freedom and hence it is 6n-dimensional. First integrals are: the center of mass, the linear momentum, the angular momentum, and the energy. Hence there are 10 independent algebraic integrals. This allows the reduction of variables to 6n 10. It was proved in 1887 by Bruns that these are the only linearly independent integrals of the n-body problem, which are algebraic with respect to phase and time variables. This theorem was later generalized by Poincaré. A second very natural problem is whether there exists a power series expressing – if not every – at least a large and relevant class of trajectories for the n-body problem. In 1912, after pioneering works of Mittag–Leffler and Levi– Civita, Sundman proved the existence of a series solution in powers of t 1/3 for the 3-body problem. This series is convergent for all real t, except for those initial data which correspond to vanishing angular momentum. These initial data have Lebesgue measure zero and therefore are not generic. An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Since then, the study of singularities became the main point of interest in the study of the n-body problem. Sundman’s result was later generalised to the case of n > 3 bodies by Q. Wang in [88]. However, the rate and domain of convergence of this series are so limited to make it hardly applicable to practical and theoretical purposes. Finally, the n-body problem can be faced from the point of view of the theory of perturbations and represents both its starting point and its most relevant application. Delaunay and Poincaré described the spatial three-body problem as a four-dimensional Hamiltonian system and already encountered some trajectories featuring a chaotic behavior. When one of the bodies is much heavier than the other two (a system of one “star” and two “planets”), one can neglect the interaction between the small planets; hence the system can be seen as a perturbation of two decoupled two-body problems whose motions are known to be all periodic from Kepler’s laws. In the planetary three-body problem, hence, two harmonic oscillators interact nonlinearly through the perturbation. Resonances between the two oscillators can be held responsible of high sensitivity with respect to initial data and other chaotic features. A natural question regards the coexistence of the irregular trajectories with regular (periodic or quasi-periodic) ones. A modern approach to the problem of stability of solutions to nearly integrable systems goes through the
application of Kolmogorov–Arnold–Moser (KAM) Theorem [8,56], whose main object indeed is indeed the persistence, under perturbations, of invariant tori. The n-body problem is paradigmatic of any complex system of many interacting objects, and it can neither be solved nor it can be simplified in an efficient way. A possible starting point for its analysis is to seek selected trajectories whose motion is particularly simple, in the sense that it repeats after a fixed period: the periodic solutions. Following Poincaré in his Méthodes Nouvelles de la Mécanique Céleste, tome I, 1892, “. . . D’ailleurs, ce qui nous rends ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule brèche par où nous puissons essayer de pénetrer dans une place jusqu’ici réputée inabordable.” Indeed, just before, Poincaré conjectured that periodic trajectories are dense in the phase space: “. . . Voici un fait que je n’ai pas pu démontrer rigourousement, mais qui me parait pourtant très vraisemblable. Étant donneées des équations de la forme définie dans le n.131 et une solution quelconque de ces équations, on peut toujours trouver une solution périodique (dont la période peut, il est vrai, être trés longue), telle que la différence entre les deux solutions soit aussi petite qu’on le veut, pendant un temps aussi long qu’on le veut.”
Singular Hamiltonian Systems From an abstract point of view, the n-body problem is a Hamiltonian System of the form m i x¨ i D
@U (t; x) ; @x i
i D 1; : : : ; n ;
(3)
@U are undefined on a singular set , where the forces @x i the set of collisions between two or more particles in the nbody problem. Such singularities play a fundamental role in the phase portrait (see, e. g. [43]) and strongly influence the global orbit structure, as they can be held responsible, among others, of the presence of chaotic motions (see, e. g. [39]) and of motions becoming unbounded in a finite time [64,90]. Two are the major steps in the analysis of the impact of the singularities in the n-body problem: the first consists in 1 Formula N. 13 quoted by Poincaré is Hamilton equation and covers our class od Dynamical Systems Eq. (3)
n-Body Problem and Choreographies
performing the asymptotic analysis along a single collision (total or partial) trajectory and goes back, in the classical case, to the works by Sundman ([84]), Wintner ([89]) and, in more recent years by Sperling, Pollard, Saari and other authors (see for instance [40,47,75,76,79,83]). The second step consists in blowing-up the singularity by a suitable change of coordinates introduced by McGehee in [65] and replacing it by an invariant boundary – the collision manifold – where the flow can be extended in a smooth manner. It turns out that, in many interesting applications, the flow on the collision manifold has a simple structure: it is a gradient-like, Morse–Smale flow featuring a few stationary points and heteroclinic connections (see, for instance, the surveys [39,68]). The analysis of the extended flow allows us to obtain a full picture of the behavior of solutions near the singularity, despite the flow fails to be fully regularizable (except for binary collisions).
Simple Choreographies and Relative Equilibria A possible starting point for the study of the n-body problem is to find selected trajectories which are particularly simple, when regarded from some point of view. Examples of such particular solutions are the collinear periodic orbits (found in 1767 by Euler), in which three bodies of any masses move such that they oscillate along a rotation line, and the Lagrange triangular solutions, where the bodies lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically, discovered in 1772. Both these trajectories are stationary in a rotating frame. A second remarkable class of trajectories – the choreographies – can be found when the masses are all equal, by exploiting the fact that particles can be interchanged without changing the structure of the system. Among all periodic solutions of the planar 3-body problem, the relative equilibrium motions – the equilateral Lagrange and the collinear Euler–Moulton solutions – are definitely the simplest and most known. In general such simple periodic motions exist for any number of bodies. Definition 1 A relative equilibrium trajectory is a solution of (1) whose configuration remains constant in a rotating frame. A homographic trajectory is a solution of (1) whose configuration remains constant up to homotheties. The normalized configurations of such trajectories are named central and are the critical points of the potential X mi m j U(x) D ; jx i x j j i< j
constrained to the ellipsoid of unitary momentum of inertia: X m i jx i j2 D 1 : I(x) D i
Relative equilibria feature an evident symmetry (SO(2) and O(2) respectively), that is, they are equivariant with respect to the symmetry group of dimension 1 acting as SO(2) (resp. O(2)) on the time circle and on the plane, and trivially on the set of indexes f1; 2; 3g. In fact, they are minimizers of the Lagrangian action functional in the space of all loops having their same symmetry group. Therefore, generally speaking, G-equivariant minimizers for the action functional (given a symmetry group G) can be thought as the natural generalization of relative equilibrium motions. This perspective has known a wide popularity in the recent literature and has produced a new boost in the study of periodic trajectories in the n-body problem; the recent proof of the existence of the Chenciner–Montgomery eight-shaped orbit is emblematic of this renewed interest (see [7,20,22,25,29,52] and the major part of our bibliographical references). In all these papers, periodic and quasi-periodic solutions of the n-body problem can be found as critical points of the Lagrangian action functional restricted to suitable spaces of symmetric paths. Basic Definitions and Notations Let us consider n point particles with masses m1 ; m2 ; : : : ; m n and positions x1 ; x2 ; : : : ; x n 2 Rd , with d 2. We denote by X the space of configurations with center of mass in 0, and by Xˆ D X n the set of collision-free configurations (collision means x i D x j for some i ¤ j). On the configuration space we define the homogeneous (Newton) potential of degree ˛ < 0: X mi m j U(x) D U i; j (jx i x j j); U i; j (jx i x j j) ' : jx i x j j˛ i< j
In many cases one can simply require the U i; j to be asymptotically homogeneous at the singularity. Furthermore, a major part of our analysis can be extended to logarithmic potentials. Relative equilibria correspond to those configurations (termed central) which are critical for the restriction of the potential U to the ellipsoid I D where I denotes the momentum of inertia: X I(x) D m i jx i j2 : i
On collisions the potential U D C1. We are interested in (relative) periodic (such that 8t : x(t C T) D x(t))
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solutions to the system of differential equations: m i x¨ i D
@U : @x i
We associate with the equation the Lagrangian integrand L(x; x˙ ) D L D KCU D
X1 i
2
2
m i j x˙ i j C
X
U i; j (jx i x j j)
i< j
A(x) D
T
L(x(t); x˙ (t))dt :
Then G acts on time (translation and reversal) T via and it acts on the configuration space X via and in the following way 8i D 1 : : : n : (gx) i D (g)x (g)1 (i) :
and the action functional: Z
an orthogonal representation of dimension 2 : G ! O(2) (on cyclic time T Š S 1 ), an orthogonal representation (on the euclidean space Rd ) : G ! O(d), and an homomorphism on the symmetric group on n elements (n D f1; 2; : : : ; ng) : G ! ˙n .
(4)
0
Sometimes it will be preferable to consider the problem in a frame rotating uniformly about the vertical axis, with an angular speed !; the corresponding action A! then contains a gyroscopic term, associated with Coriolis force. We shall seek periodic solutions as critical points of the action functional on the Sobolev space of T-periodic trajectories: D H 1 (T ; X ) or, to be more precise, of the action constrained on suitable linear subspaces 0 . Two are the major difficulties to be faced in following the variational approach; the first is due to the lack of coercivity (or of Palais–Smale) due to the vanishing at infinity of the force fields: indeed sequences of almost-critical points (such as minimizing sequences) may very well diverge. Furthermore, as the potential U is singular on collisions, minimizers or other critical points can a priori be collision trajectories. Compactness can be successfully recovered by the symmetry of the problem, as we are going to explain here below. We shall expose in the last part of this article some of the strategies which has been developed in order to overcome the problem of collisions. Symmetry Groups and Equivariant Orbits We can generalize the concept of relative equilibria as follows: we impose the permutation of the positions of the particles, after a given time, up to isometries of the space. This gives rise to a class of symmetric trajectories (the generalized choreoghapies). It is worthwhile noticing that these trajectories arise as a common feature of any system of interacting objects, regardless whether they come from models in Celestial mechanics or not. It applies to atoms, or molecules, galaxies or whatever system, provided the objects can be freely interchanged. Let us start by introducing some basics concepts and definitions from [52]. Let G be a finite group endowed with:
As a consequence we have an action on the space of trajectories: Definition 2 A continuous function x(t) D (x1 (t); : : : ; x n (t)) is G-equivariant if 8g 2 G : x(g t) D (gx)(t) : The linear subspace 0 D G denotes the set of periodic trajectories in which are equivariant with respect to the G-action: The Palais Principle of symmetric criticality [73] ensures that critical points of an invariant functional restricted to the space of equivariant trajectories are indeed free critical points. We stress that by the particular form of the interaction potentials, in our setting invariance is simply implied by equality of those masses which are interchanged by the action of G on the set of the indices. When the action of is transitive, all the masses must be equal and the associated G–equavariant trajectories give rise to generalized choreographies. Cyclic and Dihedral Actions Consider the normal subgroup ker G G and the quotient G¯ D G/ker. Since G¯ acts effectively on T , it is either a cyclic group or a dihedral group. If the group G¯ acts trivially on the orientation of T , then G¯ is cyclic and we say that the action of G on is of cyclic type.
If the group G¯ consists of a single reflection on T , then we say that action of G on is of brake type.
n-Body Problem and Choreographies
Otherwise, we say that the action of G on is of dihedral type.
With this first classification, we can easily associate to our minimization problem some proper boundary conditions.
Proposition 1 Let I be the fundamental domain (for a dihedral type), or any interval having as length the minimal angle of time-rotation in G¯ (for a cyclic type). Then the G-equivariant minimization problem is equivalent to problem of minimizing the action over all paths x : I ! X ker subject to the boundary conditions x(0) 2 X H 0 and x(1) 2 X H 1 , where H 0 and H 1 are the maximal isotropy subgroups of the boundary of I.
Of course, beside seeking minimizers, one can look for other type of critical points, such as mountain pass or others. As the potential U is singular on collisions, minimizers (or other critical points) can a priori be collision trajectories. Many strategies were proposed in the literature in order to overcome these obstacles. The development of a suitable Critical Point theory taking into account of the contribution of fake periodic solutions (the critical points at infinity) was proposed by some authors [9,60,77] and returned a good estimate of the number of periodic trajectories satisfying an appropriate bound on the length, with the main disadvantage of requiring a strong order of infinity at the collisions (the strong force condition). Another strategy to recover coercivity of action functional consists in imposing a symmetry constraint on the loop space. Surprisingly enough, once coercivity is recovered, also the problem of collisions becomes much less dramatic. This fact was remarked for the first time in [34] and widely exploited in the literature, also thanks to the neat idea, due to C. Marchal, of averaging over all possible variations (generalized and exposed in Sect. “Generalized Orbits and Singularities”) to avoid the occurrence of collisions for extremals of the action (Marchal idea was first exposed in [25]). This argument can be used in most of the known cases to prove that absence of collisions for minimizing trajectories and will be outlined in the last section. The Eight Shaped Three-Body Solution
The Variational Approach Let us consider the A restricted to the space of symmetric loops G . We recall that the action functional is said to be coercive if limjxj!C1 A(x) D C1. Coercivity implies the validity of the direct method of Calculus of Variations and, consequently, the existence of a minimizer. Proposition 2 The action functional A is coercive in G if and only if X G D 0. Consequently, if X G D 0 then a minimizer of AG in G exists. Given and , we can compute dimX G 1 X dimX G D Tr( (g))#Fix( (g)) d : jGj g2G
In the frame rotating with constant angular speed ! the action A! is generally coercive, except for a (possible) discrete set values of !.
In their paper [29], Chenciner and Montomery exploited a variational argument in order to prove the existence of a periodic trajectory for the three body problem, where the three particles move one a single eight-shaped curve, interchanging their positions after a fixed time. C. Moore [71] was the first to find numerically the eight, lead by topological reasons which turned out to be insufficient to insure its existence. One of the the simplest symmetries giving rise to the eight shaped trajectory is the following. Denote x(t) D (x1 (t); x2 (t); x3 (t)) 2 R6 : Let G D D6 be the dihedral group generated by two following reflections: the first x1 (t) D x1 (t) ; x3 (t) D x2 (t) ;
x2 (t) D x3 (t) ;
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with g1 D (1 ; 1 ; 1 ) is 1 (t) D t, 1 (x) D x and 1 (1; 2; 3) D (1; 3; 2). And the second x1 (1 t) D x2 (t) ;
x2 (1 t) D x1 (t) ;
x3 (1 t) D x3 (t) ; with g2 D (2 ; 2 ; 2 ) is 2 (t) D 1 t, 2 (x) D x and 2 (1; 2; 3) D (2; 1; 3). There are three possible groups yielding an eightshaped trajectory. First, we consider the group of cyclic action type C6 (the cyclic eight having order 6), which acts cyclically on T (i. e. by a rotation of angle /3), by a reflection in the plane E, and by the cyclic permutation (1; 2; 3) in the index set. The second group, which we denote by D12 , is the group of order 12 obtained by extending C6 with the element h defined as follows: (h) is a reflection in T , (h) is the antipodal map in E (thus, the rotation of angle ), and (h) is the permutation (1; 2). This is the symmetry group used by Chenciner and Montgomery. The third group is the subgroup of D12 generated by h and the subgroup C3 of order 3 of C6 D12 . We denote this group D6 (since it is a dihedral group of order 6). The symmetry groups D12 and D6 are of dihedral type. The choreography group C3 is a subgroup of all the three groups, thus the action is coercive on G-equivariant loops.
Definition 3 A group G acts with the rotating circle property if for every T -isotropy subgroup G t G and for at least n 1 indexes i 2 n there exists in Rd a rotating circle S under Gt for i. In most of the known examples the property is fulfilled. In [52] the following results were proved. Theorem 1 Consider a finite group K acting on with the rotating circle property. Then a minimizer of the K-equivariant fixed-ends (Bolza) problem is free of collisions. Corollary 1 For every ˛ > 0, minimizers of the fixed-ends (Bolza) problem are free of interior collisions. Corollary 2 If the action of G on is of cyclic type and ker has the rotating circle property then any local minimizer of AG in G is collisionless. Corollary 3 If the action of G on is of cyclic type and ker D 1 is trivial then any local minimizer of AG in G is collisionless. Theorem 2 Consider a finite group G acting on so that every maximal T -isotropy subgroup of G either has the rotating circle property or acts trivially on the index set n. Then any local minimizer of AG yields a collision-free periodic solution of the Newton equations for the n-body problem in Rd .
The Rotating Circle Property (RCP) As a second step of the variational approach to the n-body problem, one has to prove that the output of the minimization (or some other variational method) procedure is free of collisions. This point involves a deep analysis of the structure of the possible singularities and will be outlined in the last part of this article. After performing the analysis of collisions, we find that if the action of G on T and X fulfills some conditions (computable) then (local) minimizers of the action functional A in G do not have collisions. For a group H acting orthogonally on Rd , a circle S
d R (with center in 0) is termed rotating under H if S is invariant under H (that is, for every g 2 H gS D S) and for every g 2 H the restriction gjS : S ! S is a rotation (the identity is meant as a~rotation of angle 0). Let i 2 n be an index and H G a subgroup. A circle S Rd D V (with center in 0) is termed rotating for i under H if S is rotating under H and S V H i V D Rd ; where H i H denotes the isotropy subgroup of the index i in H relative to the action of H on the index set n induced by restriction (that is, the isotropy H i D fg 2 Hjg i D ig).
Examples In this section, for the sake of illustrating the power and the limitations of the approach through the G-equivariant minimization, we include a few examples fitting in our theoretical framework and we give some hints of which symmetry groups satisfy or not assumptions of Theorem 2. Well-known examples are the celebrated Chenciner–Montgomery “eight” [29], Chenciner– Venturelli “Hip–Hop” solutions [30], Chenciner “generalized Hip–Hops” solutions [26], Chen’s orbit [20,22] and Terracini–Venturelli generalized Hip–Hops [85]. One word about the pictures of planar orbits: the configurations at the boundary points of the fundamental domain I are denoted with an empty circle (starting point x i (0)) and a black disc (ending point x i (t), with t appropriate), with a label on the starting point describing the index of the particle. The trajectories of the particles with the times in I are painted as thicker lines (thus it is possible to recover the direction of the movement from x i (0) to x i (t)). Unfortunately this feature was not possible with the threedimensional images. Also, in all the following examples but 4 and 5 existence of the orbits follows directly from the results of The-
n-Body Problem and Choreographies
orem 2. The existence of the orbits described in Examples 4 and 5, which goes beyond the scope of this article, has been recently proved by Chen in [22]. Thousands of other suitable actions and the corresponding orbits have been found by a special-purpose computer program based on GAP [53]. Example 1 (Choreographies) Consider the cyclic group G D Z n of order n acting trivially on V, with a cyclic permutation of order n on the index set n D f1; : : : ; ng and with a rotation of angle 2/n on the time circle T . Since X G D 0, by Proposition 2 the action functional AG is coercive. Moreover, since the action of G on T is of cyclic type and ker D 1, by Corollary 2 the minimum exists and it has no collisions. For several numerical results and a description of choreographies we refer the reader to [32]. An insight of the variational properties of choreographies is provided in Sect. “Minimizing Properties of Simple Choreographies”. Example 2 Let n be odd. Consider the dihedral group G D D2n of order 2n, with the presentation ˛ ˝ G D g1 ; g2 jg12 D g2n D (g1 g2 )2 D 1 : Let be the homomorphism defined by
0 1 ; and (g1 ) D 1 0
cos 2 sin 2 n n (g2 ) D : sin 2 cos 2 n n Furthermore, let the homomorphism be defined by
1 0 ; and
(g1 ) D 0 1
1 0 :
(g2 ) D 0 1 Finally, let G act on n by the homomorphism defined as (g1 ) D (1; n 1)(2; n 2) : : : ((n 1)/2; (n C 1)/2), (g2 ) D (1; 2; : : : ; n), where (i1 ; i2 ; : : : ; i k ) means the standard cycle-decomposition notation for permutation groups. By the action of g 2 it is easy to show that all the loops in G are choreographies, and thus that, since X G D 0, the action functional is coercive. The maximal T -isotropy subgroups are the subgroups of order 2 generated by the elements g1 g2i with i D 0 : : : n 1. Since they are all conjugated, it is enough to show that one of them acts with the rotating circle property. Thus consider H D hg1 i G. For every index i 2 f1; 2; : : : ; n 1g the isotropy H i H relative to the action of H on n is trivial, and g 1 acts by rotation on V D R2 . Therefore for every i 2 f1; 2; : : : ; n1g it is possible to choose a circle rotating
n-Body Problem and Choreographies, Figure 1 The (D6 -symmetric) eight for n D 3
n-Body Problem and Choreographies, Figure 2 The (D10 -symmetric) eight with n D 5
under H for i, since, being H i trivial, V H i D V. The resulting orbits are not homographic (since all the particles pass through the origin 0 at some time of the trajectory and the configurations are centered). For n D 3 this is the eight with less symmetry of [25]. Possible trajectories are shown in Figs. 1 and 2. Example 3 As in the previous example, let n 3 be an odd integer. Let G D C2n Š Z2 C Z n be the cyclic group of order 2n, presented as G D hg1 ; g2 jg12 D g2n D g1 g2 g11 g21 D 1i. The action of G on T is given by (g1 g2 ) D 2n , where 2n denotes the rotation of angle /n (hence the action will be of cyclic type). Now, G can act on the plane V D R2 by the homomorphism defined by
1 0 ; and
(g1 ) D 0 1
1 0
(g2 ) D : 0 1 Finally, the action of G on n D f1; 2; : : : ; ng is given by the homomorphism : G ! ˙n defined by (g1 ) D (), (g2 ) D (1; 2; : : : ; n). The cyclic subgroup H2 D hg2 i
G gives the symmetry constraints of the choreographies, hence loops in G are choreographies and the functional is coercive. Furthermore, since the action is of cyclic type, by Corollary 3 the minimum of the action functional is collisionless. It is possible that such minima coincide with the minima of the previous example: this would imply that the symmetry group of the minimum contains the two groups above.
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n-Body Problem and Choreographies, Figure 3 Another symmetry constraint for an eight-shaped orbit (n D 5)
Example 4 Consider four particles with equal masses and an odd integer q 3. Let G D D4q C2 be the direct product of the dihedral group of order 4q with the group C2 of order 2. Let D4q be presented by D4q D hg1 ; g2 jg12 D 2q g2 D (g1 g2 )2 D 1i, and let c 2 C2 be the non-trivial element of C2 . Now define the homomorphisms , and as follows:
1 0 ;
(g1 ) D (g1 ) D 0 1 " # cos 2 sin 2 2q 2q
(g2 ) D (g2 ) D ; sin 2 cos 2 2q 2q
1 0 1 0 ; ; (c) D
(c) D 0 1 0 1 (g1 ) D (1; 2)(3; 4) ;
(g2 ) D (1; 3)(2; 4) ;
(c) D (1; 2)(3; 4) : It is not difficult to show that X G D 0, and thus the action is coercive. Moreover, ker D C2 , which acts on R2 with the rotation of order 2, hence ker acts with the rotating circle property. Thus, by Proposition 2 and Theorem 1 the minimizer exists and does not have interior collisions. To exclude boundary collisions we cannot invoke Theorem 2, since the maximal T -isotropy subgroups do not act with the rotating circle property. A possible graph for such a minimum can be found in Fig. 4, for q D 3 (one needs to prove that the minimum is not the homographic solution – with a level estimate – and that there are no boundary collisions – with an argument similar to [20]). See also [22] for an updated and much generalized treatment of such orbits. Example 5 Consider four particles with equal masses and an even integer q 4. Let G D D q C2 be the direct product of the dihedral group of order 2q with the group C2 of q order 2. Let D4q be presented by D4q D hg1 ; g2 jg12 D g2 D (g1 g2 )2 D 1i, and let c 2 C2 be the non-trivial element of C2 . As in Example 4, define the homomorphisms , and as follows.
1 0 ;
(g1 ) D (g1 ) D 0 1
n-Body Problem and Choreographies, Figure 4 The orbit of Example 4 with q D 3
"
# cos 2 sin 2 q q
(g2 ) D (g2 ) D ; sin 2 cos 2 q q
1 0 1 0
(c) D ; (c) D ; 0 1 0 1 (g1 ) D (1; 2)(3; 4) ;
(g2 ) D (1; 3)(2; 4) ;
(c) D (1; 2)(3; 4) : Again, one can show that a minimizer without interior collisions exists since ker D C2 acts with the rotating circle property (a possible minimizer is shown in Fig. 5). This generalizes Chen’s orbit [22]. See also [20]. Example 6 (Hip–Hops) If G D Z2 is the group of order 2 acting trivially on n, acting with the antipodal map on V D R3 and on the time circle T , then again X G D 0, so that Proposition 2 holds. Furthermore, since the action is of cyclic type Corollary 2 ensures that minimizers have no collisions. Such minimizers were called generalized Hip–
n-Body Problem and Choreographies, Figure 5 A possible minimizer for Example 5
n-Body Problem and Choreographies
n-Body Problem and Choreographies, Figure 6 The Chenciner–Venturelli Hip–Hop n-Body Problem and Choreographies, Figure 7 The planar equivariant minimizer of Example 7
Hops in [25]. See also [26]. A subclass of symmetric trajectories leads to a generalization of such a Hip–Hop. Let n 4 an even integer. Consider n particles with equal masses, and the group G D C n C2 direct product of the cyclic group of order n (with generator g 1 ) and the group C2 of order 2 (with generator g 2 ). Let the homomorphisms , and be defined by 2
sin 2 cos 2 n n 2 4
(g1 ) D sin n cos 2 n 0 0 2 3 1 0 0
(g2 ) D 4 0 1 0 5 ; 0 0 1
1 0 (g2 ) D ; 0 1 (g1 ) D (1; 2; 3; 4) ;
3 0 05 ; 1
(g1 ) D
1 0
0 ; 1
(g2 ) D () :
Example 7 Consider the direct product G D D6 C3 of the dihedral group D6 (with generators g 1 and g 2 of order 3 and 2 respectively) of order 6 and the cyclic group C3 of order 3 generated by c 2 C3 . Let us consider the planar n-body problem with n D 6 with the symmetry constraints given by the following G-action.
1 0 1 ; (g2 ) D 0 1 0
2 2 cos 3 sin 3
(c) D ; sin 2 cos 2 3 3
(g1 ) D (1; 3; 2)(4; 5; 6) ;
0 ; 1
(g2 ) D (1; 4)(2; 5)(3; 6) ;
(c) D (1; 2; 3)(4; 5; 6) :
It is easy to see that X G D 0, and thus a minimizer exists. Since the action is of cyclic type, it suffices to exclude interior collisions. But this follows from the fact that ker D C n has the rotating circle property. This example is the natural generalization of the Hip–Hop solution of [30] to n 4 bodies. We can see the trajectories in Fig. 6.
(g1 ) D
cos 2 sin 2 3 3 ; sin 2 cos 2 3 3
0 1 1 ; (c) D (g2 ) D 1 0 0
(g1 ) D
0 ; 1
By Proposition 2 one can prove that a minimizer exists, and since G acts with the rotating circle property (actually, the elements of the image of are rotations) on T -maximal isotropy subgroups, the conclusion of Theorem 2 holds. It is not difficult to see that configurations in X ker are given by two centered equilateral triangles. Now, to guarantee that the minimizer is not a homographic solution, of course it suffices to show that there are no homographic solutions in G (like in the case of Example 2). This follows from the easy observation that at some times t 2 T with maximal isotropy it happens that x1 D x4 , x2 D x5 and x3 D x6 , while at some other times it happens that x1 D x5 , x2 D x6 and x3 D x4 or that x1 D x6 , x2 D x4 and x3 D x5 and this implies that there are no homographic loops in G . With no difficulties the same action can be defined for n D 2k, where k is any odd integer. We can see a possible trajectory in Fig. 7. Also, it is not difficult to consider a similar example in dimension 3. With n D 6 and the notation of D6 and C3 as above, consider the group G D D6 C3 C2 . Let g 1 , g 2 , c be as above, and let c2 be the generator of C2 . The homomorphisms , and are defined in a similar way by 2 3 2 3 1 0 0 1 0 0
(g1 ) D 40 1 05 ; (g2 ) D 4 0 1 05 ; 0 0 1 0 0 1 2 3 2 2 cos 3 sin 3 0
(c) D 4 sin 2 cos 2 05 ; 3 3 0 0 1
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n-Body Problem and Choreographies, Figure 8 The three-dimensional equivariant minimizer of Example 7
2
3
1 0 0
(c2 ) D 40 1 0 5 ; 0 0 1
cos 2 0 sin 2 3 3 (g1 ) D ; (g2 ) D 2 sin 2 cos 1 3 3
1 0 1 0 ; ; (c2 ) D (c) D 0 1 0 1 (g1 ) D (1; 3; 2)(4; 5; 6) ; (c) D (1; 2; 3)(4; 5; 6) ;
1 ; 0
(g2 ) D (1; 4)(2; 5)(3; 6) ; (c2 ) D () :
In the resulting collisionless minimizer (again, it follows by Proposition 2 and Theorem 2) two equilateral triangles rotate in opposite directions and have a “brake” motion on the third axis. The likely shape of the trajectories can be found in Fig. 8. Example 8 (Marchal’s P12 –symmetry revisited) Let k 2 be an integer, and consider the cyclic group G D C6k of order 6k generated by the element c 2 G. Now consider orbits for n D 3 bodies in the space of dimension d D 3. With a minimal effort and suitable changes the example can be generalized for every n 3. We leave the details to the reader. The homomorphisms , and are defined by 3 2 0 cos k sin k cos k 05;
(c) D 4 sin k 0 0 1
2 2 cos 6k sin 6k ; (c) D sin 2 cos 2 6k 6k (c) D (1; 2; 3) : Straightforward calculations show that X G D 0 and hence Proposition 2 can be applied. Furthermore, the action is of cyclic type with ker D 1, and hence by Corollary 2 the minimizer does not have collisions. It is left to show that this minimum is not a homographic motion. The
n-Body Problem and Choreographies, Figure 9 The non-planar choreography of Example 8 for k D 4
only homographic motion in G is a Lagrange triangle y(t) D (y1 ; y2 ; y3 )(t), rotating with angular velocity 32k (assume that the period is 2, i. e. that T D jT j D 2) in the plane u3 D 0 (let u1 ; u2 ; u3 denote the coordinates in R3 ). To be a minimum it needs to be inscribed in the horizontal circle of radius ((˛3˛/2 )/(2(32k)2 ))1/(2C˛) . Now, for every function (t) defined on T such that (c 3 t) D (t), the loop given by v1 (t) D (0; 0; (t)), v2 (t) D (0; 0; (c 2 t)) and v3 (t) D (0; 0; (c 2 t)) is G-equivariant, and thus belongs to G . If one computes the value of Hessian of the Lagrangian action A in y and in the direction of the loop v one finds that ˇ Dv2 Aˇ y D 3
Z
2
˙ 2 (t)dt 2(3 2k)2
0
Z
2
((t) C (ct))2 dt :
0
In particular, if we set the function (t) D sin(kt), which has the desired property, elementary integration yields Dv2 Aj y D 3(k 2 2(3 2k)2 ) ; which does not depend on ˛ and is negative for every k 3. Thus for every k 3 the minimizer is not homographic. We see a possible trajectory in Fig. 9. Remark 1 In the previous example, if k 6 0 mod 3, the cyclic group G can be written as the sum C3 C C2k . The generator of C3 acts trivially on V, acts with a rotation of order 3 on T and with the cyclic permutation (1; 2; 3) on f1; 2; 3g. This means that for all k 6 0 mod 3 the orbits of Example 8 are non-planar choreographies. Furthermore, it is possible to define a cyclic action of the same kind by setting and as above and 2 3 p p cos 3k sin 3k 0 6 7 p
(c) D 4 sin p cos 3k 05; 3k 0 0 1
n-Body Problem and Choreographies
n-Body Problem and Choreographies, Figure 10 A non-planar symmetric orbit of Remark 1, with k D 3 and p D 1
where p is a non-zero integer. If p D 3 one obtains the same action as in Example 8. One can perform similar computations and obtain that the Lagrange orbit (with angular velocity p 2k, this time) is not a minimizer for all (p; k) such that 0 < p < 3k and k 2 2(p 2k)2 < 0.
1 Y
(1; 1) 1
is isomorphic to the direct product I Y of order 120, and acts on the euclidean space R3 as the full icosahedron group. The action on T is cyclic and given by the fact that ker D 1 Y. The isotropy is generated by the central inversion 1, and hence the set of bodies is G/I Š Y. Thus at any time t the 60 point particles are constrained to be a Y-orbit in R3 (which does not mean they are vertices of a icosahedron, simply that the configuration is Y-equivariant). After half period every body is in the antipodal position: x i (t C T/2) D x i (in other words, the group contains the anti-symmetry), also known as Italian symmetry – see [25,26]. Of course, the group Y is just an example: one can choose also the tetrahedral group T or the octahedral O and obtain anti-symmetric orbits for 12 (tetrahedral) or 24 (octahedral) bodies, as depicted in Fig. 11. The action is by its definition transitive and coercive; local minimizers are collisionless since the maximal T -isotropy group acts as a subgroup of SO(3) (i. e. orientation–preserving).
More Examples with non Trivial Core
Example 10 Let G be the group with Krh data 1 (1; 1) ; Dk 1
Now consider the core of G, ker D K G. With an abuse of notation, using the factorization of the action, we can identify K with a subgroup of O(3). A very remarkable class of symmetry groups with non trivial core has been defined by Ferrario in [50], through its Krh decomposition:
where Dk is the rotation dihedral group of order 2k. As in the previous example, the action is such that the action functional is coercive and its local minima collisionless. At every time instant the bodies are Dk -equivariant in R k and the anti-symmetry holds. Approximations of minima can be seen in Fig. 12.
Definition 4 Let G be a symmetry group. Then define
Example 11 To illustrate the case of non-transitive symmetry group, consider the following (cyclic) Krh data: 1 (3; 1) ; 1 1
1. K D ker, 2. [r] 2 WO(3) K as the image in the Weyl group of the generator mod K of ker det G/K (corresponding to the time-shift with minimal angle). If ker det D K, then [r] D 1. 3. [h] 2 WO(3) K as the image in the Weyl group of one of the time-reflections mod K in G/K, in the cases such an element exists. Otherwise it is not defined. In short, the triple (K; [r]; [h])
which yields a group of order 6 acting cyclically on 3 bodies, and with the antipodal map on R3 . Since ker is trivial and the group is of cyclic type, local minima are collisionless. Now, by adding k copies of such group one obtains a symmetry group having k copies of it as its transitive components, where still local minimizers are collisionless and the restricted functional is coercive. Some possible minima can be found in Fig. 13, for k D 3; 4.
is said the Krh data of G. Example 9 Consider the icosahedral group Y of order 60. The group G with Krh data
The 3-Body Problem The major achievement of [14] is to give the complete description of the outcome of the equivariant minimization
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n-Body Problem and Choreographies
n-Body Problem and Choreographies, Figure 11 60-icosahedral Y, 12-tetrahedral T and 24-octahedral O periodic minimizers (chiral)
n-Body Problem and Choreographies, Figure 12 4-dihedral D2 and 6-dihedral D6 symmetric periodic minimizers
n-Body Problem and Choreographies, Figure 13 9 and 12 bodies in anti-choreographic constraints grouped by 3
n-Body Problem and Choreographies
procedure for the planar three-body problem. First we can ensure that minimizers are always collisionless. Theorem 3 Let G a symmetry group of the Lagrangian in the 3-body problem (in a rotating frame or not). If G is not bound to collision (i. e. every equivariant loop has collisions), then any (possible) local minimizer is collisionless. A symmetry group G of the Lagrangian functional A is termed bound to collisions if all G-equivariant loops actually have collisions, fully uncoercive if for every possible rotation vecG in the frame rotating tor ! the action functional A! around ! with angular speed j!j is not coercive in the space of G-equivariant loops (that is, its global minimum escapes to infinity); homographic if all G-equivariant loops are constant up to orthogonal motions and rescaling. The core of the group G is the subgroup of all the elements which do not move the time t 2 T . If, for every angular velocity, G is a symmetry group for the Lagrangian functional in the rotating frame, then we will say that G is of type R. This is a fundamental property for symmetry groups. In fact, if G is not of type R, it turns out that the angular momentum of all G-equivariant trajectories vanishes. The Classification of Planar Symmetry Groups for 3-body Theorem 4 Let G be a symmetry group of the Lagrangian action functional in the planar 3-body problem. Then, up to a change of rotating frame, G is either bound to collisions, fully uncoercive, homographic, or conjugated to one of the symmetry groups listed in Table 1 (RCS stands for Rotating
Circle Property and HGM for Homographic Global Minimizer). Planar Symmetry Groups The trivial symmetry Let G be the trivial subgroup of order 1. It is clear that it is of type R, it has the rotating circle property. It yields a coercive functional on G D only when ! is not an integer. If ! D 12 mod 1 then the minimizers are minimizers for the anti-symmetric symmetry group (also known as Italian symmetry) x(at) D ax(t), where a is the antipodal map on T and E. The masses can be different. Proposition 3 For every ! … Z and every choice of masses the minimum for the trivial symmetry occurs in the relative equilibrium motion associated to the Lagrange central configuration. The line symmetry Another case of symmetry group that can be extended to rotating frames with arbitrary masses is the line symmetry: the group is a group of order 2 acting by a reflection on the time circle T , by a reflection on the plane E, and trivially on the set of indexes. That means, at time 0 and the masses are collinear, on a fixed line l E. It is coercive only when ! 62 Z. In this case the Lagrangian solution cannot be a minimum, while it can the relative equilibrium associated with the Euler configuration. The 2 1-choreography symmetry Consider the group of order 2 acting as follows: (g) D 1, (g) D 1 (that is, the translation of half-period) and (g) D (1; 2) (that is, (g)(1) D 2, (g)(2) D 1,
n-Body Problem and Choreographies, Table 1 Planar symmetry groups with trivial core Name jGj Type R Trivial 1 yes Line 2 yes 2 1-choreography 2 yes Isosceles 2 yes Hill 4 yes 3-choreography 3 yes Lagrange 6 yes C6 6 no D6 6 no D12 12 no
Action type trans. dec. 1C1C1 brake 1C1C1 cyclic 2C1 brake 2C1 dihedral 2C1 cyclic 3 dihedral 3 cyclic 3 dihedral 3 dihedral 3
RCP yes (no) yes no no yes no yes yes no
HGM yes no no yes no yes yes no no no
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n-Body Problem and Choreographies, Figure 14 The poset of symmetry groups for the planar 3-body problem
n-Body Problem and Choreographies, Figure 15 Action levels for the line symmetry
and (g)(3) D 3). That is, it is a half-period choreography for the bodies 1 and 2. It can be extended to rotating frames and coercive for a suitable choice of ! ¤ 0; 1 mod 2. The Euler’s orbit with k D 1 and the Hill’s orbits with k D ˙1 are equivariant for the 2 1-choreography symmetry, while the Euler’s orbit with k D 0 is not equivariant for this symmetry. In Figures 15 and 16 Euler 1 represents the action levels on the Euler’s orbit with k D 1, Hill 1–2 the ones on the Hill’s with k D ˙1. The isosceles symmetry The isosceles symmetry can be obtained as follows: the group is of order 2, generated by h; (h) is a reflection
n-Body Problem and Choreographies, Figure 16 Action levels for the 2 1-choreography symmetry
in the time circle T , (h) is a reflection along a line l in E, and (h) D (1; 2) as above. The constraint is therefore that at time 0 and the 3-body configuration is an isosceles triangle with one vertex on 1 (the third). Proposition 4 For every ! … Z and every choice of masses the minimum for the isosceles symmetry occurs in the relative equilibrium motion associated to the Lagrange configuration. The Euler–Hill symmetry Now consider the symmetry group with a cyclic generator r of order 2 (i. e. (r) D 1) and a time reflection h (i. e. (h) is a reflection of T ) given by (r) D 1,
n-Body Problem and Choreographies
(r) D (1; 2), (h) is a reflection and (h) D (). It contains the 2 1-choreography (as the subgroup ker det()), the isosceles symmetry (as the isotropy of /2 2 T ) and the line symmetry (as the isotropy of 0 2 T ) as subgroups. Proposition 5 The minimum of the Euler–Hill symmetry is not homographic, provided that the angular velocity ! is close to 0.5 and the values of the masses are close to 1. The choreography symmetry The choreography symmetry is given by the group C3 of order 3 acting trivially on the plane E, by a rotation of order 3 in the time circle T and by the cyclic permutation (1; 2; 3) of the indices. Proposition 6 For every ! the minimal choreography of the 3 body problem is a rotating Lagrange configuration. The Lagrange symmetry The Lagrange symmetry group is the extension of the choreography symmetry group by the isosceles symmetry group. Thus, it is a dihedral group of order 6, the action is of type R. Hence, the relative equilibrium motions associated to the Lagrange configuration are admissible motions for this symmetry and, again, the minimizer occurs in the relative equilibrium motion associated to the Lagrange configuration. The Chenciner–Montgomery symmetry group and the eights There are three symmetry groups (up to change of coordinates) that yield the Chenciner–Montgomery figure eight orbit: they are the only symmetry groups which do not extend to the rotating frame and we have already described them in Subsect. “The Eight Shaped Three-Body Solution”. One can prove that all G-equivariant trajectories have vanishing angular momentum, whenever the group is not of type R. Moreover, we were able to partially answer to the open question (posed by Chenciner) whether their minimizers coincide or not: for two of them (D6 and D12 ) the minimizer is necessarily the same. Space Three-body Problem Based on the classification of planar groups, by introducing a natural notion of space extension of a planar group, Ferrario gave in [51] a complete answer to the classification problem for the three-body problem in the space and at the same time to determine the resulting minimizers and describe its more relevant properties.
n-Body Problem and Choreographies, Table 2 Space extensions of planar symmetry groups with trivial core Name Trivial Line Isosceles Hill 3-choreography Lagrange D6 D12
Extensions C1 LC; , L;C 2 2 C; H2 , H2;C H4C; , H4;C C3C , C3 LC;C , LC; , L;C 6 6 6 C; ;C D6 , D6 D;C 12
Theorem 5 Symmetry groups not bound to collisions, not fully uncoercive and not homographic are, up to a change of rotating frame, either the three-dimensional extensions of planar groups (if trivial core) listed in table below or the vertical isosceles triangle (if non-trivial core). The next theorem is the answer to the natural questions about collisions and description of some main features of minimizers. Theorem 6 Let G be a symmetry group not bound to collisions and not fully uncoercive. Then 1. Local minima of A! do not have collisions. 2. In the following cases minimizers are planar trajectories: ;C a) If G is not of type R: D6C; , D6;C and D12 (and then G-equivariant minimizers are Chenciner– Montgomery eights). b) If there is a G-equivariant minimal Lagrange rotating solution: C1 , H2C; , C3C , LC;C , LC; (and then 6 6 the Lagrange solution is of course the minimizer). c) If the core is non-trivial and it is not the vertical isosceles (and then minimizers are homographic). 3. In the following cases minimizers are always non-planar: a) The groups L;C and C3 for all ! 2 (1; 1) C 6Z, 6 ! ¤ 0 (the minimizers for L;C are the elements of 6 Marchal family P12 , and minimizers of C3 are a less0 2 ). symmetric family P12 b) The extensions of line and Hill–Euler type groups, for on open subset of mass distributions and angular speeds !: LC; , L;C , H4C; and H4;C (for L;C 2 2 2 this happens also with equal masses). c) The vertical isosceles for suitable choices of masses and !. 2 Highly likely they are not distinct families: this is the recurring phenomenon of “more symmetries than expected” in n-body problems.
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n-Body Problem and Choreographies, Figure 17 and LC; , plotted in the fixed frame Non planar minimizers for the groups L;C 6 2
n-Body Problem and Choreographies, Figure 18 Examples for Theorem 10 close to an integer that divides n
Minimizing Properties of Simple Choreographies In the space of symmetric (choregraphical) loops, the action takes the form A(x) D
n1 Z 1X 1 j x˙ (t C h)j2 dt C 2 2 0 hD0
n1 Z X h;l D0
0
dt : jx(t C l) x(t C h)j˛
h¤l
A natural question concerns the nature of the minimizers under the sole choregraphical constraint. Unfortunately, the bare minimization among choreographical loops gives returns only trivial motions:
Theorem 7 For every ˛ 2 RC and d 2, the absolute minimum of A on is attained on a relative equilibrium motion associated to the regular n-gon. This theorem extends some related result for the italian symmetry by Chenciner and Desolneux [27], and the results in Sect. “The 3-Body Problem”. The proof is based on a (quite involved) convexity argument together with the analysis of some spectral properties related to the choreographical constraint. Now, in order to find nontrivial minimizers, we look at the same problem in a rotating frame. In order to take into account of the Coriolis force, the new action functional has to contain a gyroscopic term: A(y) D
1 2
Z
2 0
j y˙(t) C J! y(t)j2dt
n-Body Problem and Choreographies
C
n1 Z dt 1 X 2 : 2 jy(t) y(t C h)j˛ 0 hD1
Consider the function h : RC ! N, h(!) D minn2N (! n)2 /n2 and let ! D 43 . The same technique used for the inertial system extends to rotating systems having small angular velocity; this gives the following result. Theorem 8 If ! 2 (0; ! )nf1g, then the action attains its minimum on a circle with minimal period 2 and radius depending on n, ˛ and !.
undergoes some transitions (for example it has to pass from relative equilibrium having different winding numbers). This scenario suggests the presence of other critical points, such as local minimizers or mountain pass. This was indeed discovered numerically in [12] and then proved by a computer assisted proof in [7]. To begin with, let us look at the following figure where the values of the action functional A! on the branches of circular orbits L!k are plotted:
When ! is Close to an Integer The situation changes dramatically when ! is close to some integer. To understand this phenomenon, let us first check the result of the minimization procedure when ! is an integer: Proposition 7 (1) If ! D n, then the action has a continuum of minimizers. (2) If ! D k, coprime with n, then the action does not achieve its infimum (escape of minimizing sequences). (3) If ! D k and k divides n, then the action does not achieve its infimum (clustering of minimizing sequences). As a consequence, we have the following result: Theorem 9 Suppose that n and k are coprime. Then there exist D (˛; n; k) such that if ! 2 (k ; k C ) the minimum of the action is attained on a circle with minimal period 2/k that lies in the rotating plane with radius depending on n, ˛ and !. An interesting situation appears when the integer closest to the angular velocity is not coprime with the number of bodies. In this case we prove that the minimal orbit it is not circle anymore, as the following theorem states. Theorem 10 Take k 2 N and g:c:d:(k; n) D k˜ > 1, k˜ ¤ n. Then there exists D (˛; n; k) > 0 such that if ! 2 (k ; k C )nfkg the minimum of the action is attained on a planar 2-periodic orbit with winding number k which is not a relative equilibrium motion. Also, it has be noticed that, for large number of bodies and angular velocities close to the half on an integer, the minimizer apparently is not anymore planar. Mountain Pass Solutions for the Choreographical 3-Body Problem The discussion carried in the previous section shows that, as the angular velocity varies, the minimizer’s shape must
The analysis of this picture suggests the presence of critical points different from the Lagrange motions. Indeed, let us take the angular velocity ! D 1:5: in this case there are two distinct global minimizers, the uniform circular motions with minimal period 2 and , lying in the plane orthogonal to the rotation direction. This is a well known structure in Critical Point Theory, referred as the Mountain Pass geometry and gives the existence of a third critical point, provided the Palais–Smale condition is fulfilled, with an additional information on the Morse index. Next theorem follows from the application of the Mountain Pass Theorem to the action functional A3/2 : Theorem 11 There exists a (possibly collision) critical point for the action functional A3/2 with Morse index smaller then 1 and distinct from any Lagrange motion. Once the existence of a Mountain Pass critical point was theoretically established, we studied its main properties in order to understand whether it belonged to some known families of periodic trajectories. To this aim we applied the bisection algorithm proposed in [13] to approximate the maximal of a locally optimal path joining the two strict global minimizers, finding in this a good numerical candidate. Of course, there could be a gap between the mountain pass solution whose existence is ensured by Theorem 11
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n-Body Problem and Choreographies, Figure 19 Non planar minimizers of the action with angular velocities close to the half on an integer
and the numerical candidate found by applying the bisection algorithm. In order to fill this gap proved the existence of an actual solution very close to the numerical output of the Mountain Pass algorithm. The argument was based upon a fixed point principle and involved a rigorous computer assisted proof. As a consequence, we obtained the existence of a new branch of solution for the spatial 3-body problem (see Fig. 8). Here are some relevant features of the new solution: the orbit is not planar, its winding number with respect, for instance, to the line x D 0:2, y D 0 is 2 and it does not intersect itself. A natural question is whether this solution can be continued as a function of the parameter !. We were able to extend the numerical–rigorous argument to cover a full interval of values of the angular velocity, providing the existence of a full branch of solution. Theorem 12 There exists a smooth map B(!) giving the (locally unique, up to symmetries) branch of solution of the choreographical 3-body problem for all ! 2 [1; 2], starting at the Mountain Pass solutions for ! D 1:5. A natural question is whether this mountain pass branch meets one of the known branches of choreographical periodic orbits: either one of the Lagrange or Marchal’s P12 (described by Marchal in [62]) families. Surprisingly enough, it turns out from further numerical computation that this branch does not emanates from any of Lagrange motions L1 or L2 ; apparently, it emanates from the branch of P12 solutions. The details of the bifurcation diagram are depicted in Fig. 8. Generalized Orbits and Singularities The existence of a G-equivariant minimizer of the action is a simple consequence of the direct method of the Calculus
of Variations. These trajectories, however, solve the associated differential equations only where they are far from the singular set. Indeed, generally speaking, G-equivariant minimizers may present singularities, even though the set of collision instants must clearly be of vanishing Lebesgue measure. A first natural question concerns the number of possible collision instants. Here below we follow [15]. Definition 5 A path x : (a; b) ! X is called a locally minimizing solution of the N-body problem if, for every t0 2 (a; b), there exists ı > 0 such that the restriction of x to [t0 ı; t0 C ı] is a local minimizer for the action with the same boundary conditions. A path which is the uniform limit of a sequence of locally minimal solutions will be called a generalized solution. We remark that: The definition can be modified in a obvious manner to include symmetries. Equivariant minimizers are generalized solutions. If the potential is of class C 2 outside collisions then every non collision solution is a generalized solution. The generalized solutions possess an index: the minimal number of intervals I j needed to cover (a; b) such that the restriction of x to I j is a local minimizer for the action. There is a natural notion of maximal existence interval for generalized solutions, even without the unique extension property. Singularities and Collisions Generally speaking we are dealing with systems of the form M x¨ D rU(t; x) ;
t 2 (a; b) ;
M i j D mi ıi j ;
n-Body Problem and Choreographies
n-Body Problem and Choreographies, Figure 20 Mountain pass solution with angular velocities close to the half on an integer 12
7.8 mountain pass L1 L2
10
mountain pass P12 L2
7.75 7.7
8
7.65 6
7.6
4
7.55
2 0
a
7.5 0
0.5
1
1.5
2
2.5
3
7.45 0.93 0.94 0.95 0.96 0.97 0.98 0.99
b
1
1.01
n-Body Problem and Choreographies, Figure 21 Action levels for the Lagrange and the mountain pass solution in the 3-body problem. On the x-axes the angular velocity varies in the interval [0; 3). The left picture focuses on the bifurcation from the P12 family
where the potential U possesses a singular set (a; b) , in the sense that [U0]
lim U(t; x) D C1 ;
x!
uniformly in t :
The set is clearly the collision set. We assume to be a cone in Rnd : x 2 H) x 2 : Definition 6 We say that a generalized solution x on the interval (a; b), has a singularity at t < C1 if
The Theorems of Painlevé and Von Zeipel In the usual mathematical language a classical solution x on the interval (a; b), has a singularity at t < C1 if it is not possible to extend x as a (classical) solution to a larger interval (a; t C ı). A classical results relates singularities of solutions with those of the potential. Theorem 13 (Painlevé’s Theorem) Let x¯ be a classical solution for the n-body dynamical system on the interval [0; t ). If x¯ has a singularity at t < C1, then lim U(x¯ (t)) D C1 :
t!t
lim sup U(t; x(t)) D C1 : t!t
When t 2 (a; b) we will say that x has an interior singularity at t D t , while when t D a or t D b (when finite) we will talk about a boundary singularity.
Painlevé’s Theorem does not necessarily imply that a collision (i. e. that is a singularity such the configuration has a definite limit) occurs when there is a singularity at a finite time (on this subject we refer to [66,75,76]). The next
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result has been stated by Von Zeipel in 1908 and definitely proved by Sperling in 1970.
With these assumptions, we can extend then Von Zeipel’s Theorem to generalized solutions:
Theorem 14 (Von Zeipel’s Theorem) If x¯ is a classical solution for the n-body dynamical system on the interval (a; t ) with a singularity at t < C1 and
Theorem 15 Let x¯ be a generalized solution on the bounded interval (a; b). If x¯ is bounded on the whole interval (a; b) then the singularities of x¯ are collisions.
lim kx¯ (t)k < C1 ;
t!t
Asymptotic Estimates at Collisions
then x¯ (t) has a definite limit configuration x a s t tends to t .
From now on, we require our potential U to satisfy some one-side homogeneity conditions:
Von Zeipel’s Theorem and the Structure of the Collision Set To the aim of extending the Von Zeipel’s Theorem to our notion of generalized solutions, we need to introduce some assumptions on the potential U and its singular set . [ D V ; (5) 2M
[U1] there exists C1 0 such that for every (t; x) 2 ((a; b) Rnd n) ˇ ˇ ˇ ˇ @U ˇ C1 U(t; x) : ˇ (t; x) ˇ ˇ @t [U2] there exist ˛ 2 (0; 2), > 0 and C2 0 such that
Rk
and M where the V ’s are distinct linear subspaces of is a finite set; observe that the set is a cone, as required before. We endow the family of the V ’s with the inclusion partial ordering and we assume the family to be closed with respect to intersection (thus we are assuming that M is a semilattice of linear subspaces of R k : it is the intersection semilattice generated by the arrangement of maximal subspaces V ’s). With each 2 we associate \ V : () D minf : 2 V g i. e. ; V() D 2V
Fixed 2 M we define the set of collision configurations satisfying D f 2 : () D g and we observe that this is an open subset of V and its closure is V . We also notice that the map ! dim(V() ) is lower semicontinuous. We denote by p the orthogonal projection onto V and we write x D p (x) C w (x) ; where, of course, w D I p . We assume that, near the collision set, the potential depends, roughly, only on the projection orthogonal to the collision set: more precisely we assume [U5] For every 2 ; there is " > 0 such that U(t; x) U(t; w() (x)) D W(t; x) 2 C 1 ((a; b) B" ()) ; where B" () D fx : jx j < "g :
One Side Conditions on the Potential and Its Radial Derivative
rU(t; x) x C ˛U(t; x) C2 jxj U(t; x) ; whenever jxj is small : We observe that when U is homogeneous functions of degree ˛, the equality in condition (U2) is attained with C2 D 0; this assumption is satisfied also when we take into account potentials of the form U(t; x) D U˛ (x) C Uˇ (x) where U˛ is homogeneous of degree ˛, Uˇ is homogeneous of degree ˇ, 0 < ˇ < ˛, and U˛ is positive. Isolatedness of Collisions Instants In order to study the behavior of the solution at a collision, we firs perform a translation in such a way that the collision holds at the origin; next we introduce polar coordinates: p p x r D Mx x D I ; s D ; r where s 2 E D fx : I 2 (x) D Mx x D 1g belongs the ellipsoid of all the configurations having unitary moment of inertia. [U3h ] there exists a function U˜ defined on (a; b) (E n), such that (on compact subsets of ((a; b) E n)): ˜ s) ; lim r˛ U(t; rs) D U(t;
r!0
[U3l ] there exists a function U˜ defined on (a; b) (E n), such that (on compact subsets of ((a; b) E n)): ˜ s) ; lim U(t; x) C M(t) log jxj D U(t; jxj!0
n-Body Problem and Choreographies
The following regularity theorem is proved in [15] based on a suitable variant of Sundman’s inequality and the asymptotic analysis of possible collisions outlined in the sequel. Theorem 16 Let x : (a; b) ! X be a generalized solution of the N-body problem, with a potential U satisfying [U0– U3]. Then collision instants are isolated in (a; b). Furthermore, if (a; b) is the finite maximal extension interval of x and no escape in finite time occurs then the number of collision instants is finite. Some remarks are in order: A generalized solution does not solve the Euler–Lagrange equation in a distributional sense (the force field can not be locally integrable). Moreover its action needs not to be finite: one needs to prove it. a priori our solution can have a huge set of collision instants (more than countable, but null measure);
the ˛ is bounded variation and d 2 ˛ 1C˛ 2 r˙j˙sj C1 r˛ U(t; rs) ˛ (r; s) r dt 2 r˙ C C2 r˛C U(t; rs) : r Generalized Sundman–Sperling Estimates The asymptotic analysis along a single collision (total or partial) trajectory goes back, in the classical case, to the works by Sundman ([84]), Wintner ([89]) and, in more recent years by Sperling, Pollard, Saari, Diacu and other authors (see for instance [40,47,75,76,79,83]). Now we are in a position to extend such estimate also to generalized solutions. Theorem 18
there is no a priori bound on the total action on the whole interval (a; b), nor on the energy;
The following asymptotic estimates hold: p 2 r ( t) 2C˛ (r jtj log(jtj)) 2˛ 2 1 ¨ 2 ( t) 2C˛ I K U 4 2˛ (2 C ˛)2 ( log jtj) :
we have very weak assumptions on the potential and only a one side inequality on the radial component
Let s be the normalized configuration of the colliding cluster s D x/r. Then
there may be accumulation of partial collisions at a collision having more bodies;
the theorem extends to the logarithmic potentials.
lim r2 j˙sj2 D 0
t!0
lim U(s(t)) D b < C1
t!0
Conservation Laws Even though they can not satisfy the differential equations in a distributional sense, generalized solutions satisfy a number of conservation laws. Theorem 17 Let x be a generalized solutions on (a; b). Then The action A(x; [a; b]) on (a; b) is finite The energy h is bounded and belongs to the Sobolev space W 1;1 ((a; b); R). Lagrange–Jacobi inequality holds in the sense of measures: 1¨ ¯ I(x¯ (t)) 2h(t)(x¯ (t)) C (2 ˛)U(t; x(t)) 2 ¯ ; 8t 2 (a; b) : C2 j x¯ j U(t; x) A monotonicity formula holds (extending Sundman’s inequality): let us consider the angular energy ˛ (r; s) :D r˛
1 2 2 r j˙sj U(t; rs) 2
Moreover there exists the angular blow-up, that is angular scaled family (s(t)) is precompact for the topology of uniform convergence on compact sets of Rnf0g. As a further consequence we have the vanishing of the total angular momentum. In the same setting of the Theorem, assume that the potential U verifies the further assumption: ˜ s). [U4] limr!0 r˛C1 rT U(t; x) D rT U(t; Then we have lim dist(C b ; s(t)) D lim inf js(t) s¯j D 0 ;
t!t 0
t!t 0 s¯2C b
where C b is the set of central configurations for U˜ at level b. Dissipation and McGehee Coordinates The main tool in proving the asymptotic estimates follow is the monotonicity formula, which is somehow equivalent to Sundman’s inequality. To fix our mind, let us think to a homogeneous potential U˛ . A possible way to to see this
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dissipation is to perform the change of variables (reminiscent of the McGehee coordinates):
Dr
2˛ 4
;
0 D
2˛ 2C˛ 4 r0 4 r
to obtain the action functional depending on ( ; s); Z 4 2 0 2 1 A( ; s) D ( ) 2 2˛ 0 C 2 12 js 0 j2 C U˛ (s) ˇ d ; where ˇ :D 2(2 C ˛)/(2 ˛) > 2. Here we have reparametrized the time as dt D r
2C˛ 2
d ;
vDr
t 2R;
where D ( i ) i2k is a central configuration with k bodies. Proposition 8 The sequences x n and (dx n )/(dt) converge to the blow-up x¯ and its derivative x˙¯ respectively, in the H 1–topology. Moreover x¯ is a minimizing trajectory in the sense of Morse. Z T ¯ 0]dt : [L(x¯ C ') L(q)
Logarithmic Type Potentials
This is the coupling between a Duffing equation and the N-body angular system. When is increasing, it acts as a viscosity coefficient on the angular Lagrangian. A classical framework for the study of collisions is given by the McGehee coordinates [65] (here and below we assume, for simplicity of notations, all the masses be equal to one): ˛/2
i 2k;
for any compactly supported variation '.
0
r D jxj D I
x¯ i (t) D jtj2/(2C˛) i ;
0
one proves that Z 1 2C˛ r 2 dt D C1 : D
1/2
The blow-up x¯ is parabolic: where a parabolic collision trajectory for the cluster k is the path
;
(y s) ;
sD
x r
uDr
lim j x¯ n (t)j D lim
n !0
n !0
(y y ss) :
r(n t) p n t 2M(0) log(n t)
p t 2M(0) log(n t) D C1
; ˛/2
The equation of motions become (here 0 denotes differentiation with respect to the new time variable ): ˛ r0 D rv ; v 0 D v 2 C juj2 ˛U˛ (s) ; s 0 D u ; ˛ 2 0 1 vu juj2 s C ˛U˛ (s)s C rU˛ (s) : u D 2 It is worthwhile noticing that the three last equations do not depen on r and hence are defined also for r D 0. In this constext, the monotonicity formula means that v is a Lyapunov function for the system.
for every t > 0. On the other, hand, looking at the differential equation, the (right) blow-up should be: ¯ :D t¯s ; q(t)
i 2k;
where s¯ is a central configuration for the system limit of a sequence s(n ) where (n )n is such that n ! 0. This limiting function is the pointwise limit of the normalized sequence x¯ n (t) :D
1 x¯ (n t) : p n 2M(0) log n
Unfortunately this path is not locally minimal for the limiting problem, indeed since, the sequence (x¨¯ n )n converges to 0 as n tends to C1, and hence this blow-up minimizes only the kinetic part of the action functional.
Blow-ups For every > 0 let x (t) D 2/(2C˛) x(t) : If fn gn is a sequence of positive real numbers such that s(n ) converges to a normalized configuration s¯, then 8t 2 (0; 1) : lim s(n t) D lim s(n ) D s¯ : n!1
It is not possible to define a blow-up suitable for logarithmic type potentials. Indeed, the natural scaling should be ¯ (n t), which x¯ n (t) :D 1 n x pdoes not converge, since, by the asymptotics jx(t)j ' jtj log jtj, we have:
n!1
Hence the rescaled sequence will converge uniformly to the blow-up of x(t) relative to the colliding cluster k n (in t D 0).
Absence of Collision for Locally Minimal Paths As a matter of fact, solutions to the Newtonian n-body problem which are minimals for the action are, very likely, free of any collision. This fact was observed by the construction of suitable local variation arguments for the 2 and 3-body cases by Serra and Terracini [81,82]. The 4body case was treated afterward by Dell’Antonio ([35],
n-Body Problem and Choreographies
with a non completely rigorous argument) and then by A. Venturelli [87]. In general, the proof goes by the sake of the contradiction and involves the construction of a suitable variation that lowers the action in presence of a collision. A recent breakthrough in this direction is due of the neat idea, due to C. Marchal, [63], of averaging over a family of variations parametrized on a sphere. The method of averaged variations for Newtonian potentials has been outlined and exposed by Chenciner [25], and then fully proved and extended to ˛-homogeneous potentials and various constrained minimization problems by Ferrario and Terracini in [52]. This technique can be used in many of the known cases to prove that minimizing trajectories are collisionless. Of course, in some specific situations, other arguments can be useful, such as level estimates, on the infimum of the action on colliding paths [17,20,21,25]. However, these argument require global conditions on the potentials and can not be applied in the present setting, where we work under local assumptions about the singularities. We are in a position to prove the absence of collisions for locally minimal solutions when the potentials have quasi-homogeneous or logarithmic singularities. The first case is simpler, because one can take advantage the blow-up technique already exploited in [52]. On the other hand, when dealing with logarithmic potentials, the blowup technique is no longer available and we conclude proving directly some averaging estimates that can be used to show the nonminimality of large classes of colliding motions. The following results are exposed in [15]. Quasi-Homogeneous Potentials Let U˜ be the C 1 function defined on (a; b)(R k n) in the following way: ˜ rx/jxj) : ˜ x) D jxj˛ lim r˛ U(t; U(t; r!0
˜ With a slight abuse of notation, we denote U(x) D ˜ ; x). U˜ is homogeneous of degree ˛. U(t ? , [U6] there is a 2-dimensional linear subspace of V() say W, where U˜ is rotationally invariant:
˜ i w) D U(w) ˜ U(e ;
8w 2 W ;
Theorem 19 In addition to [U0], [U1], [U2 h ], [U3 h ], [U4 h ], [U5], assume that, for all 2 [U6] and [U7] hold. Then generalized solutions do not have collisions at the time t . Remark 2 As our potential U˜ is homogeneous of degree ˛ the function '(x) D U˜ 1/˛ (x) is a non negative, homogeneous of degree one function, having now as zero set. In most of our applications ' will be indeed a quadratic form. Assume that ' 2 splits in the following way: ' 2 (x) D KjW (x)j2 C ' 2 (W ? (x)) for some positive constant K. Then [U6] and [U7] are satisfied. Indeed, denoting w D W (x) and z D x w we have, for every ı 2 W, ' 2 (x C ı) D Kjw C ıj2 C ' 2 (z) ˇ ˇ ˇ '(x) ' 2 (z) '(w) ˇˇ2 ˇ D Kˇ wC ı ˇ C K 2 jıj2 '(w) '(x) ' (x) ˇ ˇ2 ˇ '(x) ˇ '(w) K ˇˇ wC ıˇ '(w) '(x) ˇ '(w) 2 '(x) D' wC ı ; '(w) '(x) which is obviously equivalent to [U7]. ˜ Proposition 9 Assume U(x) D Q˛/2 (x) for some non negative quadratic form Q(x) D hAx; xi. Then assumptions [U6] and [U7] are satisfied whenever W is included in an eigenspace of A associated with a multiple eigenvalue. Given two potentials satisfying [U6] and [U7] for a common subspace W, their sum enjoys the same properties. On the other hand, the class of potentials satisfying [U6] and [U7] is not stable with respect to the sum of potentials. In order to deal with a class of potentials which is closed with respect to the sum, we introduce the following variant of the last Theorem. Theorem 20 In addition to [U0], [U1], [U2h], [U3h], [U4h], [U5], assume that U˜ has the form
8 2 [0; 2] ;
Rk
˜ U(x) D
N X
K ; (dist(x; V ))˛ D1
and ı 2 W there holds: [U7] for every x 2 1/˛ ˜ U(W (x)) ˜ ˜ W (x) U(x C ı) U ˜ U(x) 1/˛ ˜ U(x) C ı ; ˜ W (x)) U(
where K are positive constants and V is a family of linear subspaces, with codim(V ) 2, for every D 1; : : : ; N. Then locally minimizing trajectories do not have collisions at the time t .
where W denotes the orthogonal projection onto W.
These two theorems extend to logarithmic potentials.
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Neumann Boundary Conditions and G-equivariant Minimizers Our analysis allows to prove that minimizers to the fixedends (Bolza) problems are free of collisions: indeed all the variations of our class have compact support. However, other type of boundary conditions (generalized Neumann) can be treated in the same way. Indeed, consider a trajectory which is a (local) minimizer of the action among all paths satisfying the boundary conditions x(0) 2 X 0 ;
x(T) 2 X 1 ;
where X 0 and X 1 are two given linear subspaces of the configuration space. Consider a (locally) minimizing path x¯ : of course it has not interior collisions. In order to exclude boundary collisions we have ensure that the class of variations preserve the boundary conditions. This can be achieved by imposing assumptions [U6] and [U7] to be fulfilled also by the restriction of the potential to the boundary subspaces X i . The analysis of boundary conditions was a key point in the paper [52], where symmetric periodic trajectories where constructed by reflections about given subspaces. Our results can be used to prove the absence of collisions also for G-equivariant (local) minimizers, provided the group G satisfies the Rotating Circle Property defined in Definition 3. Hence, existence of G-equivariant collisionless periodic solutions can be proved for the wide class of symmetry groups described in [14,52], for a much larger class of interacting potentials, including quasi-homogeneous and logarithmic ones. On the other hand, our results can be applied to prove that G-equivariant minimals are collisionless for many relevant symmetry groups violating the rotating circle property, such as the groups of rotations recently introduced in [50,51]. The Standard Variation In order to prove that local minimizers are free of collisions we are going to make use of the following class of variations: Definition 7 The standard variation associated to ı 2 ) and T is defined as 8 if 0 jtj T jıj ˆ 0, 2 R3 nf0g and for every circle S Rd with center in 0, Z ˜ S) D 1 S(; ı)dı S(; jSj S Z 2 1 ˚( )d < 0 : D jj1˛/2 jıj1˛/2 2 0 Consider D x i x j and ı ranging in a circle. Then the above inequality implies the following principle, a generalization of Marcha’s statement in [25]: “it is more convenient (from the point of view of the integral of the potential on the time line) to replace one of the point particles with a homogeneous circle of same mass and fixed radius which is moving keeping its center in the position of the original particle.” Future Directions This article mainly focuses on the variational approach to the search of selected trajectories to the n-body problem. In our examples, the masses can very well be equal: hence the problem can not be regarded as a small perturbation of a simpler one and a full picture of the dynamics is out of reach. In contrast, the planetary n-body problem deals with systems where one of the bodies (a “star”) is much heavier than the others (the “planets”) and can be seen as a perturbation of a decoupled systems of one center problems. A small parameter " represents the order of the ratio between the mass of the planets and that of the star. The system is then called nearly integrable as it can be associated with a Hamiltonian of the form: H(I; ') D h(I) C " f (I; ') ; where I 2 R N and ' 2 T N (N is the number of degree of freedom) and " is a small parameter. In the three body problem we have N D 4, but the integrable limit possesses motions lying on T 2 (the product of two Keplerian orbits) so the integrable limit depends on less action variables than the number of degrees of freedom. For this reason the system is properly degenerate. The integrable limit possesses only quasi-periodic motions: in [56], the question whether these motions survive for positive values of " is settled. KAM Theory deals with the problem of persistence of such invariant tori but can not be directly applied to the planetary n-body problem, for its degeneracy. Nevertheless, the
existence of invariant tori has been recently achieved in the planar and spatial three body problem [8,16,18,57,78] and in the planetary many body problem [19,48]. The future researches will extend these results to some non perturbative settings, finding both regular motions, such as periodic, quasi–periodic or quasi–periodic trajectories and irregular, chaotic ones, through the application of suitable variational methods taking into account of collision trajectories.
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Navier–Stokes Equations: A Mathematical Analysis
Navier–Stokes Equations: A Mathematical Analysis GIOVANNI P. GALDI University of Pittsburgh, Pittsburgh, USA Article Outline Glossary Definition of the Subject Introduction Derivation of the Navier–Stokes Equations and Preliminary Considerations Mathematical Analysis of the Boundary Value Problem Mathematical Analysis of the Initial-Boundary Value Problem Future Directions Acknowledgment Bibliography Glossary Steady-State flow Flow where both velocity and pressure fields are time-independent. Three-dimensional (or 3D) flow Flow where velocity and pressure fields depend on all three spatial variables. Two-dimensional (or planar, or 2D) flow Flow where velocity and pressure fields depend only on two spatial variables belonging to a portion of a plane, and the component of the velocity orthogonal to that plane is identically zero. Local solution Solution where velocity and pressure fields are known to exist only for a finite interval of time. Global solution Solution where velocity and pressure fields exist for all positive times. Regular solution Solution where velocity and pressure fields satisfy the Navier–Stokes equations and the corresponding initial and boundary conditions in the ordinary sense of differentiation and continuity. At times, we may interchangeably use the words “flow” and “solution”. Basic notation N is the set of positive integers. R is the field of real numbers and RN , N 2 N, is the set of all N-tuple x D (x1 ; : : : ; x N ). The canonical base in R N is denoted by fe 1 ; e 2 ; e 3 ; : : : ; e N g fe i g. For a; b 2 R, b > a, we set (a; b) D fx 2 R : a < x < bg, [a; b] D fx 2 R : a x bg, [a; b) D fx 2 R : a x < bg and (a; b] D fx 2 R : a < x bg. ¯ we indicate the closure of the subset A of R N . By A
A domain is an open connected subset of R N . Given a second-order tensor A and a vector a, of components fA i j g and fa i g, respectively, in the basis fe i g, by a A [respectively, A a] we mean the vector with components A i j a i [respectively, A i j a j ]. (We use the Einstein summation convention over repeated indices, namely, if an index occurs twice in the same expression, the expression is implicitly summed over all possible p values for that index.) Moreover, we set jAj D A i j A i j . If h(z) fh i (z)g is a vector field, by r h we denote the second-order tensor field whose components fr hg i j in the given basis are given by f@h j /@z i g. Function spaces notation If A R N and k 2 N [ f0g, ¯ )] we denote the class of by C k (A) [respectively, C k (A functions which are continuous in A up to their kth derivatives included [respectively, are bounded and uniformly continuous in A up to their kth derivatives included]. The subset of C k (A) of functions vanishing outside a compact subset of A is indicated by C0k (A). If u 2 C k (A) for all k 2 N [ f0g, we shall write ¯) u 2 C 1 (A). In an analogous way we define C 1 (A and C01 (A). The symbols L q (A), W m;q (A), m 0, 1 q 1, denote the usual Lebesgue and Sobolev spaces, respectively (W 0;q (A) D L q (A)). Norms in L q (A) and W m;q (A) are denoted by k kq;A , k km;q;A . The trace space on the boundary, @A, of A for functions from W m;q (A) will be denoted by W m1/q;q (@A) and its norm by k km1/q;q;@A . By D k;q (A), k 1, 1 < q < 1, we indicate the homogeneous Sobolev space of order (m; q) on A, that is, the class of functions u that are (Lebesgue) locally integrable in A and with Dˇ u 2 L q (A), jˇj D k, where ˇ ˇ ˇ Dˇ D @jˇ j /@x1 1 @x2 2 : : : @x NN , jˇj D ˇ1 C ˇ2 C C ˇ N . For u 2 D k;q (A), we put 0 juj k;q;A D @
X Z jˇ jDk
A
11/q j Dˇ u jq A
:
Notice that, whenever confusion does not arise, in the integrals we omit the infinitesimal volume or surface elements. Let D(A) D f' 2 C01 (A) : div ' D 0g : q
By L (A) we denote the completion of D(A) in the norm k kq . If A is any domain in R N we have L2 (A) D L2 (A) ˚ G(A), where G(A) D fh 2 L2 (A) : h D r p ; for some p 2 D1;2 (A)g; (see Sect. III.1 in [31]). We denote by P the orthogonal projection operator from L2 (A) onto L2 (A). By D1;2 0 (A) we mean the completion of D(A) in the norm j
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j1;2;A . D1;2 0 (A)R is a Hilbert space with scalar product [v 1 ; v 2 ] :D A (@v 1 /@x i ) (@v 2 /@x i ). Furthermore, 1;2 1;2 D0 (A) is the dual space of D0 (A) and h; iA is the associated duality pairing. If g fg i g and h fh i g are vector fields on A, we set Z (g; h)A D g i hi ; A
whenever the integrals make sense. In all the above notation, if confusion will not arise, we shall omit the subscript A. Given a Banach space X, and an open interval (a; b), we denote by L q (a; b; X) the linear space of (equivalence classes of) functions f : (a; b) ! X whose X-norm is in L q (a; b). Likewise, for r a nonnegative integer and I a real interval, we denote by C r (I; X) the class of continuous functions from I to X, which are differentiable in I up to the order r included. If X denotes any space of real functions, we shall use, as a rule, the same symbol X to denote the corresponding space of vector and tensor-valued functions. Definition of the Subject The Navier–Stokes equations are a mathematical model aimed at describing the motion of an incompressible viscous fluid, like many common ones as, for instance, water, glycerin, oil and, under certain circumstances, also air. They were introduced in 1822 by the French engineer Claude Louis Marie Henri Navier and successively re-obtained, by different arguments, by a number of authors including Augustin-Louis Cauchy in 1823, Siméon Denis Poisson in 1829, Adhémar Jean Claude Barré de Saint-Venant in 1837, and, finally, George Gabriel Stokes in 1845. We refer the reader to the beautiful paper by Olivier Darrigol [17], for a detailed and thorough analysis of the history of the Navier–Stokes equations. Even though, for quite some time, their significance in the applications was not fully recognized, the Navier– Stokes equations are, nowadays, at the foundations of many branches of applied sciences, including Meteorology, Oceanography, Geology, Oil Industry, Airplane, Ship and Car Industries, Biology, and Medicine. In each of the above areas, these equations have collected many undisputed successes, which definitely place them among the most accurate, simple and beautiful models of mathematical physics. Notwithstanding these successes, up to the present time, a number of unresolved basic mathematical questions remain open – mostly, but not only, for the physically relevant case of three-dimensional flow.
Undoubtedly, the most celebrated is that of proving or disproving the existence of global 3D regular flow for data of arbitrary “size”, no matter how smooth (global regularity problem). Since the beginning of the 20th century, this notorious question has challenged several generations of mathematicians who have not been able to furnish a definite answer. In fact, to date, 3D regular flows are known to exist either for all times but for data of “small size”, or for data of “arbitrary size” but for a finite interval of time only. The problem of global regularity has become so intriguing and compelling that, in the year 2000, it was decided to put a generous bounty on it. In fact, properly formulated, it is listed as one of the seven $1M Millennium Prize Problems of the Clay Mathematical Institute. However, the Navier–Stokes equations present also other fundamental open questions. For example, it is not known whether, in the 3D case, the associated initialboundary value problem is (in an appropriate function space) well-posed in the sense of Hadamard. Stated differently, in 3D it is open the question of whether solutions to this problem exist for all times, are unique and depend continuously upon the data, without being necessarily “regular”. Another famous, unsettled challenge is whether or not the Navier–Stokes equations are able to provide a rigorous model of turbulent phenomena. These phenomena occur when the magnitude of the driving mechanism of the fluid motion becomes sufficiently “large”, and, roughly speaking, it consists of flow regimes characterized by chaotic and random property changes for velocity and pressure fields throughout the fluid. They are observed in 3D as well as in two-dimensional (2D) motions (e. g., in flowing soap films). We recall that a 2D motion occurs when the relevant region of flow is contained in a portion of a plane, $ , and the component of the velocity field orthogonal to $ is negligible. It is worth emphasizing that, in principle, the answers to the above questions may be unrelated. Actually, in the 2D case, the first two problems have long been solved in the affirmative, while the third one remains still open. Nevertheless, there is hope that proving or disproving the first two problems in 3D will require completely fresh and profound ideas that will open new avenues to the understanding of turbulence. The list of main open problems can not be exhausted without mentioning another outstanding question pertaining to the boundary value problem describing steadystate flow. The latter is characterized by time-independent velocity and pressure fields. In such a case, if the flow region, R, is multiply connected, it is not known (neither in 2D nor in 3D) if there exists a solution under a given ve-
Navier–Stokes Equations: A Mathematical Analysis
locity distribution at the boundary of R that merely satisfy the physical requirement of conservation of mass. Introduction Fluid mechanics is a very intricate and intriguing discipline of the applied sciences. It is, therefore, not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and difficult. Complexities and difficulties, of course, may be more or less challenging depending on the mathematical model chosen to describe the physical situation. Among the many mathematical models introduced in the study of fluid mechanics, the Navier–Stokes equations can be considered, without a doubt, the most popular one. However, this does not mean that they can correctly model any fluid under any circumstance. In fact, their range of applicability is restricted to the class of so-called Newtonian fluids; see Remark 3 in Sect. “Derivation of the Navier–Stokes Equations and Preliminary Considerations”. This class includes several common liquids like water, most salt solutions in water, aqueous solutions, some motor oils, most mineral oils, gasoline, and kerosene. As mentioned in the previous section, these equations were proposed in 1822 by the French engineer Claude Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than two decades later, in 1845, that the same equations were re derived by the 26-year-old George Stokes in a quite general way, by means of the theory of continua. The role of mathematics in the investigation of the fluid properties and, in particular, of the Navier–Stokes equations, is, as in most branches of the applied sciences, twofold and aims at the accomplishment of the following objectives. The first one, of a more fundamental nature, is the validation of the mathematical model, and consists in securing conditions under which the governing equations possess the essential requirements of well-posedness, that is, existence and uniqueness of corresponding solutions and their continuous dependence upon the data. The second one, of a more advanced character, is to prove that the model gives an adequate interpretation of the observed phenomena. This paper is, essentially, directed toward the first objective. In fact, its goals consist of (1) formulating the primary problems, (2) describing the related known results, and (3) pointing out the remaining basic open questions,
for both boundary and initial-boundary value problems. Of course, it seemed unrealistic to give a detailed presentation of such a rich and diversified subject in a few number of pages. Consequently, we had to make a choice which, we hope, will still give a fairly good idea of this fascinating and intriguing subject of applied mathematics. The plan of this work is the following. In Sect. “Derivation of the Navier–Stokes Equations and Preliminary Considerations”, we shall first give a brief derivation of the Navier–Stokes equations from continuum theory, then formulate the basic problems and, further on, discuss some basic properties. Section “Mathematical Analysis of the Boundary Value Problem” is dedicated to the boundary value problem in both bounded and exterior domains. Besides existence and uniqueness, the main topics include the study of the structure of solution set at large Reynolds number, as well as a condensed treatment of bifurcation issues. Section “Mathematical Analysis of the InitialBoundary Value Problem” deals with the initial-boundary value problem in a bounded domain and with the related questions of well-posedness and regularity. Finally, in Sect. “Future Directions” we shall outline some future directions of research. Companion to this last section, is the list of a number of significant open questions that are mentioned throughout the paper. Derivation of the Navier–Stokes Equations and Preliminary Considerations In the continuum mechanics modeling of a fluid, F , one assumes that, in the given time interval, I [0; T], T > 0, of its motion, F continuously fills a region, ˝, of the three-dimensional space, R3 . We call points, surfaces, and volumes of F , material points (or particles), material surfaces, and material volumes, respectively. In most relevant applications, the region ˝ does not depend on time. This happens, in particular, whenever the fluid is bounded by rigid walls like, for instance, in the case of a flow past a rigid obstacle, or a flow in a bounded container with fixed walls. However, there are also some significant situations where ˝ depends on time as, for example, in the motion of a fluid in a pipe with elastic walls. Throughout this work, we shall consider flow of F where ˝ is time-independent. Balance Laws In order to describe the motion of F it is convenient to represent the relevant physical quantities in the Eulerian form. Precisely, if x D (x1 ; x2 ; x3 ) is a point in ˝ and t 2 [0; T], we let D (x; t), v D v(x; t) D (v1 (x; t); v2 (x; t); v3 (x; t)) and a D a(x; t) D (a1 (x; t); a2 (x; t); a3 (x; t)) be the density, velocity, and acceleration, re-
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spectively, of that particle of F that, at the time t, passes through the point x. Furthermore, we denote by f D f (x; t) D ( f1 (x; t); f2 (x; t); f3 (x; t)) the external force per unit volume (body force) acting on F . In the continuum theory of non-polar fluids, one postulates that in every motion the following equations must hold 9 @ @ > > ( (x; t)v i (x; t)) D 0 ; (x; t) C > > @t @x i > > > > for all =
(x; t) a i (x; t) (x; t) 2 ˝ @T ji > > C (x; t) f i (x; t) ; i D 1; 2; 3 ;> D > (0; T) : > @x j > > > ; Ti j (x; t) D T ji (x; t) ; i; j D 1; 2; 3 (1) These equations represent the local form of the balance laws of F in the Eulerian description. Specifically, (1)1 expresses the conservation of mass, (1)2 furnishes the balance of linear momentum, while (1)3 is equivalent to the balance of angular momentum. The function T D T(x; t) D fT ji (x; t)g is a second-order, symmetric tensor field, the Cauchy stress tensor, that takes into account the internal forces exerted by the fluid. More precisely, let S denote a (sufficiently smooth) fixed, closed surface in R3 bounding the region RS , let x be any point on S, and let n D n(x) be the outward unit normal to S at x. Furthermore, let R(e) S be the region exterior to RS , with R(e) \ ˝ ¤ ;. Then the S vector t D t(x; t) defined as t(x; t) :D n(x) T(x; t) ;
(2)
represents the force per unit area exerted by the portion of the fluid in R(e) S on S at the point x and at time t; see Fig. 1. Constitutive Equation An important kinematical quantity associated with the motion of F is the stretching tensor field, D D D(x; t), whose components, Dij , are defined as follows: 1 @v j @v i ; i; j D 1; 2; 3 : (3) C Di j D 2 @x i @x j
The stretching tensor is, of course, symmetric and, roughly speaking, it describes the rate of deformation of parts of F . In fact, it can be shown that a necessary and sufficient condition for a motion of F to be rigid (namely, the mutual distance between two arbitrary particles of F does not change in time), is that D(x; t) D 0 at each (x; t) 2 ˝ I. @v i Moreover, div v(x; t) trace D(x; t) @x (x; t) D 0 for i all (x; t) 2 ˝ I, if and only if the motion is isochoric, namely, every material volume does not change with time. A noteworthy class of fluids whose generic motion is isochoric is that of fluids having constant density. In fact, if is a positive constant, from (1)1 we find div v(x; t) D 0, for all (x; t) 2 ˝I. Fluids with constant density are called incompressible. Herein, incompressible fluids will be referred to as liquids. During the generic motion, the internal forces will, in general, produce a deformation of parts of F . The relation between internal forces and deformation, namely, the functional relation between T and D, is called constitutive equation and characterizes the physical properties of the fluid. A liquid is said to be Newtonian if and only if the relation between T and D is linear, that is, there exist a scalar function p D p(x; t) (the pressure) and a constant (the shear viscosity) such that T D pI C 2D ;
(4)
where I denotes the identity matrix. In a viscous Newtonian liquid, one assumes that the shear viscosity satisfies the restriction > 0:
(5)
We will discuss further the meaning of this assumption in Remark 2. Navier–Stokes Equations In view of the condition div v D 0, it easily follows from (4) that @Tji @p D C v i @x j @x i where :D @x@@x is the Laplace operator. Therefore, by l l (1) and (4) we deduce that the equations governing the motion of a Newtonian viscous liquid are furnished by 2
9 @p > C v i C f i ; i D 1; 2; 3> = @x i in ˝(0; T) : > @v i > ; D0 @x i (6)
a i D
Navier–Stokes Equations: A Mathematical Analysis, Figure 1 Stress vector at the point x of the surface S
Navier–Stokes Equations: A Mathematical Analysis
It is interesting to observe that both equations in (6) are linear in all kinematical variables. However, in the Eulerian description, the acceleration is a nonlinear functional of the velocity, and we have ai D
@v i @v i ; C vl @t @x l
or, in a vector form, aD
@v C v rv : @t
Replacing this latter expression in (6) we obtain the Navier–Stokes equations: 9 @v > >
C v rv > = @t in ˝ (0; T) : (7) D r p C v C f ;> > > ; divv D 0 In these equations, the (constant) density , the shear viscosity [satisfying (5)], and the force f are given quantities, while the unknowns are velocity v D v(x; t) and pressure p D p(x; t) fields. Some preliminary comments about the above equations are in order. Actually, we should notice that the unknowns v; p do not appear in a “symmetric” way. In other words, the equation of conservation of mass (7)2 does not involve the pressure field. This is due to the fact that, from the mechanical point of view, the pressure plays the role of reaction force (Lagrange multiplier) associated with the isochoricity constraint div v D 0. In other words, whenever a portion of the liquid “tries to change its volume” the liquid “reacts” with a suitable distribution of pressure to keep that volume constant. Thus, the pressure field must be generally deduced in terms of the velocity field, once this latter has been determined; see Remark 1. Initial-Boundary Value Problem In order to find solutions to the problem (7), we have to append suitable initial, at time t D 0, and boundary conditions. As a matter of fact, these conditions may depend on the specific physical problem we want to model. We shall assume that the region of flow, ˝, is bounded by rigid walls, @˝, and that the liquid does not “slip” at @˝. The appropriate initial and boundary conditions then become, respectively, the following ones v(x; 0) D v 0 (x) ; v(x; t) D v 1 (x; t) ;
x2˝ (x; t) 2 @˝ (0; T) ;
where v 0 and v 1 are prescribed vector fields.
(8)
Steady-State Flow and Boundary-Value Problem An important, special class of solutions to (7), called steadystate solutions, is that where velocity and pressure fields are independent of time. Of course, a necessary requirement for such solutions to exist is that f does not depend on time as well. From (7) we thus obtain that a generic steady-state solution, (v D v(x); p D p(x)), must satisfy the following equations )
v rv D r p C v C f ; in ˝ : (9) divv D 0 Under the given assumptions on the region of flow, from (8)2 it follows that the appropriate boundary conditions are v(x) D v (x) ;
x 2 @˝ ;
(10)
where v is a prescribed vector field. Two-Dimensional Flow In several mathematical questions related to the unique solvability of problems (7)–(8) and (9)–(10), separate attention deserve two-dimensional solutions describing the planar motions of F . For these solutions the fields v and p depend only on x1 , x2 (say) [and t in the case (7)–(8)], and, moreover, v3 0. Consequently, the relevant (spatial) region of motion, ˝, becomes a subset of R2 . Remark 1 By formally operating with “div” on both sides of (7)1 and by taking into account (8)2 , we find that, at each time t 2 (0; T) the pressure field p D p(x; t) must satisfy the following Neumann problem p D (v rv f ) in ˝ @p D [v (v 1 rv f )] n @n
at @˝
(11)
where n denotes the outward unit normal to @˝. It follows that the prescription of the pressure at the bounding walls or at the initial time independently of v, could be incompatible with (8) and, therefore, could render the problem ill-posed. Remark 2 We can give a simple qualitative explanation of the assumption (5). To this end, let us consider a steadystate flow of a viscous liquid between two parallel, rigid walls ˘ 1 , ˘ 2 , set at a distance d apart and parallel to the plane x2 D 0 see Fig. 2. The flow is induced by a force per unit area F D F e1 , F > 0, applied to the wall ˘ 1 , that moves ˘ 1 with a constant velocity V D V e 1 , V > 0, while ˘ 2 is kept fixed; see Fig. 2. No external force is acting on the liquid. It is then easily checked that the velocity
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presentation into two parts, depending on the “geometry” of the region of flow ˝. Specifically, we shall treat the cases when ˝ is either a bounded domain or an exterior domain (flow past an obstacle). For each case we shall describe methods, main results, and fundamental open questions. Navier–Stokes Equations: A Mathematical Analysis, Figure 2 Pure shear flow between parallel plates induced by a force F
and pressure fields v D V (x2 /d)e 1 ; p D p0 D const. (pure shear flow) satisfy (9) with f 0, along with the appropriate boundary conditions v(0) D 0, v(d) D V e 1 . From (2) and (4), we obtain that the force t per unit area exerted by the fluid on ˘ 1 is given by t(x1 ; d; x3 ) D e 2 T(x1 ; d; x3 ) D p0 e 2
V e1 ; d
that is, t has a “purely pressure” component t p D p0 e 2 , and a “purely shear viscous” component t v D (V /d)e 1 . As expected in a viscous liquid, t v is directed parallel to F and, of course, if it is not zero, it should also act against F, that is, t v e 1 < 0. However, (V /d) > 0, and so we must have > 0. Since the physical properties of the fluid are independent of the particular flow, this simple reasoning justifies the assumption made in (5). Remark 3 As mentioned in the Introduction, the constitutive Eq. (4) and, as a consequence, the Navier–Stokes equations, provide a satisfactory model only for a certain class of liquids, while, for others, their predictions are at odds with experimental data. These latter include, for example, biological fluids, like blood or mucus, and aqueous polymeric solutions, like common paints, or even ordinary shampoo. In fact, these liquids, which, in contrast to those modeled by (4), are called non-Newtonian, exhibit a number of features that the linear relation (4) is not able to predict. Most notably, in liquids such as blood, the shear viscosity is no longer a constant and, in fact, it p decreases as the magnitude of shear (proportional to D i j D i j ) increases. Furthermore, liquids like paints or shampoos show, under a given shear rate, a distribution of stress (other than that due to the pressure) in the direction orthogonal to the direction of shear (the so-called normal stress effect). Modeling and corresponding mathematical analysis of a variety of non-Newtonian liquids can be found, for example, in [42].
Flow in Bounded Domains In this section we shall analyze problem (9)–(10) where ˝ is a bounded domain of R3 . A similar (and simpler) analysis can be performed in the case of planar flow, with exactly the same results. Variational Formulation and Weak Solutions To show existence of solutions, one may use, basically, two types of methods that we shall describe in some detail. The starting point of both methods is the so-called variational formulation of (9)–(10). Let ' 2 D(˝). Since div ' D div v D 0 in ˝ and 'j@˝ D 0, we have (by a formal integration by parts) Z p' n D 0 ; ('; r p) D @˝ Z n rv ' D [v; '] (v; ') D [v; '] C @˝ Z (v rv; ') D v nv ' (v r'; v) D (v r'; v) @˝
(12) where [; ] is the scalar product in D1;2 0 (˝). Thus, if we dot-multiply both sides of (9)1 by ' 2 D(˝) and integrate by parts over ˝, we (formally) obtain with :D / [v; '](v r'; v) D h f ; 'i ;
for all ' 2 D(˝) ; (13)
where we assume that f belongs to D1;2 (˝). Equa0 tion. (13) is the variational (or weak) form of (9)1 . We observe that (13) does not contain the pressure. Moreover, 1;2 (˝). every term in (13) is well-defined provided v 2 Wloc Definition 1 A function v 2 W 1;2 (˝) is called a weak solution to (9)–(10) if and only if: (i) divv D 0 in ˝; (ii) vj@˝ D v (in the trace sense); (iii) v satisfies (13). If, in particular, v 0, we replace conditions (i), (ii) with the single requirement: (i)0 v 2 D1;2 0 (˝) . Throughout this work we shall often use the following result, consequence of the Hölder inequality and of a simple approximating procedure. Lemma 1 The trilinear form
Mathematical Analysis of the Boundary Value Problem We begin to analyze the properties of solutions to the boundary-value problem (9)–(10). We shall divide our
(a; b; c) 2 L q (˝) W 1;2 (˝) Lr (˝) 7! (a rb; c) 2 R ;
1 1 1 C D ; q r 2
Navier–Stokes Equations: A Mathematical Analysis
is continuous. Moreover, (a rb; c) D (a rc; b), for any b; c 2 D01;2 (˝), and for any a 2 L2 (˝). Thus, in particular, (a rb; b) D 0. Regularity of Weak Solutions If f is sufficiently regular, the corresponding weak solution is regular as well and, moreover, there exists a scalar function, p, such that (9) is satisfied in the ordinary sense. Also, if @˝ and v are smooth enough, the solution (v; p) is smooth up to the boundary and (10) is satisfied in the classical sense. A key tool in the proof of these properties is the following lemma, which is a special case of a more general result due to Cattabriga [12]; (see also Lemma IV.6.2 and Theorem IV.6.1 in [31]). Lemma 2 Let ˝ be a bounded domain of R3 , of m;q (˝), u 2 class C mC2 , m 0, and let g 2 W R mC21/q;q (@˝), 1 < q < 1, with @˝ u n D 0. W Moreover, let u 2 W 1;2 (˝) satisfy the following conditions (a) [u; '] D (g; ') for all ' 2 D(˝) ; (b) div u D 0 ; (c) u D u at @˝ in the trace sense. Then, u 2 W mC2;q R (˝) and there exists a unique 2 W mC1;q (˝), with ˝ D 0, such that the pair (u; ) satisfies the following Stokes equations ) u D r C g in ˝ : div u D 0 Furthermore, there exists a constant C D C(˝; m; q) > 0 such that kukmC2;q C kkmC1;q C kgkm;q C ku kmC21/q;q;@˝ : To give an idea of how to prove regularity of weak solutions by means of Lemma 2, we consider the case f 2 ¯ ˝ of class C 1 and v 2 C 1 (@˝). (For a general C 1 (˝), regularity theory of weak solutions, see Theorem VIII.52 in [32].) Thus, in particular, by the embedding Theorem II.2.4 in [31] W 1;2 (˝) L q (˝) ;
for all q 2 [1; 6] ;
(14)
and by the Hölder inequality we have that g :D f vrv 2 L3/2 (˝). From (13) and Lemma 2, we then deduce that v 2 W 2;3/2 (˝) and that there exists a scalar field p 2 W 1;3/2 such that (v; p) satisfy (9) a.e. in ˝. Therefore, because of the embedding W 2;3/2 (˝) W 1;3 (˝) Lr (˝), arbitrary r 2 [1; 1), Theorem II.2.4 in [31], we obtain the improved
regularity property g 2 W 1;s (˝), for all s 2 [1; 3/2). Using again Lemma 2, we then deduce v 2 W 3;s (˝) and p 2 W 2;s (˝) which, in particular, gives further regularity for ¯ g. By induction, we then prove v; p 2 C 1 (˝). Existence Results. Homogeneous Boundary Conditions As we mentioned previously, there are, fundamentally, two kinds of approaches to show existence of weak solutions to (9)–(10), namely, the finite-dimensional method and the function-analytic method. We will first describe these methods in the case of homogeneous boundary conditions, v 0, deferring the study of the non-homogeneous problem to Subsect. “Existence Results. NonHomogeneous Boundary Conditions”. In what follows, we shall refer to (9)–(10) with v D 0 as (9)–(10)hom . A. The Finite-Dimensional Method This popular approach, usually called Galerkin method, was introduced by Fujita [28] and, independently, by Vorovich and Yudovich [96]. It consists in projecting (13) on a suitable finite dimensional space, V N , of D1;2 0 (˝) and then in finding a solution, v N 2 VN , of the “projected” equation. One then passes to the limit N ! 1 to show, with the help of an appropriate uniform estimate, that fv N g contains at least one subsequence converging to some v 2 D1;2 0 (˝) that satisfies condition (13). Precisely, let f k g D(˝) be an orthonormal basis of D1;2 0 (˝), and PN set v N D c , where the coefficients ciN are reiD1 i N i quested to be solutions of the following nonlinear algebraic system [v N ;
k]
(v N r
k; vN )
D hf;
ki ;
k D 1; : : : ; N :
(15)
By means of the Brouwer fixed point theorem, it can be shown (see Lemma VIII.3.2 in [32]) that a solution (c1N ; : : : ; c N N ) to (15) exists, provided the following estimate holds jv N j1;2 M
(16)
where M is a finite, positive quantity independent of N. Let us show that (16) indeed occurs. Multiplying through both sides of (15) by ckN , summing over k from 1 to N, and observing that, by Lemma 1, (v N rv N ; v N ) D 0, we obtain jv N j21;2 D h f ; v N i j f j1;2 jv N j1;2 ; which proves the desired estimate (16) with M :D j f j1;2 . Notice that the validity of this latter estimate can be obtained by formally replacing in (13) ' with v 2 D1;2 0 (˝) and by using Lemma 1. From classical properties of
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Hilbert spaces and from (16), we can select a subsequence fv N 0 g and find v 2 D1;2 0 (˝) such that lim [v N 0 ; '] D [v; '] ;
N 0 !1
for all ' 2 D1;2 0 (˝) : (17)
By Definition 1, to prove that v is a weak solution to (9)– (10)hom it remains to show that v satisfies (13). In view of the Poincaré inequality (Theorem II.4.1 in [31]): '2
k'k2 c P j'j1;2
1;2 D0 (˝) ;
c P D c P (˝) > 0 ;
by (16) it follows that fv N g is bounded in W01;2 (˝) and so, by Rellich compactness theorem (see Theorem II.4.2 in [31]) we can assume that fv N 0 g converges to v in L4 (˝): lim kv N 0 vk4 D 0 :
(19)
N 0 !1
We now consider (15) with N D N 0 and pass to the limit N 0 ! 1. Clearly, by (17), we have for each fixed k lim [v N 0 ;
k]
D [v;
k] :
(20)
Moreover, by Lemma 1 and by (19), lim (v N 0 r
N 0 !1
k ; vN0 )
D (v r
k ; v) ;
and so, from this latter relation, from (20), and from (15) we conclude that v;
k
(v r
k ; v)
D hf;
1;2 D0 (˝) L4 (˝) ;
(21)
and so, from Lemma 1 and from the Riesz representation theorem, we have that, for each fixed v 2 D1;2 0 (˝), there exists N (v) 2 D1;2 (˝) such that 0 (v r'; v) D [N (v); '] ;
(18)
N 0 !1
the advantage of furnishing, as a byproduct, significant information on the solutions set. By (14) and (18) we find
ki ;
for all k 2 N ;
Since f k g is a basis in D1;2 0 (˝), this latter relation, along Lemma 1, immediately implies that v satisfies (13), which completes the existence proof. Remark 4 The Galerkin method provides existence of a weak solution corresponding to any given f 2 1;2 D0 (˝). Moreover, it is constructive, in the sense that the solution can be obtained as the limit of a sequence of “approximate solutions” each of which can be, in principle, evaluated by solving the system of nonlinear algebraic Eqs. (15). B. The Function-Analytic Method This approach, that goes back to the work of Leray [61,63] and of Ladyzhenskaya [58], consists, first, in re-writing (13) as a nonlinear equation in the Hilbert space D1;2 0 (˝), and then using an appropriate topological degree theory to prove existence of weak solutions. Though more complicated than, and not constructive like the Galerkin method, this approach has
for all ' 2 D1;2 0 (˝) : (22)
Likewise, we have h f ; 'i D [F; '], for some F 2 D1;2 0 (˝) and for all ' 2 D1;2 0 (˝). Thus, Eq. (13) can be equivalently re-written as N(; v) D F ;
in D1;2 0 (˝) ;
(23)
where the map N is defined as follows N : (; v) 2 (0; 1) D1;2 0 (˝) 7! v C N (v) 2 D1;2 0 (˝) :
(24)
In order to show the above-mentioned property of solutions, we shall use some basic results related to Fredholm maps of index 0 [84]. This approach is preferable to that originally used by Leray – which applies to maps that are compact perturbations of homeomorphism – because it covers a larger class of problems, including flow in exterior domains [36]. Definition 2 A map M : X 7! Y, X, Y Banach spaces, is Fredholm, if and only if: (i) M is of class C1 (in the sense of Fréchet differentiability) and denoted by D x M(x) its derivative at x 2 X (iii) the integers. The integers ˛ :D dim fz 2 X : [D x M(x)](z) D 0g and ˇ :D codimfy 2 Y : [D x M(x)](z) D yg for some z 2 X are both finite. The integer m :D ˛ ˇ is independent of the particular x 2 X (Sect. 5.15 in [100]), and is called the index of M. Definition 3 A map M : X 7! Y, is said to be proper if K1 :D fx 2 X : M(x) D y ; y 2 Kg is compact in X, whenever K is compact in Y. By using the properties of proper Fredholm maps of index 0 [84], and of the associated Caccioppoli–Smale degree [10,84], one can prove the following result (see Theorem I.2.2 in [37]) Lemma 3 Let M be a proper Fredholm map of index 0 and of class C2 , satisfying the following. (i) There exists y¯ 2 Y such that the equation M(x) D y¯ has one and only one solution x¯ ; (ii) [D x M(x¯ )](z) D 0 ) z D 0.
Navier–Stokes Equations: A Mathematical Analysis
Then: (a) M is surjective ; (b) There exists an open, dense set Y0 Y such that for any y 2 Y0 the solution set fx 2 X : M(x) D yg is finite and constituted by an odd number, D (y), of points ; (c) The integer is constant on every connected component of Y 0 . The next result provides the required functional properties of the map N. Proposition 1 The map N defined in (24) is of class C 1 . 1;2 Moreover, for any > 0, N(; ) : D1;2 0 (˝) 7! D0 (˝) is proper and Fredholm of index 0. Proof It is a simple exercise to prove that N is of class C 1 (see Example I.1.6 in [37]). By the compactness of the embedding W01;2 (˝) L4 (˝) Theorem II.4.2 in [31], from Lemma 1 and from the definition of the map N [see (22)], one can show that N is compact, that is, it maps bounded sequences of D1;2 0 (˝) into relatively compact sequences. Therefore, N(; ) is a compact perturbation of a multiple of the identity operator. Moreover, by (22) and by Lemma 1 we show that [N (v); v] D 0, which implies, [N(; v); v] D jvj21;2 :
(25)
Using the Schwartz inequality on the left-hand side of (25), we infer that jN(; v)j1;2 jvj1;2 and so, for each fixed > 0 we have jN(; v)j1;2 ! 1 as jvj1;2 ! 1, that is, N(; ) is (weakly) coercive. Consequently, N(; ) is proper (see Theorem 2.7.2 in [7]). Finally, since the derivative of a compact map is compact (see Theorem 2.4.6 in [7]), the derivative map Dv N(; v) I C Dv N(v); I identity operator, is, for each fixed > 0, a compact perturbation of a multiple of the identity, which implies that N(; ) is Fredholm of index 0 (see Theorem 5.5.C in [100]). It is easy now to show that, for any fixed > 0, N(; ) satisfies all the assumptions of Lemma 3. Actually, in view of Proposition 1, we have only to prove the validity of (i) and (ii). We take y¯ 0, and so, from (25) we obtain that the equation N(; v) D 0 has only the solution x¯ v D 0, so that (i) is satisfied. Furthermore, from (22) it follows that the equation Dv N(; 0)(w) D 0 is equivalent to w D 0, and so also condition (ii) is satisfied for > 0. Thus, we have proved the following result (see also [22]). Theorem 1 For any > 0 and F 2 D1;2 0 (˝), Eq. (23) has at least one solution v 2 D1;2 (˝). Moreover, for each fixed 0 > 0, there exists open and dense O D O() D1;2 0 (˝)
Navier–Stokes Equations: A Mathematical Analysis, Figure 3 Sketch of the manifold S(F)
with the following properties: (i) For any F 2 O the number of solutions to (23), n D n(F; ) is finite and odd ; (ii) the integer n is constant on each connected component of O. Next, for a given F 2 D1;2 0 (˝), consider the solution manifold n o 1;2 S(F) D (; v) 2 (0; 1) D0 (˝) : N(; v) D F : By arguments similar to those used in the proof of Theorem 1, one can show the following “generic” characterization of S(F) see Example I.2.4 in [37] and also Chap. 10.3 in [93]. Theorem 2 There exists dense P D1;2 0 (˝) such that, for every F 2 P , the set S(F) is a C 1 1-dimensional manifold. Moreover, there exists an open and dense subset of (0; 1), D (F), such that for each 2 , Eq. (23) has a finite number m D m(F; ) > 0 of solutions. Finally, the integer m is constant on every open interval contained in . In other words, Theorem 2 expresses the property that, for every F 2 P , the set S(F) is the union of smooth and non-intersecting curves. Furthermore, “almost all” lines D 0 D const. intersect these curves at a finite number of points, m(0 ; F), each of which is a solution to (23) corresponding to 0 and to F. Finally, m(0 ; F) D m(1 ; F) whenever 0 and 1 belong to an open interval of a suitable dense set of (0; 1). A sketch of the manifold S(F) is provided in Fig. 3. Existence Results. Non-Homogeneous Boundary Conditions Existence of solutions to (9)–(10) when v 6 0 leads to one of the most challenging open questions in the mathematical theory of the Navier–Stokes equations, in the case when the boundary @˝ is constituted by more than one connected component. In order to explain the problem, let Si , i D i; : : : ; K, K 1, denote these components. Conservation of mass (9)2 along with Gauss theorem imply the following compatibility condition on the
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data v K Z X iD1
v n i Si
K X
˚i D 0 ;
(26)
iD1
where n i denotes the outward unit normal to Si . From the physical point of view, the quantity ˚ i represents the mass flow-rate of the liquid through the surface Si . Now, assuming v and ˝ sufficiently smooth (for example, v 2 W 1/2;2 (@˝) and ˝ locally Lipschitzian Sect. VIII.4 in [32]), we look for a weak solution to (9)–(10) in the form v D u C V, where V 2 W 1;2 (˝) is an extension of v with div V D 0 in ˝. Thus, if we use, for example, the Galerkin method of Subsect. IV.1.1(A), from (15) we obPN tain that the “approximate solution” u N D iD1 c i N i must satisfy the following equations (k D 1; : : : ; N) [u N ;
k]
(V r
(u N r k; uN )
k ; uN )
(V r
(u N r
k; V)
k; V)
C [V ; D hf;
k] k i:
(27)
Therefore, existence of a weak solution will be secured provided we show that the sequence fv N :D u N C V g satisfies the bound (16). In turn, this latter is equivalent to showing ju N j1;2 M1 ;
(28)
where M 1 is a finite, positive quantity independent of N. Multiplying through both sides of (27) by ckN , summing over k from 1 to N, and observing that, by Lemma 1, (u N ru N ; u N ) D (V ru N ; u N ) D 0, we find ju N j21;2 D (u N ru N ; V ) C (V ru N ; V ) [V ; u N ] C h f ; u N i : By using (14), the Hölder inequality and the Cauchy– Schwartz inequality ab "a2 C (1/4")b2 ;
a; b; " > 0
(29)
on the last three terms on the right-hand side of this latter equation, we easily find, for a suitable choice of ", ju N j21;2 (u N ru N ; V ) C C ; 2
(30)
where C D C(V ; f ; ˝) > 0. From (30) it follows that, in order to obtain the bound (28) without restrictions on the magnitude of , it suffices that V meets the following requirement: Given ";
there is V D V("; x) 2 (0; /4)
such that (' r'; V ) "j'j21;2
for all ' 2 D(˝) : (31)
As indicated by Leray pp. 28–30 in [61] and clarified by Hopf [52], if ˝ is smooth enough and if K D 1, that is, if @˝ is constituted by only one connected component, S1 , it is possible to construct a family of extensions satisfying (31). Notice that, in such a case, condition (26) reduces to the single condition ˚1 D 0. If K > 1, the same construction is still possible but with the limitation that ˚ i D 0, for all i D 1; : : : ; K. It should be emphasized that this condition is quite restrictive from the physical point of view, in that it does not allow for the presence, in the region of flow, of isolated “sources” and “sinks” of liquid. Nevertheless, one may wonder if, by using a different construction, it is still possible to satisfy (31). Unfortunately, as shown by Takeshita [90] by means of explicit examples, in general, the existence of extensions satisfying (31) implies ˚ i D 0, for all i D 1; : : : ; K; see also § VIII.4 in [32]. We wish to emphasize that the same type of conclusion holds if, instead of the Galerkin method, we use the function-analytic approach; see [61], notes to Chap. VIII in [32]. Finally, it should be remarked that, in the special case of two-dimensional domains possessing suitable symmetry and of symmetric boundary data, Amick [1] and Fujita [29] have shown existence of corresponding symmetric solutions under the general assumption (26). However, we have the following. Open Question Let ˝ be a smooth domain in Rn , n D 2; 3, with @˝ constituted by K > 1 connected components, Si , i D 1; : : : ; K, and let v be any smooth field satisfying (26). It is not known if the corresponding problem (9)–(10) has at least one solution. Uniqueness and Steady Bifurcation It is a well-established experimental fact that a steady flow of a viscous incompressible liquid is “observed”, namely, it is unique and stable, if the magnitude of the driving force, usually measured through a dimensionless number 2 (0; 1), say, is below a certain threshold, c . However, if > c , this flow becomes unstable and another, different flow is instead observed. This latter may be steady or unsteady (typically, time-periodic). In the former case, we say that a steady bifurcation phenomenon has occurred. From the physical point of view, bifurcation happens because the liquid finds a more “convenient” motion (than the original one) to dissipate the increasing energy pumped in by the driving force. From the mathematical point of view, bifurcation occurs when, roughly speaking, two solutioncurves (parametrized with the appropriate dimensionless number) intersect. (As we know from Theorem 2, the intersection of these curves is not “generic”.) It can happen
Navier–Stokes Equations: A Mathematical Analysis
that one curve exists for all values of , while the other only exists for > c (supercritical bifurcation). The point of intersection of the two curves is called the bifurcation point. Thus, in any neighborhood of the bifurcation point, we must have (at least) two distinct solutions and so a necessary condition for bifurcation is the occurrence of nonuniqueness. This section is dedicated to the above issues.
Theorem 4 (Non-Uniqueness) Let ˝ be a bounded smooth body of revolution around an axis r, that does not include points of r. For example, ˝ is a torus of arbitrary bounded smooth section. Then there are smooth fields f and v and a value of > 0 such that problem (9)–(10) corresponding to these data admits at least two distinct and smooth solutions.
Uniqueness Results It is simple to show that, if jvj1;2 is not “too large”, then v is the only weak solution to (9)–(10) corresponding to the given data.
Some Bifurcation Results By using the functional setting introduced in Sect. “B. The Function-Analytic Method”, it is not difficult to show that steady bifurcation can be reduced to the study of a suitable nonlinear eigenvalue problem in the space D1;2 0 (˝). To this end, we recall certain basic definitions. Let U be an open interval of R and let
Theorem 3 (Uniqueness) Let ˝ be locally Lipschitzian and let jvj1;2 < / ;
(32)
with D (˝) > 0. Then, there is only one weak solution to (9)–(10). Proof Let v, v 1 D v C u be two different solutions. From (13) we deduce [u; '] D (u r'; v)C(v 1 r'; u) ;
for all ' 2 D(˝) : (33)
If ˝ is locally Lipschitzian, then u 2 D1;2 0 (˝) (see Theorem II.3.2 and § II.3.5 in [31]), and since D(˝) is dense in 1;2 D0 (˝), with the help of Lemma 1 we can replace ' with u in (33) to get juj21;2 D (u ru; v) : We now use (14) and Lemma 1 on the right-hand side of this equation to find ( jvj1;2 )juj21;2 0 ; with D (˝) > 0, which proves u D 0 in D1;2 0 (˝), namely, uniqueness, if v satisfies (33).
M : (x; ) 2 X U 7! Y :
(34)
Definition 4 The point (x0 ; 0 ) is called a bifurcation point of the equation M(x; ) D 0
(35)
if and only if (a) M(x0 ; 0 ) D 0, and (b) there are (at least) ; )g, to two sequences of solutions, f(x m ; m )g and f(x m m (35), with x m ¤ x m , for all m 2 N, such that (x m ; m ) ! ; ) ! (x ; ) as m ! 1. (x0 ; 0 ) and (x m m 0 0 If M is suitably smooth around (x0 ; 0 ), a necessary condition for (x0 ; 0 ) to be a bifurcation point is that D x M(x0 ; 0 ) is not a bijection. In fact, we have the following result which is an immediate corollary of the implicit function theorem (see, e. g., Lemma III.1.1 in [37]). Lemma 4 Suppose that D x M exists in a neighborhood of (x0 ; 0 ), and that both M and D x M are continuous at (x0 ; 0 ). Then, if (x0 ; 0 ) is a bifurcation point of (34), D x M(x0 ; 0 ) is not a bijection. If, in particular, D x M(x0 ; 0 ) is a Fredholm operator of index 0 (see Definition 2), then the equation D x M(x0 ; 0 )x D 0 has at least one nonzero solutions.
Remark 5 As we have seen previously, if v 0, weak solutions satisfy jvj1;2 j f j1;2 . Thus, (32) holds if j f j1;2 2 /. If v ¤ 0, then one can show that (32) holds if j f j1;2 C (1 C )kv k1/2;2;@˝ C kv k21/2;2;@˝ 2 /1 , where 1 has the same properties as ; (see Theorem VIII.4.2 in [32]). Notice that these conditions are satisfied if a suitable non-dimensional parameter j f j1;2 C kv k1/2;2;@˝ is “sufficiently small”.
Let v 0 D v 0 (), 2 (0; 1) be a family of weak solution to (9)–(10) corresponding to given data f and v . We assume that f and v are fixed. Denoting by v any other solution corresponding to the same data, by an argument completely similar to that leading to (23), we find that u :D v v 0 , satisfies the following equation in D1;2 0 (˝)
Remarkably enough, one can give explicit examples of nonuniqueness, if condition (32) is violated. More specifically, we have the following result (see Theorem VIII.2.2 in [32]).
with B(v 0 ) :D Dv N (v 0 ) and N defined in (22). Obviously, (v 0 (0 ); 0 ) is a bifurcation point for the original equation if and only if (0; 0 ) is such for (36). Thus, in view of Lemma 4, a necessary condition for (0; 0 ) to be
u C B(v 0 )(u) C N (u) D 0 ;
(36)
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a bifurcation point for (36) is that the equation 0 v C B(v 0 (0 ))(v) D 0
(37)
has at least one nonzero solution v 2 D1;2 0 (˝). In several significant situations it happens that, after a suitable non-dimensionalization of (36), the family of solutions v 0 (), 2 (0; 1) is independent of the parameter which, this time, has to be interpreted as the inverse of an appropriate dimensionless number (Reynolds number), like, for instance, in the Taylor–Couette problem; see the following subsection. Now, from Proposition 1, we know that B(u) is compact at each u 2 D1;2 0 (˝), so that I C B(u) is Fredholm of index 0, at each u 2 D1;2 0 (˝), for all > 0 (see Theorem 5.5.C in [100]). Therefore, whenever v 0 does not depend on , in a neighborhood of 0 , from Lemma 4 we find that a necessary condition for (v 0 ; 0 ) to be a bifurcation point for (35) is that 0 is an eigenvalue of the (compact) linear operator B(v 0 ). The stated condition becomes also sufficient, provided we make the additional assumptions that 0 is a simple eigenvalue of B(v 0 ). This is a consequence of the following theorem (see, e. g., Lemma III.1.2 in [37]). Theorem 5 Let X Y and let the operator M in (34) be of the form M D I C T, where I is the identity in X and T is of class C1 . Furthermore, set L :D D x T(0). Suppose that 0 I C L is Fredholm of index 0 (Definition 2), for some 0 2 U, and that 0 is a simple eigenvalue for L, namely, the equation 0 x C L(x) D 0 has one and only one (nonzero) independent solution, x1 , while the equation 0 x C L(x) D x1 has no solutions. Then, (0; 0 ) is a bifurcation point for the equation M(x; ) D 0.
Navier–Stokes Equations: A Mathematical Analysis, Figure 4 Sketch of the streamlines of Taylor vortices, at the generic section =const
solve (9), with f D 0, for all values of the Reynolds number :D ! r12 /. Moreover, u0 satisfies the boundary conditions u0 (r) D 0
at r D 1 ;
u 0 (r) D e
at r D R :
Experiments show that, if exceeds a critical value, c , a flow with entirely different features than the flow (38) is observed. In fact, this new flow is dominated by large toroidal vortices, stacked one on top of the other, called Taylor vortices. They are periodic in the z-direction and axisymmetric (independent of ); see Fig. 4. Therefore, we look for bifurcating solutions of the form u 0 (r) C w(r; z), p0 (r) C p(r; z), satisfying (39), and where w and p are periodic in the z-direction with period P, and w satisfies the following parity conditions: wr (r; z) D wr (r; z) ;
w (r; z) D w (r; z) ; wz (r; z) D wz (r; z) :
Bifurcation of Taylor–Couette Flow A notable application of Theorem 5 is the bifurcation of Taylor–Couette flow. In this case the liquid fills the space between two coaxial, infinite cylinders, C1 and C2 , of radii R1 and R2 > R1 , respectively. C1 rotates around the common axis, a, with constant angular velocity !, while C2 is at rest. Denote by (r; ; z) a system of cylindrical coordinates with z along a and oriented as ! and let (e r ; e ; e z ) the associated canonical base. The components of a vector w in such a base are denoted by wr ; w and wz , respectively. If we introduce the non-dimensional quantities u D v/(!r1 ) ; x¯ D x/r1 ; p D p/ ! 2 r12 ; R D R2 /R1 ; with ! D j!j, we see at once that the following velocity and pressure fields 1 u 0 D 1 R2 r R2 /r e ;
p0 D ju0 j2 ln rCconst ; (38)
(39)
(40)
Moreover, the relevant region of flow becomes the periodicity cell ˝ :D (1; R) (0; P). If we now introduce the stream function @ D wr ; @z
@(r ) D r wz ; @r (r; z) D (r; z) ;
(41)
and the vector u :D ( ; w ), it can be shown (see § 72.7 in [99]) that u satisfies an equation of the type (36), namely, ¯ 0 )(u) C N ¯ (u) D 0 in H (˝) u C B(v (42) ¯ 0 ) and N ¯ obey the same funcwhere the operators B(v tional properties as B(v 0 ) and N , and where H (˝) is the Banach space of functions u :D ( ; w ) 2 C 4;˛ (˝) C 2;˛ (˝); ˛ 2 (0; 1), such that: (i) u(r; 0) D
Navier–Stokes Equations: A Mathematical Analysis
u(r; P) for all r 2 (1; R), (ii) u satisfies the parity conditions in (40), (41), and (iii) D @ /@r D w D 0 at r D 1; R. Thus, in view of Theorem 5 and of the properties of the operators involved in (42), we will obtain that (0; 0 ) is a bifurcation point for (41) if the following two conditions are met ¯ 0 )(u) D 0 (a) u C 0 B(v has one and only one independent solution, u1 2 H (˝) ; (b)
¯ 0 )(u) D u1 the equation u C 0 B(v has no solution in H (˝) :
(43)
It is in Lemma 72.14 in [99] that there exists a period P for which both conditions in (43) are satisfied. In addition, for ¯ 0 )(u) D 0 has only all 2 (0; 0 ) the equation u C B(v the trivial solution in H (˝), which, by Lemma 4, implies that no bifurcation occurs for 2 (0; 0 ), which, in turn, means that the bifurcation is supercritical.
Extension of the Boundary Data If ˝ isRsmooth enough (locally Lipschitzian, for example), since @˝ e 1 n D 0, by what we observed in Sect. “Existence Results. NonHomogeneous Boundary Conditions”, for any > 0 we find V D V (; x), with div V D 0 in ˝, and satisfying (31) with " D 1/(2), say. Actually, as shown in Proposition 3.1 in [36], the extension V can be chosen, in particular, to be of class C 1 ((0; 1) ˝) and such that the support of V(; ) is contained in a bounded set, independent of . The proof given in [36] is for ˝ R3 , but it can be easily extended to the case ˝ R2 . Variational Formulation and Weak Solutions Setting u D w V in (44)1 , dot-multiplying through both sides of this equation by ' 2 D(˝), integrating by parts and taking into account 12, we find @u ; ' (u r'; u) f(u r'; V) [u; '] @x1 C (V r'; u)g C (H; ') D h f ; 'i ;
for all ' 2 D(˝) :
Flow in Exterior Domains
(45)
One of the most significant questions in fluid mechanics is to determine the properties of the steady flow of a liquid past a body B of simple symmetric shape (such as a sphere or cylinder), over the entire range of the Reynolds number :D U L/ 2 (0; 1); see, e. g., [§ 4.9] in [4]. Here L is a length scale representing the linear dimension of B, while v 1 D U e 1 , U D const > 0, is the uniform velocity field of the liquid at large distances from B. In the mathematical formulation of this problem, one assumes that the liquid fills the whole space, ˝, outside the closure of the domain B Rn , n D 2; 3. Thus, by scaling v by U and x by L, from (9)–(10) we obtain that the steady flow past a body consists in solving the following non-dimensional exterior boundary-value problem 9 @w w C w rw D r p C f = @x1 in ˝ ; divw D 0 w(x) D e 1 ;
x 2 @˝ ;
lim w(x) D 0 ; (44)
jxj!1
where w :D v C e 1 and where we assume, for simplicity, that v 0. (However, all the stated results continue to hold, more generally, if v belongs to a suitable trace space.) Whereas in the three-dimensional case (e. g., flow past a sphere) the investigation of (44) is, to an extent, complete, in the two-dimensional case (e. g., flow past a circular cylinder) there are still fundamental unresolved issues. We shall, therefore, treat the two cases separately. We need some preliminary considerations.
where H D H(V) :D V C
@V V rV : @x1
Definition 5 A function w 2 D1;2 (˝) is called a weak solution to (44) if and only if w D u C V where u 2 D1;2 0 (˝) and u satisfies (45). Regularity of Weak Solutions (A) Local Regularity. The proof of differentiability properties of weak solutions can be carried out in a way similar to the case of a bounded domain. In particular, if f and ˝ are of class C 1 , one can show that w 2 ¯ and there exists a scalar field p 2 C 1 (˝) ¯ C 1 (˝) such that (44)1;2;3 holds in the ordinary sense. For this and other intermediate regularity results, we refer to Theorem IX.1.1 in [32]. (B) Regularity at Infinity. The study of the validity of (44)4 and, more particularly, of the asymptotic structure of weak solutions, needs a much more involved treatment, and the results depend on whether the flow is three- or two-dimensional. In the three-dimensional case, if f satisfies suitable summability assumptions outside a ball of large radius, by using a local representation of a weak solution, (namely, at all points of a ball of radius 1 and centered at x 2 ˝, dist (x; @˝) > 1) one can prove that w and all its derivatives tend to zero at infinity uniformly pointwise and that a similar property holds for p (see Theorem X.6.1 in [32]). The starting point of this analysis
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Navier–Stokes Equations: A Mathematical Analysis
to (45) of the form u N D
is the crucial fact that if ˝ R ; 3
1;2 D0 (˝)
is continuously embedded in L6 (˝) ;
(46)
which implies that w tends to zero at infinity in a suitable sense. However, the existence of the “wake” behind the body along with the sharp order of decay of w and p requires a more complicated analysis based on the global representation of a weak solution (namely, at all points outside a ball of sufficiently large radius) by means of the Oseen fundamental tensor, along with maximal regularity estimates for the solution of the linearized Oseen problem, this latter being obtained by suppressing the nonlinear term w rw in (44) [§§ IX.6, IX.7 and IX.8] in [32]. In particular, one can prove the following result concerning the behavior of w; (see Theorems IX.7.1, IX.8.1 and Remark IX.8.1 in [32]). Theorem 6 Let ˝ be a three-dimensional exterior domain of class C2 , and let w be a weak solution to (44) corresponding to a given f 2 L q (˝), for all q 2 (1; q0 ], q0 > 3. Then w 2 Lr (˝) ;
if and only if r > 2 :
(47)
If, in addition, f is of bounded support, then, denoting by the angle made by a ray starting from the origin of coordi¯ with the nates (taken, without loss of generality, in R3 ˝) positively directed x1 -axis, we have jw(x)j
M ; jxj 1 C jxj(1 C cos )
x 2˝;
(48)
where M D M(; ˝) > 0. Remark 6 Two significant consequences of Theorem 6 are: (i) the total kinetic energy of the flow past an obstacle ( 12 kwk22 ) is infinite , see (47) ; and (ii) the asymptotic decay of w is faster outside any semi-infinite cone with its axis coinciding with the negative x1 -axis (existence of the “wake”); see (48). The study of the asymptotic properties of solutions in the two-dimensional case is deferred till Sect. “Two-Dimensional Flow. The Problem of Existence”. Three-Dimensional Flow. Existence of Solutions and Related Properties As in the case of a bounded domain, we may use two different approaches to the study of existence of weak solutions. A. Finite-Dimensional Method Assume f 2 D1;2 (˝). 0 With the same notation as in Subsect. “A. The Finite-Dimensional Method”, we look for “approximate solutions”
uN ;
k
f(u N r
PN
@u ; @N x 1
k; V)
iD1 c i N
k
uN r
C (V r
D hf;
i , where
ki ;
k ; u N )g C
k; uN
(H;
k D 1; : : : ; N :
k)
(49)
As in the case of a bounded domain, existence to the algebraic system (49), in the unknowns ciN , will be achieved if we show the uniform estimate (28). By dot-multiplying through both sides of (49) by ckN , by summing over k between 1 and N and by using Lemma 1 we find ju N j21;2 D (u N ru N ; V ) (H; u N ) C h f ; u N i : (50) From the properties of the extension of the boundary data, we have that V satisfies (31) with " D 1/(2) and that, moreover (H; u N ) Cju N j1;2 , for some C D C(˝; ) > 0. Thus, from this and from (50) we deduce the uniform bound (28). This latter implies the existence of a subsequence fu N 0 g converging to some u 2 D1;2 0 (˝) weakly. Moreover, by Rellich theorem, u N 0 ! u in L4 (K), where K :D ˝ \ fjxj > g, all sufficiently large and finite
> 0 (see Theorem II.4.2 in [31]). Consequently, recalling that k is of compact support in ˝, we proceed as in the bounded domain case and take the limit N N 0 ! 1 in (49) to show that u satisfies (45) with ' k . Successively, by taking into account that every ' 2 D(˝) can be approximated in L3 (˝) by linear combinations of k (see Lemma VII.2.1 in [31]) by (46) and by Lemma 1 we conclude that u satisfies (45). Notice that, as in the case of flow in bounded domains, this method furnishes existence for (˝). any > 0 and any f 2 D1;2 0 B. The Function-Analytic Method As in the case of flow in a bounded domain, Subsect. “The Function-Analytic Method”ion IV.1.(B), our objective is to rewrite (45) as a nonlinear operator equation in an appropriate Banach space, where the relevant operator satisfies the assumptions of Lemma 3. In this way, we may draw the same conclusions of Theorem 1 also in the case of a flow past an obstacle. However, unlike the case of flow in a bounded domain, the map ' 2 D(˝) 7! (u r'; u) 2 R can not be extended to a linear, bounded functional in 1;2 1;2 D0 (˝), if u merely belongs to D0 (˝). Analogous conclusion holds for the map ' 2 D(˝) 7! (@u/@x1 ; ') 2 R. The reason is because, in an exterior domain, the Poincaré inequality (18) and, consequently, the embedding 21 are, in general, not true. It is thus necessary to consider the
Navier–Stokes Equations: A Mathematical Analysis
above functionals for u in a space strictly contained in 1;2 D0 (˝). Set ˇ ˇ ˇ ˇ @u ˇ @x 1 ; ' ˇ [juj] :D sup j'j1;2 '2D(˝) and let n X D X(˝) :D u 2 D1;2 0 (˝) :
o [juj] < 1 :
Clearly, X(˝) endowed with the norm j j1;2 C [j j] is a Banach space. Moreover, X(˝) L4 (˝), continuously; see Proposition 1.1 in [36]. We may thus conclude, by Riesz theorem, by Hölder inequality and by the properties of the extension V that, for any u 2 X(˝), there exist L(u), M(u), V (u), and H () in D1;2 0 (˝) such that, for all ' 2 D(˝), @u ; ' D [L(u); '] ; @x1 (u r'; u) D [M(u); '] ; (u r'; V ) (V r'; u) D [V (u); '] ; (H; ') D [H (); '] : Consequently, we obtain that (45) is equivalent to the following equation M(; u) D F
in D1;2 0 (˝) ;
(51)
where M : (; u) 2 (0; 1) X(˝) 7! uC (L(u) C V (u) C M(u))CH () 2 D1;2 0 (˝) : A detailed study of the properties of the operator M is done in § 5 in [36], where, in particular, the following result is proved. Lemma 5 The operator M is of class C 1 . Moreover, for each > 0, M(; ) : X(˝) 7! D1;2 0 (˝) is proper and Fredholm of index 0, and the two equations M(; u) D H () and D u M(; 0)(w) D 0 only have the solutions u D w D 0. From this lemma and with the help of Lemma 3, we obtain the following result analogous to Theorem 1. Theorem 7 For any > 0 and F 2 D1;2 0 (˝) the Eq. (51) has at least one solution u 2 X(˝). Moreover, for each fixed > 0, there exists open and dense Q D Q()
1;2 D0 (˝) with the following properties: (i) For any F 2 Q the number of solutions to (51), n D n(F; ) is finite and odd ; (ii) the integer n is finite on each connected component of Q.
Navier–Stokes Equations: A Mathematical Analysis, Figure 5 Visualization of a flow past a sphere at increasing Reynolds numbers ; after [91]
Finally, concerning the geometric structure of the set of pairs (; u) 2 (0; 1) X(˝) satisfying (51) for a fixed F, a result entirely similar to Theorem 2 continues to hold; see Theorem 6.2 in [36]. Three-Dimensional Flow. Uniqueness and Steady Bifurcation There is both experimental [91,97] and numerical [68,95] evidence that a steady flow past a sphere is unique (and stable) if the Reynolds number is sufficiently small. Moreover, experiments report that a closed recirculation zone first appears at around 20–25, and the flow stays steady and axisymmetric up to at least '130. This implies that the first bifurcation occurs through a steady motion. For higher values of , the wake behind the sphere becomes time-periodic, thus suggesting the occurrence of unsteady (Hopf) bifurcation; see Fig. 5. In this section we shall collect the relevant results available for uniqueness and steady bifurcation of a flow past a three-dimensional obstacle. Uniqueness Results Unlike the case of a flow in a bounded domain where uniqueness in the class of weak solutions is simply established (see Theorem 3), in the situation of a flow past an obstacle, the uniqueness proof requires the detailed study of the asymptotic behavior of a weak solution mentioned in Sect. “Flow in Exterior Domains”; see also Theorem 6. Precisely, we have the following result, for whose proof we refer to Theorem IX.5.3 in [32]. ¯ Theorem 8 Suppose f 2 L6/5 (˝) \ L3/2 (˝), 2 (0; ], ¯ > 0, and let w be the corresponding weak for some solution.(Notice that, under the given assumptions, f 2 1;2 ¯ > 0 such that, if D0 (˝).) There exists C D C(˝; ) k f k6/5 C < C ; then w is the only weak solution corresponding to f .
1023
1024
Navier–Stokes Equations: A Mathematical Analysis
Some Bifurcation Results The rigorous study of steady bifurcation of a flow past a body is a very challenging mathematical problem. However, the function-analytic framework developed in the previous section allows us to formulate sufficient conditions for steady bifurcation, that are formally analogous to those discussed in Sect. “Uniqueness and Steady Bifurcation” for the case of a flow in a bounded domain; see § VII in [36]. To this end, fix f 2 D1;2 (˝), once and for all, and let u0 D u 0 (), 0 in some open interval I (0; 1), be a given curve in X(˝), constituted by solutions to (51) corresponding to the prescribed f . If u C u0 , is another solution, from (51) we easily obtain that u satisfies the following equation in 1;2 D0 (˝) u C (L(u) C B(u 0 ())(u) C M(u)) D 0 ; u 2 X(˝) ;
(52)
where B(u 0 ) :D Du M(u 0 ). In this setting, the branch u 0 () becomes the solution u 0 and the bifurcation problem thus reduces to find a nonzero branch of solutions u D u() to (52) in every neighborhood of some bifurcation point (0; 0 ); see Definition 4. Define the map F : (; u) 2 (0; 1) X(˝) 7! u C (L(u) C B(u 0 ())(u) C M(u)) 2 D1;2 0 (˝) : (53) In § VII in [36] the following result is shown. Lemma 6 The map F is of class C 1 . Moreover, the derivative Du F(; 0)(w) D w C (L(w) C B(u0 ())(w)) ; is Fredholm of index 0. Therefore, from this lemma and from Lemma 4 we obtain that a necessary condition for (0; 0 ) to be a bifurcation point to (52) is that the linear problem w 1 C 0 (L(w 1 ) C B(u 0 (0 ))(w 1 )) D 0 ; w 1 2 X(˝) ;
(54)
has a non-zero solution w 1 . Once this necessary condition is satisfied, one can formulate several sufficient conditions for the point (0; 0 ) to be a bifurcation point. For a review of different criteria for global and local bifurcation for Fredholm maps of index 0, we refer to Sect. 6 in [33]. Here we wish to use the criterion of Theorem 5 to present a very simple (in principle) and familiar sufficient condition in the particular case when the given curve u0 can be made (locally, in a neighborhood of 0 ) independent of . This may depend
