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M and g : N —> N are two Cr, r > 0 maps on manifolds. We say that / and g are topologically conjugate if there is a homeomorphism h : M —> N such that ho f = g oh. Orbits of conjugate maps are in 1-1, continuous correspondence (given by the map h). If the map h is continuous but only surjective (and not necessarily injective), we say that g is a factor of / and we call h a semiconjugacy. Finally, if / is a diffeomorphism and if it is a factor of any homeomorphisms in a C° neighborhood of it, we say that / is topologically stable. In light of this terminology, we can say that the Aubry Mather theorem is a "weak" stability statement: All maps in a C 1 neighborhood of the completely integrable map have the completely integrable map restricted to irrational rotation invariant circles as a factor.
B. From the Annuius to the Cylinder We precede our study by a Lemma, which implies that we can reduce our study to twist maps of the cylinder. Lemma 8.2 Let f be a Ck,k
> 2, twist map of a compact annuius A- Then f can
be extended to a Ck twist map of the cylinder C, in such a way that it coincides with the shear map (x, y) i-+ (a; + cy, y) outside a compact set. In particular, generating function lim
of any lift of the extended map satisfies the growth
the
condition
S(x, X ) —• +oo.
[X— x\—>oo
To prove this lemma, one extends the generating function S from ip(A) to H 2 by interpolating it to the quadratic § (X — x)2 outside of some appropriate compact set. See Forni & Mather (1994) or Moser (1986a) . As a corollary of this lemma, we obtain the following version of the Aubry-Mather theorem:
34
2: AUBRY-MATHER THEOREM
Theorem 8.3 (Aubry-Mather on the compact annulus) Let F be the lift of a twist map of the bounded annulus and suppose that the rotation numbers of the restriction of F to the lower and upper boundaries are p^, and p+ respectively. Then F has orbits of all rotation numbers in [p_, p+]. These orbits are minimizers,
recurrent,
cycli-
cally ordered and they lie on compact invariant sets that form (uniformly)
Lipschitz
graphs over their projections.
invariant
These sets may either be periodic orbits,
circles or invariant Cantor sets on which the map is semi-conjugate
to lifts of circle
rotations.
9. Cyclically Ordered Sequences and Orbits If a map G : IR —> H is the lift of a circle homeomorphism which preserves the orientation, it is necessarily strictly increasing and must satisfy G(x+l)
= G(x) +1. Hence, if
{xk}kez
is an orbit of G, it must satisfy:
(9.1)
xk <Xj+p=>
We will say that a sequence {xk}kei
xk+i < xj+1 +p, V k,j,p
€ Z.
in IR Z is Cyclically Ordered, (or CO in short) if it
satisfies (9.1). Clearly the CO sequences form a closed set for the topology of pointwise convergence in M z : x^
—» x whenever x?k —* xk for all k. Note that this topology is
the same as the product topology on the space of sequences. Using the partial order on sequences (it comes with three degrees of strictness): x < y *> {\fk, xk < yk} x < y o {Vk, xk < yk x p(x) is a continuous function on CO sequences, with the topology of pointwise
convergence.
Define: CO[aM
=
{x€CO\p(x)e[a,b}}.
The following lemma shows that it is easy to find limits of CO sequences, as long as their rotation numbers are bounded. Lemma 9.2 The sets C O ^ / T ^ O and CO[a^ f l { i 6 M z | x0 e [0,1]} are compact for the topology of pointwise
convergence.
We give the proofs of both these lemmas in the appendix to this chapter. The fact, given by these lemmas, that the rotation number behaves well under limits of CO-sequences is one of the essential points in the theory of twist maps that does not generalize to higher dimensional maps: to our knowledge, there is no dynamically natural definition of CO sequences in 1R™, n > 2 which ensures the existence of rotation vectors which behave well under limits. Note that there is, however, a natural generalization of CO sequences in the context of maps 2 d —> M, see Chapter 9. There is a visual way to describe CO sequences, which we now come to. A sequence x in IR is a function 7L —> H . One can interpolate this function linearly and obtain a piecewise affine function IR —> IR that we denote by 11—> xt. The graph of this function is sometimes This is not an indictment of the authors who have used these terminologies: the author of this book has himself used them all in various publications...
36
2: AUBRY-MATHER THEOREM
called the Aubry diagram of the sequence. We say that two sequences x and w cross if their corresponding Aubry diagrams cross. There are two types of crossing: at an integer k, in which case (xk-i
- Wk-\){xk+i
— w/t+i) < 0 or at a non integer t 6 (k,k + 1), in
which case (xk — Wk){xk+\ — W/H-I) < 0. These inequalities can be taken as a definition of crossings. Non-crossing of two sequences can be put in terms of the partial order on sequence: x, y do not cross if and only if x < y . In particular a sequence x is CO if and only if it has no crossing with any of its translates
Tm^nx.
F i g . 9.0. Aubry diagrams of sequences and their crossings: in this example the sequences x and w have crossings at the integer fc and between the integers j and j + 1.
10. Minimizing Orbits Throughout the rest of this chapter, we consider a lift F of a given twist map / of the cylinder, and its corresponding generating function S, action function W, periodic action function Wmn
and change of variable ip. A sequence segment (xk,. • •, xm) is (action)
minimizing if W(xk,•••,xm)
< W(yk,•
••,ym)
for any other sequence segment (yk, • • •, ym) with same endpoints: xk = yk,xm
= ym.
Since minimizing segments are necessarily critical for W, they correspond to orbit segments called (action) minimizing orbit segment. A bi-infinite sequence is called a (global action) minimizer if any of its segments is minimizing. The orbit it corresponds to is a minimizing orbit, or simply minimizer, when the context is clear. Note that the set of minimizers is closed under the topology of pointwise limit (see Exercise 10.5). Finally a is a periodic sequence in Xm^n that minimizes the function
Wmn.
Wmn-minimizer
10. Minimizing Orbits
37
A recurrent theme in the Calculus of Variation is that minimizers have regimented crossings. In the case of geodesies on a Riemannian manifold, geodesies that (locally) minimize length cannot have conjugate points, i.e. small variations with fixed endpoints of a minimizing geodesic only intersect that geodesic at the endpoints ( Milnor (1969)), and geodesies that minimize length globally cannot have self intersections (Mane (1991), page 102 ). We will see, in the present theory, that minimizers satisfy a non-crossing condition, which implies that Wm n -minimizers (and more generally, recurrent minimizers) are CO.
Lemma 10.1 (crossing) Suppose that (x — w)(X — W) < 0. Then: S(x, X) + S(w, W) - S{x, W) - S(w, X) < 0, and equality occurs iff {x — w)(X — W) = 0 Proof.
We can write: S(x,X)-S{x,W)=
f d2S{x,Xs){X Jo
-W)ds,
where Xs — (1 — s)W + sX. Applying the same process to h(x) = S(x, X) — S(x, W), we get: S(x, X) + S{w, W) - S(x, W) - S{w, X) = h{x) - h(w) =
-L
f di2S(xr, Jo
XS){X - W)(x - w)dsdr = \{X - W){x - w)
for some strictly negative A, by the positive twist condition and for xr = (1 — r)w + rx.
D
The following is a watered down version of the Fundamental Lemma in Aubry & Le Daeron (1983). We follow Meiss (1992): Lemma 10.2 (Aubry's Fundamental Lemma) Two distinct minimizers
cross at most
once. Proof.
Suppose that x and w are two distinct minimizers who cross twice. We perform
some surgery on finite segments of x and w to get two new sequences x' and w' with at least one of them of lesser action, contradicting minimality. There are three cases to consider: (i)
38
2: AUBRY-MATHER THEOREM
both crossings are at non integers, (ii) one crossing is at an integer, (iii) both crossings are at integers.
Fig. 10.2. A crossing of Case (ii) Case (i):Letti € (i — l,i)andt 2 € (j,j + 1) be the crossing times. Define: x, = k
(wk if k e [i,j] \ Xk otherwise
fc
w,
=
(xk if ke [i,j] \ Wk otherwise
Letting W denote the action over an interval [N, M] containing [j — 1, k + 1], we easily compute that:
W(x') + W(w')-W(x)
- W(w) =
S(Xi-l,Wi)
+S{xj,wj+i)
+ S(Wi-!,Xi)
+ S(wj,xj+i)
- S(Xi-i,Xi)
- S(xj,xj+1)
-
S(Wi-i,Wi)
-
S(wj,wj+i).
The Crossing Lemma 10.1 shows that this difference of actions is negative, contradicting the minimality of x and w. Case (ii): In this case, only one crossing will contribute negatively to the difference of action of new and old sequences. We still get a contradiction. Case (iii) Let i — 1 and j + 1 be the crossing times of x and w, and construct x' and w' as before. In this case the difference in action between old and new segments is null. The sequences x',w' must be minimizing, and hence correspond to orbits. But we have Xj_2 = u^_2,
Xi-i = w'i_1. Hence the points ip~1(xi^2,Xi-i)
and i>~1(w'i_2,w'i_1)
2
of IR are the same and thus generate the same orbit under F. This in turn implies that x = w, a contradiction to our assumption.
D
11. Minimizing Orbits Corollary 10.3 Wmn-minimizing
39
sequences are CO and their set is completely ordered
for the partial order on sequences. Proof.
Since the proof of Aubry's Lemma deals with finite segments of sequences only,
it also applies to show that two Wmn-minimizers in X m n , may not cross twice within one period n. But two m, n-periodic sequences that cross once must necessarily cross twice within one period. Hence two Wmn-minimizers cannot cross at all. It is easy to check that Wmn is invariant under Tjj for all integers i, j . Thus, if a: is a Wmn minimizer, ritjX is also a Wmn-minimizer. Since they do not cross, one must have either x < Ti^x or n^x
< x,
for all i, j e Z , i. e. x is CO.
•
We end this section by a proposition which we will need in Chapter 3. Proposition 10.4 Any Wmn-minimizer Proof.
is a
minimizer.
We will show that if m is a W mn -minimizer, it is also a Wkmkn minimizer for
any k. This implies that x is a minimizer on segments of arbitrary length: if x is a Wkmkn minimizer, any segment of x of length less than kn is minimizing. Hence x is a minimizer. Now, take a Wfcmfcn-minimizer w. Ifw is not m, n-periodic, then w and Tm,nw are distinct. By Corollary 10.3, they cannot cross. Suppose, say, that rm w. Since r m n trivially preserves the (strict) order on sequences, we must also have rm nw > w, a contradiction to the fact that w is km, kn- periodic. Hence w is in Xmn
and its action over intervals of any
length multiple of n cannot be less than that of x. Hence x is also a Wkmkn minimizer.
•
Exercise 10.5 Show t h a t the set of minimizers (either sequences or orbits) is closed under pointwise limits. Exercise 10.6 a) Show t h a t the set of recurrent minimizers of rotation number u> is completely ordered. {Hint. Mimic the proof of Proposition 10.4: if an appropriate inequality is not satisfied, there must be a crossing. By recurrence, there is another one, a contradiction to Aubry's Lemma). b) Show t h a t a minimizer corresponding to a recurrent (not necessarily periodic) orbit of the twist map is CO. (Remember t h a t the orbit zk of a dynamical system is called recurrent if zo is the limit of a subsequence Zkj • Equivalently, zo is in its own u;-limit set).
40
2: AUBRY-MATHER THEOREM
11. CO Orbits of All Rotation Numbers A. Existence of CO Periodic Orbits We prove that the set of Wmn-minimizers is not empty. By Corollary 10.3 this will show the existence of CO orbits of all rational rotation numbers. Proposition 11.1 Let F be the lift of a twist map with a generating function satisfies the coercion condition Ymi\x-x\^>oo S{x,X) Wmn Proof.
has a minimum
on
which
—> +oo. Then, for all m,n,
Xmn.
Note that, by periodicity of S, the ranges of Wmn on X m n and on its subset
Xm,n n {xi G [0,1]} are the same: we can translate any sequence of Xm^n by an integer to bring it to that sub>> i without changing its action. Now, if S satisfies the coercion condition, then for x € Xm,n D \x\ G [0,1]}, limn-,.^^ Wmn(x)
—> +oo: if ||x|| —> oo and xi
remains bounded, at least one \xk — Xk-\\ must tend to +oo. In particular, for any large enough K G H , W^(—oo, K] is bounded and not empty. Since, by continuity, this set is also closed, it must be compact. Thus Wmn attains its minimum on that set.
D
An interesting sufficient condition for S to satisfy the coercion condition is that the "twist" of the map be uniformly bounded below (see MacKay & al. (1989)): Proposition 11.2 Let the twist condition for the lift of a twist map F be uniform: ^ p y l > dy
a >
0
V(x,y)GlR 2 .
Then there is a constant a, and two strictly positive constants (3 and 7 such that : S(x, X) > a - 0 \X - x\ + 7 \X - x\2 .
Proof.
We can write:
S(x,X) = S(x,x)+
f Jo
d2S(x,Xs)(X-x)ds,
where Xs = (1 — s)x + sX. Applying the same process to 92-5, we get:
12. Aubry-Mather Sets
S(x,X) = S(x,x) + f Jo
41
d2S(Xs,Xs)(X-x)ds d12S(Xr,Xs)(X~x)2.dr
- f ds [ Jo Jo
We can conclude the proof of the lemma by taking a = mmS(x,x),
f3 = max \d2S(x, x)\
x£lR
ieIR
(a, /? exist by periodicity of S) and 7 = a/2.
•
B. Existence of CO Orbits of Irrational Rotation Numbers
The existence of CO orbits of irrational rotation numbers is a simple consequence of the existence of CO periodic orbits: pick a sequence x^> of VFmfc]nfc-minimizers, with rrik/nk —> u ask —» 00. By using appropriate translations ofthe type rm>o on x^ (which neither change their rotation numbers, nor the fact that they are minimizers) we can assume
that 4*° e [0,1]. The sequence mk/rik is bounded and hence, by Corollary 10.3 the sequences x^ are in CO[atb] n {a: e IRZ | xo G [0,1]} for some a, b G IR. Lemma 9.2 guarantees the existence of a converging subsequence in CO[a^] and Lemma 9.1 shows that the limit of this subsequence has rotation number LJ. Finally, note that the periods nk go to infinity as k goes to infinity. In particular, any finite segment of a limit x of x^ is the limit of minimizing segments, hence minimizing itself (Exercise 10.5). •
12. Aubry-Mather Sets We have proven Part (1) and (3) of the Aubry-Mather theorem: existence of cyclically ordered, minimizing orbits of all rotation numbers. We now prove Part (2): the cyclically ordered orbits that we found in the previous section lie on Aubry-Mather sets, which we describe in this section. We say that a set M in IR2 is F-ordered if, for z, z' in M, n(z) < ir{z') => w(F{z)) < n(F{z')), where 7r is the a;-projection. A set is F-ordered invariant if it is JP-ordered and invariant under both F and F _ 1 . On such a set, the sequences x, x' of ^-coordinates of z and z'
42
2: AUBRY-MATHER THEOREM
must satisfy x -< x'. An example of F-ordered invariant set is the set of points in a CO orbit and all their integer translates. In fact, this can be used to give an alternative definition of CO orbits: an orbit is CO if and only if its points form an F-ordered invariant set. Note that an invariant circle which is a graph is F-ordered invariant (we will see in Chapter 6 that all invariant circles are graphs). We now want to explore the properties of F-ordered invariant sets. Crucial to the properties of these sets is the following ratchet phenomenon (I owe this terminology to G.R. Hall), which is a somewhat quantitative expression of the twist condition. This phenomenon, or condition is best described by the following picture: a
='•
G F ^
Ft 72)
.
n.,
•"*>
F i g . 1 2 . 0 . T h e ratchet phenomenon for the lift of a positive twist map F: there are two cones (shaded in this picture) „, then F(z') e F(z) + 0/,. More precisely, for a positive twist map z' e z + Gi => F(z') e F(z) + 6>+, where the half cones 0^,0+ have the obvious meaning. The same holds for the half cones 0^ and 0^ • If g is negative twist (eg. F _ 1 ) , then the signs are reversed. T h e same cones can be used for F'1 as for F.
Lemma (Ratchet) 12.1 Let F be the lift of a twist map satisfying ^ > a > 0 in some region. Then, in that region, F satisfies the ratchet phenomenon for some cones 0v,&h whose angles only depend on a. Proof. See Exercise 12.9. Proposition 12.2 The closure of an F-ordered invariant set is F-ordered and invariant. Proof. The invariance is by continuity of F. Let M be an F-ordered invariant set. We let the reader prove that the uniform twist condition^ > a > 0 is automatically satisfied on an
12. Aubry-Mather Sets
43
F-ordered invariant set (essentially, such a set is necessarily bounded in the y direction, see Exercise 12.9). Suppose that, in the closure M of M there are z, z' in M , with -K{Z) < -ir(z') but n(F(z))
= 7r(F(z')) (the worst case scenario). By the ratchet phenomenon for F ~ \
F(z) must be above F(z') and TT(F2(Z'))
< TT(F2(Z)),
i.e. the x orbits of z and z' switched
order. This is impossible since in M the (strict) order is preserved by F.
Proposition 12.3 If M is an F-ordered invariant set, then it is a Lipschitz
graph
over its projection: there exists a constant K depending only on F such that, if (x, y) and (x1 ,y') are two points of M,
then:
\y' - y\ < K\x' - x\ with K only depending on the twist constant a = inf M ^ •
Note that a, and hence K could also be chosen the same for all F-ordered sets in a compact region.
Proof.
The proof of Lemma 12.2 shows that if M is F-ordered, we cannot have z, z'
in M and ir(z) = TT(Z') unless z = z'. Hence IT is injective on M , and M is a graph. To show that M forms a Lipschitz graph over its projection, let z and z' be two points of M and x and x' the corresponding sequences of x-coordinates of their orbits. Assuming 7r(z) < ir(z'), we must have x -< x'. If z' € z + 6>+, the ratchet phenomenon implies that F~1(z')
€ F~l{z)
+ O^, i.e. x'_x > x~\, a contradiction. Likewise z' cannot be in
the cone z + 0~, and hence it must be in the cone complementary to Ov at z. This cone condition is easily transcribed into a uniform Lipschitz condition \y' — y\ < K\x' — x\.
O
Remark 12.4 Applied to the special case of invariant circles, Proposition 12.3 shows that any invariant circle for a twist map which is a graph is Lipschitz. This is a theorem originally due to Birkhoff, who also proved (see Chapter 6) that all non-homotopically trivial invariant circles for twist maps must be graphs.
Lemma 12.5 All points in an F'-ordered set have the same rotation
number.
44
2: AUBRY-MATHER THEOREM
Proof.
This is a consequence of the simple fact (Lemma 13.3 in the appendix) that if
x < x' are two CO sequences, they must have the same rotation number.
D
Definition 12.6 An Aubry-Mather set M for the lift F of a twist map / of the cylinder is a closed, F-ordered invariant set which is also invariant under the integer translation T.
Note that some authors call Aubry-Mather sets the projections of the above sets to the annulus. Exercise 12.9 shows that these projections are necessarily compact. Taking the closure of all the integer translates of the points in the CO orbits found in the previous section, we immediately get:
Theorem 12.7 Let F be the lift of a twist map of the cylinder. Then F has AubryMather sets of all rotation numbers in IR. Any CO orbit is in an
Aubry-Mather
set.
Note that this theorem gives part (b) of the Aubry-Mather theorem.
Theorem 12.8 (Properties of Aubry-Mather sets) Let M be an Aubry-Mather
set for
a lift F of a twist map of the cylinder. (a) M forms a graph over its projection TV(M),
which is Lipschitz
with
Lipschitz
constant only depending on the twist constant a = infM ^~ • (b) All the orbits in M are cyclically ordered and they all have the same
rotation
number, which is called the rotation number of M. (c) The projection ir(M) is a closed invariant set for the lift of a circle homeomorphism, and hence F restricted to M is conjugated to the lift of a circle
homeomor-
phism via -n.
Proof of Theorem 12.8. We have shown in Lemmas 12.5 and 12.6 that (a) and (b) are in fact properties of F-ordered invariant sets. As for Property (c), since ir is one to one on M, F induces a continuous (Lipschitz, in fact) increasing map G on TT{M), defined by G(TT(Z))
= TT(F(Z).
Since M and thus 7r(M) are invariant under integer translation, we
have G(x + 1) = G(x) + 1. The set ir(M) is closed and invariant under integer translation
12. Aubry-Mather Sets since M is. If 7r(M) = IR, then G is the lift of a circle homeomorphism. If n(M)
45 ^ IR,
then its complement is made of open intervals. Extend G by linear interpolation on each interval in the complement of n(M). Since G is increasing on n(M), its extension to IR (call it G) is increasing as well, continuous and G(x + 1) = G(x) + 1, hence the lift of a circle homeomorphism. By construction G(TT(Z)) = n(F(z)), and 7r | M is a continuous, 1 -1 map on the compact set M, hence a homeomorphism M —> 7r(M). Thus 7r is a conjugacy between i*1 on M and G on 7r(M), which is a closed and invariant set under G and G _ 1 .
•
Recapitulation on the Dynamics of Aubry-Mather Sets. If G is the lift of a circle homeomorphism constructed in the proof of Theorem 12.7, the possible dynamics for invariant sets of circle maps described in the appendix become, under the conjugacy, possible dynamics on Aubry-Mather sets M for F. Hence an Aubry-Mather set M is either: (i) an ordered collection of periodic orbits with (possibly) heteroclinic orbits joining them, or (ii) the lift of an /-invariant circle, or (iii) an F-invariant Cantor set with (possibly) homoclinic orbits in its gaps. The rotation number of M is necessarily rational in Case (i), and necessarily irrational in Case (iii). In Case (ii), M may either have a rational or irrational rotation number, as the example of the shear map shows. However, maps with rational invariant circles are non generic. Indeed, as a circle map, the restriction of the twist map to the invariant circle must have a periodic orbit. For generic twist maps, periodic orbits must be hyperbolic and the circle must be made of stable and unstable manifolds of such orbits, that coincide. But generically, such manifolds intersect transversally. See Herman (1983) and Robinson (1970) for more details. As for homoclinic and heteroclinic orbits as in (i) and (iii), they have been shown to exist each time there are no invariant circles of the corresponding rotation numbers, see Hasselblat & Katok (1995), Mather (1986). The feature that is striking in the Aubry-Mather theorem is the possible occurrence of Aubry-Mather sets as in (iii). The F-invariant Cantor sets have been called Canton by Percival (1979)who constructed them for the discontinuous sawtooth map (a standard map with sawtooth shaped potential). This type of dynamics does occur in twist map, since it can be shown that many maps have no invariant circles, and hence the irrational Aubry-Mather sets must be of type (iii), i.e. contain Cantori.
46
2: AUBRY-MATHER THEOREM Although one can construct many Aubry-Mather sets that are not made of minimizers
(Mather (1985)), the name "Aubry Mather set" is often reserved to the action minimizing Cantori Mw as defined below:
Proposition 12.9 For each irrational rotation number ui there is a unique Mu made of recurrent minimizing CO minimizing
Proof.
Cantorus
orbits of rotation number w. The closure of any
orbit of rotation number w is contained in
Mu.
A CO minimizing orbit forms an F-ordered set, contained in an Aubry-Mather set,
and hence conjugated to an orbit of a circle homeomorphism. The closure of the irrational CO minimizing orbit is therefore in a Cantorus, conjugated to the w-limit set of the circle homeomorphism. As limit of minimizers, this Cantorus is made up of minimizers. We now prove that this Cantorus is unique: suppose there are two of them. Exercise 10.6 implies that the (disjoint) union of these two Cantori forms an F-ordered set, hence conjugated to a closed invariant set of a circle homeomorphism. Each Cantorus is the w-limit set of its points. This is a contradiction to the uniqueness of ui limit sets of circle homeomorphisms proven in Theorem 13.4.
•
Exercise 12.9 a) Prove the Ratchet Lemma 12.1. b) Prove that if F is an .F-ordered invariant set, then the projection proj(M) of M to the cylinder is compact, /-invariant. Deduce from this that M satisfies the uniform twist condition dX/dy > a > 0. [Hint. Use Lemma 9.2]. Exercise 12.10 Show that a twist map / restricted to a Cantorus (irrational Aubry-Mather set) is semiconjugate to a rotation of the same rotation number.
13. Appendix: Cyclically Ordered Sequences and Circle Maps
In this section, we prove Lemma 9.1, and Lemma 9.2. We then recover important facts about circle homeomorphisms and their invariant sets using the language of CO sequences.
13. Appendix: CO Sequences
47
A. Proofs of Lemmas 9.1 and 9.2 We recall the statements of each lemma before proving it. Part of the proof below is classical, due to Poincare in his study of circle homeomorphisms. Lemma 9.1 Let {xk}kzz
be a CO sequence then p(x) = limfc_,oo Xk/k exists and:
(13.1)
\xk - x0 - kp(x)\ < 1.
Moreover x —> p(x) is a continuous function
on CO sequences, when the set of
sequences has been given the topology of pointwise
convergence.
Proof . Let x be a CO sequence. We first prove that the sequence { X r t ~ x o } r a ez is Cauchy as n —> ±oo. We do the case n —• +oo first. Given n 6 IN, let an be the integer such that: (13.2)
xo + a „ < xn < x0 + an + 1.
We prove by induction that (13.3)
xo + kan < Xkn < ^o + kan + k,
Vfc e IN.
Indeed, step 1 in the induction is just (13.2), and if we assume step k, i.e. (13.3) then, since x is CO, we get xn + kan < X(fc+i)n < xn + kan + k. Using (13.2) this gives x0 + (k + l)an < a; (fc+1)n <x0 + (k + l ) a „ + (k + 1), which is the step k + 1 and finishes the induction. Dividing (13.3) by k we get Xkn
(13.4)
an
0, we must have, for all n ^ 0, the two equivalent inequalities (13.5) Writing zn =
*£fcn
*^0
3-n
2-0
l Xn XQ n ,
< 14*
•Efcn
kn
^0
~^n
*^0
0, n > 0, the triangular inequality gives:
48
2: AUBRY-MATHER THEOREM
(13.0)
\Zn — Zm\
< \Zn — Zmn\
+ \zmn
~
z
m\
SL
I
j
and hence {-zn}neiN, is a Cauchy sequence whose limit we call p(x). Let m —> oo in (13.6), and multiply by n: (13.7)
\xn - xo - np(x)\ < 1.
To see how the case n —> — oo follows, note that in all the above we could have replaced xo by an arbitrary xm,m S 2 and obtained: (13.8)
\xn — xm — (n — m)p(x)\
oo. Constructing sequences z^
as above, and denoting
p(xW>) = Wj, (13.7) yields a)
(13.9)
1
14 -^I Zk-ojj\ z, for all A; and e > 0, |W, -
Wi|
< | W , - s£>| + \z? - 4 4 >| + | z « -
Wi|
< | +e
whenever z, j are big enough. Hence {u)k}kei. is a Cauchy sequence, whose limit we denote by u>. Letting j —> oo in (13.9) yields w = /J(:E)-
•
Lemma 13.1 T/ie sets COfa.&j/n.o a«d C,0[Ojt] PI {x € 1RZ | xo € [0,1]} are compact for the topology of pointwise Proof.
convergence.
We have already remarked that, trivially, CO is closed for pointwise convergence,
i.e. the product topology on sequences. Lemma 9.1 implies that CO [„_(,] fl{ x | XQ € [0,1]} is a closed subset of the set: {x £ E
Z
| xk = x0 + ku) + yk,(x0,u>,y)
e [0,1] x [a,b] x [ - 1 , l ] 2 , withj/o = 0 }
which is compact for the product topology, by Tychonov's theorem. We let the reader derive a similar proof for C O ^ / T ^ O .
•
13. Appendix: CO Sequences
49
B. Dynamics of Circle Homeomorphisms
Rotation Numbers and Circle Homeomorphisms. The orbits of an orientation preserving circle homeomorphism are by definition Cyclically Ordered. From Lemma 9.1, we can deduce the following theorem, due to Poincare (1885): Theorem 13.2 All the orbits of the lift F of an orientation preserving circle homeomorphism f have the same rotation number, denoted by p(F). The rotation number p is a continuous function of F, where the set of lifts of homeomorphisms of the circle is given the C° topology. Proof. We start by a simple but useful lemma. Lemma 13.3 If two CO sequences x,x' satisfy x < x' then p{x) = p(x'). Proof. The rotation numbers are the respective asymptotic slopes of the Aubry diagram of x and x'. Thus, if p{x) / p(x'), the Aubry diagrams of x and x' must cross. In this case, there must be a ko and a fci such that Xk0 > x'ko and xkl < x'ki. This contradicts x < x'. • Continuing with the proof of Theorem 13.2, since F is increasing, two CO sequences x and w corresponding to distinct orbits of F must satisfy x -< w or w -< x. From the previous lemma x and w have same rotation number. Finally, if /„ —> / in the C° topology, then the /„ orbit of a point x (a CO sequence) tends pointwise to the / orbit of x. By Lemma 9.1, ]imp(fn) = limp({/*(:r)}fc6z) = p({fk(x)}keZ) = p(f). D Dynamical classification of circle homeomorphisms. We now review the classification of circle homeomorphisms by Poincare (1885). Recall some general terminology from dynamical systems. The Omega limit set u>(x) of a point x under a dynamical system / is the set of limit points of the forward orbit, i. e. the set of limit points of all subsequences {xkj} where xk = fk(x) and kj —> +oo as j —> +oo. Likewise, the Alpha limit set a{x) is the set of limit points of the backward orbit. A minimal invariant set for a dynamical system is a closed, (forward and backward) invariant set which contains no closed invariant proper subset. A heteroclinic orbit between two invariant sets A and B is the orbit of a point x such that ot(x) C A and ui(x) C B. The term homoclinic is used when A = B.
50
2: AUBRY-MATHER THEOREM
Theorem 13.4 Let f be a circle homeomorphism then, for any x € $ , ui{x) and a(x) periodic (in which case x € u(x)
and F a lift of f. If p(F) is rational,
are periodic orbits. The orbit of x is either
= ot{x)) or it is heteroclinic between a(x)
and
If p(F) is irrational, then, for any x,x' € S 1 , a{x) = a{x') = OJ(X) = w(x').
Call
OJ(X).
this set O(f).
Then f2{f) is either the full circle, or a minimal invariant set which
is a Cantor set. In the first case any orbit is dense in the circle, and f is conjugated to a rotation by p(F). In the second case, a point x of S 1 is either in Q{f)
and
recurrent, or it is homoclinic to fi{f),
to a
a "gap orbit", and f is semi-conjugate
rotation by p(F).
We remind the reader that a Cantor set K is a closed, perfect, and totally disconnected topological set. Perfect means that each point in K is the limit of some (not eventually constant) sequence in K, and totally disconnected means that, given any two points a and b in K, one can find disjoint closed sets A and B with a G A,b 6 B and A U B = K. In the real line or the circle, a closed set is totally disconnected if and only if it is nowhere dense. A set X is nowhere dense if Interior(Closure{X))
Proof of Theorem
= 0.
13.4.
Rational rotation number. Suppose p{F) = m/n. Then Fn(-) — m must have a fixed point, otherwise for all x G H , Fn(x)
— x ^ m and we can assume Fn{x)
— x > m. By
1
compactness of S , p(F) > m/n, a contradiction. Hence F has an m, n-periodic orbit. By continuity, on any interval / where Fn — Id — m is non zero, it must stay of a constant sign. This sign describes the direction of progress of points inside the orbit of / towards its endpoints: they must be heteroclinic to the endpoint orbits. Conversely, if F has an m, n-periodic orbit, its rotation number and thus that of F must be m/n. Irrational rotation number. Suppose p(F) is irrational. Let x e S {xk}kel
its
1
and denote by x =
1
orbit under / (with x = x0). Suppose u(x) = S . We show that LU(X') = S 1
for any other x' e S 1 . Suppose not, and there is an interval (a, b) which contains no x'k = fk(x').
But (a, b) must contain some [ x n , i m ] by density of x. Again by density,
the intervals f-^™--™)[xn,xm]
must cover S 1 and hence / i ( m -™)x / £ (a, b) for some i, a
13. Appendix: CO Sequences
51
contradiction. We guide the reader through the proof that / is conjugated to a rotation by p(f) in Exercise 13.6. Suppose OJ(X) •£ S 1 . Then, since u>(x) is closed, its complement is the union of open intervals. Take another point x'. We want to show that ui(x') = u){x). We will prove that UJ(X') C u>(x): by symmetry UJ(X) C ui(x'). This is obvious if x' G w(x). Suppose not. Then x' is in an open interval I in the complement of UJ(X) whose endpoints are in ui(x). The orbit of I is made of open intervals in the complement of UJ (x) whose endpoints are orbits in w(x).
Since there is no periodic orbit, these intervals are disjoint: by the intermediate value
theorem fk(I)
C I would imply the existence of a fixed point for fk, hence a periodic
orbit. The length of these intervals must tend toward 0 under iteration. Thus the orbit of x' approaches the endpoint orbit of I arbitrarily i.e. the orbit of x' is asymptotic to UJ(X). Hence UJ(X') C u)(x). In particular UJ(X) = (2(f) is a minimal invariant set: any closed invariant subset of (2(f) must contain the w-limit set of any of its point, hence (2(f) itself. We now show that (2(f) is a Cantor set. That it is closed is a property of w-limit sets. It is perfect since x 6 (2(f) means that x e u(x) and hence fnk (x) —> x for some nk / nk
the f
oo and
(x)'s are in w(x), and are all distinct. To prove that (2(f) is nowhere dense, first note
that the topological boundary d(2(f) = (2\Interior((2(f)) or df2(f)
must satisfy d(2(f)
= (2(f)
= 0: dQ(f) is closed, invariant under / and included in fi(f) which is a minimal
set. But dil(f)
= 0 means fi(f)
= Interior(il(f))
is open, and because it is also closed,
1
it must be all of S , which we have ruled out. The alternative is dfl(f) means Interior(f2(f))
= O(f), which
= 0, which is what we wanted to prove. Exercise 13.6 walks the
reader through the proof that / is semi-conjugate to a rotation in this case.
•
Remark 13.5 A circle homeomorphism with an invariant Cantor set cannot be too smooth: Denjoy (see Hasselblat & Katok (1995), Robinson (1994)) proved that if / is a C 1 diffeomorphism of S 1 with irrational rotation number and derivative of bounded variation, then / has a dense orbit (i.e. fi(f) p(F).
= S 1 ) and is therefore conjugated to a rotation of angle
On the other hand, Denjoy did construct a C 1 diffeomorphism with O(f) a Cantor
set. The idea is simple: take a rotation by irrational angle a. Cut the circle at some point x and at all its iterate fk(x).
Glue in at these cuts intervals Ik of length going to 0 as k -+ oo,
in such a way that the new space you obtain is again a circle. Extend the map / by linear interpolation on the Ik. You get a circle homeomorphism with rotation number a. With
52
2: AUBRY-MATHER THEOREM
some care, one can make this homeomorphism differentiable, but only up to a point ( C 1 with Holder derivative). T h e complement of the Ik's in the new circle is a Cantor set, which is minimal for the extended map.
Exercise 13.6 In this exercise, we prove t h a t all orientation preserving circle homeomorphism with irrational rotation number ui has the rotation of angle u a s a factor. This is sometimes called Poincare's Classification Theorem (see Hasselblat & Katok (1995)). a) Prove t h a t a; is a CO sequence with irrational p(x) iff Vn,m,p£"E,
xn < xm + p •*=> np(x) < mp{x) + p
(Hint. Use Formula (13.8) for multiples of m and n). W h a t is t h e proper corresponding statement for CO sequences of rational rotation number? b) Suppose the circle homeomorphism / has a dense orbit, which lifts t o an orbit x of some F. Build a monotone map h : 1R —• IR by first defining it on x by: i t + m i - > kp(x) + m,
Vm, k £ 7L.
Use a) to show t h a t h is order preserving and show t h a t its extension by continuity is well defined, has continuous inverse and preserves orbits of F, and it commutes with the translation T (Hint, density of the orbit in S 1 means density of t h e set {x/c + m}k.mez in IR). Hence in this case / is conjugate to a rotation. c) Suppose now t h a t O(f) ^ S 1 . Following the steps in b), take a dense orbit x in Q(f) and build a map h : fi(F) —> ]R as before (0(F) denotes the lift of P(f) here). Check t h a t this m a p is onto, non decreasing and extend it to a m a p IR —* IR by mapping each the gap of the Cantor set t o a single point. d) Conclude that, in both cases, h provides a (semi)-conjugacy between / and a rotation by u.
3 GHOST CIRCLES In Chapter 2, we saw how traces of the invariant circles of the completely integrable map persist, either as invariant circles, as periodic orbits or as invariant Cantor sets, in any twist map. The main result of this chapter, Theorem 18.1, provides a vertical ordering of these Aubry-Mather sets in the cylinder for each given map. Indeed, we show that each Aubry-Mather set is a subset of a circle in a family of disjoint, homotopically nontrivial circles that are graph over the circle {y = 0}. The circles in this family are ordered according to the rotation number of the Aubry-Mather sets. To prove this, we establish important properties of the gradient flow of the action functional in the space of sequences. The central property, given by the Sturmian Lemma, is that the intersection index of two sequences cannot increase under the gradient flow of the action. One consequence is that the flow is monotone: it preserves the natural partial order between sequences. This fact yields a new proof of the Aubry-Mather Theorem. It also enables us to define special invariant sets for the gradient flow that we called ghost circles, which we study in some detail here. The family of circles that neatly arranges the Aubry-Mather sets are projections of ghost circles in the cylinder. The results of this chapter come from three sources: Gole (1992 a), in which properties of ghost circles were systematically investigated; Gole (1992 b), where gradient flow techniques were used to give a proof of the Aubry-Mather theorem. There was a gap in that last paper, pointed out to me by Sinisa Slijepcevic which isfixedhere thanks to a lemma from Koch & al. (1994). Finally, the bulk of this chapter comes from Angenent & Gole (1991), in which we gave a proof of the ordering of Aubry-Mather sets via ghost circles. I am deeply indebted to Sigurd Angenent for letting me publish this work here for the first time. The notion of ghost circles originated in my thesis, in which I was looking for regularity properties for
54
3: GHOST CIRCLES
ghost tori, their higher dimensional counterparts. In Chapter 5, a link is made between ghost tori and Floer Homology.
14. Gradient Flow of the Action A. Definition of the Flow Throughout this chapter, we consider a twist map / of the cylinder and its lift F whose generating function S is C2. For simplicity, we will also assume that the second derivative of S is bounded. This mild assumption is satisfied for twist maps of the bounded annulus which are extended to maps of the cylinder as in Lemma 8.2, as well as for standard maps. In this section we investigate the property of the "gradient" flow of the action associated with the generating function S of F solution to: (14.1)
xk = -VW(x)k
= -[d1S(xk,xk+1)
+ d2S(xk-i,Xk)},
fceZ
Since this is an infinite system of ODEs, we need to set up the proper spaces to talk about such aflow.We endow 1RZ with the norm :
—oo
We let X be the subspace of IR of elements of bounded norm, which is a Banach space. On bounded subsets of X, the topology given by the above norm is equivalent to the product topology, itself equivalent to the topology of pointwise convergence. Remember from Chapter 2 that Z 2 acts on IR by:
The map
T0>I
which we also denote by T has the effect of translating each term of the
sequence by 1. The map r^o which we denote also by a is called the shift map, as it shifts the indices of a sequences by 1. We define X/~Z := X/T and we can choose as a representative of a sequence x one such that xo € [0,1). More generally, in this chapter, the quotient of any subset of IRZ by Z will be with respect to the action of the translation T = T 0> I.
Proposition 14.1 Suppose that the generating function S is C2 with bounded second derivative. The infinite system of O.D.E's
14. Gradient Flow of the Action
(14.2)
xk = -VW(x)k
= -[dxS{xk, xk+i) + d2S(xk-i,
55
xk)]
defines a C1 local flow £* on X as well as on XjlL, for the topology of pointwise convergence. The rest points of £' on X correspond to orbits of the map F. Proof. We prove that the vector field — VW is C 1 by exhibiting its differential. The proposition follows from general theorems on existence and uniqueness of solutions of ODEs in Banach spaces (Lang (1983), Theorems 3.1 and 4.3). The following map is the derivative of x — i > — WW(x): L • {vk}keJ. >-> {PkVk-i + akVk + Pk+iVk+i}keZ Oik = -d22S{xk-i,xk)
-dnS{xk,xk+1),
(3k =
~di2S{xk-i,xk)
Indeed, this map is linear with (uniformly) bounded coefficients, hence a continuous linear operator. Clearly: -VW{x
+ v) + VW{x) - L{v) = \\v\\ ip(v)
with lim„_o i>{v) = 0.
•
B. Order Properties of the Flow
Angenent (1988) was the first author, to my knowledge, to notice the similarity between the ODE (14.1) and the heat flow of parabolic PDEs. Indeed, when we consider the standard map with generating function S{x, X) = \(X - a:)2 + V(x), the ODE (14.1) becomes xk = (-Ac)fc
-V'(xk)
where A(x)k = Ixk — Xk-i — xk+\ is the discretized Laplacian. It is not too surprising therefore, that the gradient flow solution of (14.1) inherits analogous order properties to those of heatflows{eg. , the comparison principle). In a nice reversal of roles, de la Llave (1999) has now proven Aubry-Mather type theorems for certain PDEs, using order properties (see Chapter 9). To explore these properties in twist maps, we come back to the notion of order introduced in Chapter 2. IRZ is partially ordered by: iq comes from the periodicity of the generating function S and its derivatives: when x £ XPtQ the infinite dimensional vector field — VW for the ODE (14.1) is a sequence of period n (made of subsequences of length n equal to VWpq).
D
15. The Gradient Flow and the Aubry-Mather Theorem In this section, we show how the existence of CO orbits of all rotation numbers can be recovered from the monotonicity of the gradient flow £*. From Lemma 9.2 and Corollary 14.4, we know that the set C O w / Z is compact and invariant under the flow £'. Rest points of the flow in this set lift to CO orbits of rotation number u>. It turns out that, even though C' is not the gradient flow of any function, we can still make it gradient like when restricted to the appropriate subsets. Denote by XK = {x e X | sup f c e Z \xk — Xk-i\ < K}. Theorem 15.1 Let C C X fZ.
be a compact invariant set under a and forward
invariant under the flow £'. Then C must contain a rest point for the flow. In particular C O w / Z contains a restpoint and thus the map has a CO orbit of rotation number ui.
Proof.
Assume, by contradiction, that there are no rest points in C. We show that, for
some large enough N, the truncated energy function WN = 5Z-AT S(xk,xk+i)
is a strict
Lyapunov function for the flow £' on C. More precisely, we find a real a > 0 such that 2IWM{X)
< —a for all x in C. This immediately yields a contradiction since on one hand
58
3: GHOST CIRCLES
WN decreases to — oo on any orbit in C, on the other hand, the continuous WN is bounded on the compact K. To show that WN is a Lyapunov function for some N, we start with: Lemma 15.2 Let C be as in Theorem 15.1. Suppose that there are no rest points in C. Then, there exist a real so > 0, a positive integer No such that, for all x € C j+N
N>N0=>VjeZ,
J2 (VW{x)kf
> e0.
Proof. Suppose by contradiction that there exist sequences jn,Nn oo such that (15.1)
J2
(vW(* ( n ) )k)
and x^n~> with Nn —•
-+0.
In
Let m(n) = —jn~ [Nn/2] where [•] is the integer part function, and let x'^ This new sequence cc'(n' is still in C, and satisfies:
=
am^x^n\
Nn~[Nn/2]
rVW{x'(n))k)
Y2
->0
asn-^co.
k=~[Nn/2]
By compactness of C, it has a subsequence that converges pointwise to some x°° in C. Since S is C 2 , VW(x°°)k
= l i m ^ ^ \/W{x'^)k
= 0 for all k and thus x°° is a rest
point, a contradiction.
•
We now show that WN is a strict Lyapunov function on C. By chain rule: ,
-WN{x) d t
N
=-J2
[9iS(xk,xk+1)VW{x)k N
(15.2)
+
d2S{xk,xk+l)VW{x)k+1]
-N
= ~YldlS(xk,xk+1)VW(x)k_N = - diS(x-N, x-N+i)VW(x)^N
- £ (VW(x)kf -N+l
N+l
£ -N+i -
d2S(xk^,xk)VW{x)k d2S(xN,xN+1)\7W(x)N+1
16. Ghost Circles
59
For all x in XK, we have \xk - Xk-i\ < K and hence, by periodicity, S(xk-i,Xk), its partial derivatives as wellas VWk are bounded on XK. In particular, we can find some M depending only on K such that \-dxS(x-N)x-N+x)VW(x)^N
- d2S(xN,xN+1)VW(x)N+1\
<M
for all x in XK and all integer k. Let p = [M/2e0] and AT > (p + 1)JV0, where N0, s0 are as in Lemma 15.2. We claim that for such an N, Wjv is a Lyapunov function. Indeed, we can split the sum 2 - J V + I ( V f ( i ) k ) into 2p + 2 sums of length greater than N0- By Lemma 15.2, each of these subsums must be greater than eo, and thus the total sum must be greater than M + 2EQ, making the expression in (15.2) less than —2eotH Remark 15.3 As in Chapter 2, we can derive from Theorem 15.1 the existence of AubryMather sets of all rotation numbers. This proof does not yield the fact that the orbits found are minimizers. This apparent weakness may be an asset in considering possible generalizations of this theorem to higher dimensions (see Chapter 9). This proof is a variation of the one given in Gole (1992 b). We are very grateful to Sinisa Slijepcevic, who pointed to a gap in Section 3 of that paper. The above is essentially a rewriting of that section. It was inspired by arguments found in Koch & al. (1994), who prove an interesting generalization of the Aubry-Mather Theorem for functions on lattices of any dimensions (see Chapter 9).
16. Ghost Circles The set of critical sequences corresponding to the orbits of an invariant circle of the twist map / , is itself a circle in IR Z /Z. Trivially, this circle is invariant under £*, as it is made of rest points of theflow.This circle is one instance of a ghost circle. In general, we think of ghost circles as £' -invariant sets that are the surviving traces in the sequence space H z of such critical circles. Definition 16.1 A subset T C M 2 is a Ghost Circle, hereafter GC, if it is 1. strictly ordered: x,y e F => x ~< y or y *!, x. 2. invariant under the Z 2 action (by r m n ), as well as under theflow£*,
60
3: GHOST CIRCLES
3. closed and connected. We will see in the Section 17 that GC's can be constructed by bridging the gaps of the Aubry-Mather sets (identified to their corresponding subsets of rest points in IR Z ) with connecting orbits of the gradient flow £'. Any sequence x in a ghost circle J 1 is CO: since r m n x must also lie in r, which is ordered, we must have x -< r m n a ; or Tm>nx -< x. Moreover, the fact that r is ordered implies, by Lemma 13.3, that all sequences in r have same rotation number. We will call this number p(F), the rotation number of the ghost circle.
Proposition 16.3 Let r be a ghost circle. a) The coordinate projection map IR
— t > IR defined by x i—t xo induces a homeo-
morphism of r to IR. The corresponding projection map IR / 2 — i > I R / 2 induces a homeomorphism
between rfS.
and the circle.
b) The set of ghost circles is closed in the Hausdorff topology of closed sets ofJR
,
and it is compact in COra w/2Z. The rotation number on GCs is continuous in this topology.
Proposition 20.2 improves on part b) of this proposition by giving a sufficient condition for convergence of sequences of GCs
Proof of Proposition 16.3. We show that, for any x, y in r, the projection 5 : x i-> XQ defines a homeomorphism from [x, y] n r to the interval [xo, yo] in IR. As before, we give IR the product topology. The projection map S is continuous and the set [x, y] is compact, by Tychonov Theorem, as a product of closed intervals. Clearly 6 preserves the strict order: x -< y =>- XQ < yo and hence it is one to one on F. Take any two points x -< y in T. As a continuous injection, the map 5 defines a homeomorphism between the compact set F D [x, y] and its image. We show that 6(r n [a:, y]) = [S(x), 5(y)). For this, it suffices to show that r D [x, y] is connected. Suppose not and FC\[x,y]
= A U B where A and B are
closed and disjoint in r f~l [x, y]. There are two possibilities: either both x and y belong to the same set, say A or else x € A, y e B. In the first case, we could write J 1 as the union of two disjoint closed sets:
I 16. Ghost Circles
61
r = [(v_(x) n r) u A u (v+(y) n r)] |J s, a contradiction since r is connected. The other case yields the same contradiction. Since r is ordered, any bounded open ball for the product topology intersects r inside an interval [x,y\. Hence what we have shown above implies in particular that i-1 (in the Hausdorff topology) as k —> oo then any point x € r is limit (in the product topology of 1RZ) of points x(fc) e /&. Since Tm>n and the flow £* are continuous, r must be invariant under these maps. "Close" and "connected" are adjectives that also behave well under Hausdorff limits. Finally, to see that F is strictly ordered, note that if x ^ y are in J1, we can find sequences x^, y^> € i~fc with x = limx(fc),y = limy(k\ If Xj < y-j, we can assume xV*> -< y^> for all k sufficiently large. Since I*, is strictly ordered and C'-invariant, we must have £~'x(fc) -< £~*y(fc) and hence C - 4 ^ < C-*!/- The strict monotonicityofthe flow now implies: x -< y. The continuity of the rotation number is a direct consequences of the continuity of the rotation number on CO sequences, given by Lemma 9.1. • It follows from this proposition that any GC has a parameterization £ G IR i-> x(£) 6 f of the form (16-1)
x(0
= (• • • , z - i ( 0 , W O . S 2 K ) , • • •) •
where the a; j (£) are strictly increasing andcontinuous functions of £. In particular^ H-> XI(£) is a homeomorphism of IR. Invariance of r under the Z 2 action T implies that Xj (£ + 1) = Xj(£) + 1, so that the Xj define homeomorphisms of the circle as well; r-invariance also implies that z2(£) = ^ I C ^ I C O ) ' and more generally that the x n are alliterates of a; 1. Any GC projects naturally to a circle wT in the annulus, where the projection •K : 1RZ —> A is defined by ?r(x) = (x0,-diS{x0,xx)) Proposition 16.3 Let r be a GC for the twist map f. Then 71T and / ( ^ r ) are periodic graphs of periodic functions (£) suc/i tftat there is a constant
62
3: GHOST CIRCLES
L < oo, depending only on the map, and, where the derivatives are defined,
p'(0 > -L,
V'(0 < L.
Proof. If one parameterizes F as in (16.1), then TTF is the graph of (16-2)
y = -ihS&X! ( 0 ) d = ¥>(£)•
Likewise, the image /(nF) is the graph of y = #2,S(:r_i(f), £) = V(0- We now give a proof of the Lipschitz estimate. Using the parameterization of the projection of our GC as in (16.2), it is enough to prove that the derivative of tp is bounded below. The same proof would hold for the estimate for the image f{irr) of our circle. Applying the chain rule to (16.2), we find:
-dnS.
This last term is bounded below by our assumption on the second derivative of S. A similar argument proves the estimate for ip' (£). D Remark 16.4 As mentioned before (see also Exercise 16.6), the set of critical sequences corresponding to an invariant circle of / is a GC, call it F. In this case TTT = f{irr), and Proposition 16.3 provides a proof that invariant circles are Lipschitz, a result of Birkhoff (see also Proposition 12.3). We end this section by giving a condition that insures that GCs do not intersect. We can define a partial ordering on GCs as follows. Let J i , r2 be GCs. We say that 7"i -< r2 if (i) for all x € Ti,x' £ T2 one has x ffl x' and I(x, x') = 1; (ii) p(A) < p{r2), i.e. p(x) < p(x'). Lemma 16.5 (Graph Ordering Lemma) If T\ -< F2 then the circle -KF\ lies below -nF2. Proof. Leta;^(£) be parameterizations of the form (16.1) for .T, (j = 1,2). Then -KF3is the graph of (Gole (1992 a), Lemma 4.22. We conjecture that this remains true when the invariant circle is not transitive (i.e., of Denjoy type).
17. Construction of Ghost Circles This section will show that GCs are plentiful. In the first subsection we construct GCs whose projection passes through any given Aubry-Mather set. The next subsection will specialize to GCs with rational rotation numbers. For generic twist maps, we construct smooth GCs containing periodic minimizers. In Section 18 we will refine this construction to obtain ordered sets of GCs, whose projections do not intersect.
A*. Ghost Circles Through Any Aubry-Mather Sets Let Mu the minimal, recurrent Aubry-Mather set of rotation number u>, as defined in Proposition 12.9. It corresponds bijectively to the set, call it Ew of x sequences of orbits in Mu. By Aubry's Fundamental Lemma 10.2, Ew is a completely ordered subset of CO w . If x is a recurrent minimizer, than so is r m > n a; for any m, n e Z , so Su is invariant under T. Each point of Eu corresponds to an orbit of F, and thus is a rest point of £'. In Gole (1992 a), we proved the following theorem:
Theorem 17.1 The set Eu is included in a ghost circle r, and hence the AubryMather set Mu is included in the projection ITF of a ghost circle. Proof (Sketch).
Sw is a Cantor set whose complementary gaps are included in order
intervals of the type ]x, y[ where x,y
e Eu. A theorem of Dancer and Hess (1991) on
monotone flows implies that, in conditions that are satisfied in the present case, if x -< y are two rest points for the strictly monotone flow C* and there is no other restpoint in [x, y] then there must be a monotone orbit (i.e. completely ordered) of £' joining x and y. Hence we
64
3: GHOST CIRCLES
can bridge all the gaps of Eu with ordered orbits of £', taking care to do so in an equivariant way with respect to the r action. The resulting set is a GC. • B. Smooth, Rational Ghost Circles
We now build rational Ghost Circles by piecing together the unstable manifolds of mountain pass points for Wpq in Xpq. This construction will be crucial when we build disjoint GCs in Section 18. Let u> = p/q be given. Beginning here and throughout Sections 18 and 19 , we shall assume the following: For any p/q € Q,
Wpq is a Morse-function on Xp
(17.1)
This is a generic condition on twist maps, as will be proven in Proposition 29.6. Since a GC consists of CO sequences we may assume that p and q have no common divisor (see the proof of Proposition 10.4). Let x G Xp 0. Due to the Perron-Frbbenius theorem, the largest eigenvalue Ao of — V2Wpq{x) is simple, and the eigenvectors = ( VJ) corresponding to Ao can be chosen to be positive: Vj > 0,j = l,...,q. Moreover all other eigenvectors are in different orthants (See Angenent (1988), Proposition 3.2andLemma3.4).Ifo;isacriticalpointofindex 1, there exist two orbits a±(x;t),t € IR of the gradient flow £' of Wpq with a±(x; t) —> x as t —> —oo, and with a ± (x; t)=x±
eA°'£ + o (eXot).
These two orbits, together with x itself, form the unstable manifold of x. The orbits a± (x; i) are monotone, a+ being increasing, and a_ decreasing; since
T±I$X
= x ± 1 are also
critical points, we have x — 1 < a± (x; t) < x + 1 so that a± (x; t) is bounded. Hence the limits
17. Construction of Ghost Circles
65
u)±(x) = lim a±(x;t) t—>oo
exist and they are critical points of Wpq. Since £' is monotone, there are no other critical points y witho>_(x) < y < x o r x < y < ui+(x). If y > x is another critical point, then y > u+(x). Moreover, since the Morse index must decrease along the negative gradient flow, the points ui±(x) have index 0, i.e. they are local minima of Wpq. We now show that the orbits ce± (x; t) converge to these points along their "slow stable manifold", tangent to the largest eigenvalue of — V2Wpq(w±(x)).
Indeed, since LJ±{X) are minima, all the
eigenvalues are negative, and thus the largest one has the smallest modulus. All orbits in the stable manifold of u>± (x) except for a finite number that are tangent to the eigenspaces of the other eigenvalues, are tangent to this "slow stable manifold". But the other eigenvectors are in different orthants than the positive or negative ones. Hence a±(x;i),
which are
in the positive or negative orthant of w±(x), must converge to u>±(x) tangentially to the eigenvector of largest eigenvalue. To construct a GC in Wpq we first consider the set of critical points such a GC must contain. Definition 17.2 A subset A C XPiq is a skeleton if the following hold. Si A consists of critical points of Wpq with Morse index < 1, 52 A is invariant under the Z 2 action r, 53 A is completely ordered. A skeleton A is maximal if the only skeleton A' with A C A' C XPiq is A itself. Lemma 17.3 A maximal skeleton A for Wpq Proof.
exists.
Chooser, s with rp+qs = landdefineT = r r>s . By Aubry's fundamental lemma
the set Ao of absolute minimisers of Wpq is a skeleton. We fix some element x G Ao- Any skeleton A D Ao is completely determined by
B = An[x,T(x)]
= {zeA:x max Wpq | Q then Q 7 = Q is connected. On the other hand, Q 70 with 70 = max (Wpq(x),
Wpq(y)) is not connected, since x and y are local minima
of Wpq. Consider 71 = inf{7 > 70 : x and y are in the same connected component of Q 7 }. By compactness, x and y are connected in Q 7 1 , and hence 71 > 70. Suppose there is no critical point of Wpq in ]x,y[. Note that, by order preservation, Q = [x,y] is forward invariant under the gradient flow: C'(Q)
c
n 7 > 7 l Q 7 there is an e > 0 such that C x (2 7 1 ) 7l
£
Q for t > 0. By compactness of Q 7 1 = c
2 7 l ~ £ > which implies that x and y are
connected in Q ~ , a contradiction. Hence there is at least one critical point z
e]x,y[,
with Wpq (z) — 71. If the Morse index of all such z were 2 or more, then the Morse Lemma 61.1 would show that Q 7 with 7 slightly less than 71 would still connects x and y, so the index of at least one such z is 1. But now we have a contradiction: if x and y are both local minima, then there is a minimax point z €]x,i/[and«4u{T m > n z : m , n g 2 } is a skeleton; this cannot be since A is maximal.
18. Construction of Disjoint Ghost Circles
67
S t e p 2. Next we show that either x or y is a local minimum.If x is not a local minimum, then w+(x) = lim^oo a + ( x ; £) is a local minimum. But OJ+{X) < y, so w+(x) = y, and we find that y must be a local minimum. Likewise, if y is not a local minimum, then x = w_ (y) must be one.
•
We have all the ingredients necessary to show the following, which was proven in a slightly different form in Gole (1992 a), Theorem 3.6. Theorem 17.5 Assume Wpq is a Morse function. If A is a maximal skeleton,
then
r_A — {a±(x; t) :t G H , x e A is a minimax} U A is a C 1 ghost circle. Proof.
It is simple to check that, by maximality, r^ is connected, and a ghost circle.
As a union of unstable manifolds, i~U is smooth except perhaps where different unstable manifold meet, at the minima. But we showed above how the orbits a± (x; t) must converge tangentially to the one dimensional eigenspace in the positive-negative cone of the minima. Hence the GC constructed is also smooth at the minima.
•
Exercise 17.6 Check that rU is indeed a GC.
18. Construction of Disjoint Ghost Circles We now arrive at the main result of this chapter, which provides a vertical ordering of Aubry-Mather sets: Theorem 18.1 (Ordering of Aubry-Mather Sets) Given any interval [a, b] in IR there is a family of nontrivial circles C „ , w £ [a, b] in the cylinder such that: (a) Each Cu is the projection of a GC ru and hence is a graph over {y = 0} (as is
ncu)j. (b) The Cw are mutually disjoint and if' w > w1, Cu is above Cu*. (c) Each Cu contains the Aubry-Mather number ui.
set Mu of recurrent minimizer
of rotation
68
3: GHOST CIRCLES
This section and the next two are devoted to the proof of this theorem. We will first construct, in this section and the next one,finitefamilies of rational ghost circles. In Section 20, we will take limits of such families and conclude the proof of the theorem. Let w i , . . . , Wfc be distinct rational numbers. The construction of the preceding section provides us with maximal skeletons Ai,...,Ak and corresponding GCs rAl,..., rAk. It is not immediatly clear from this construction that the projections Cj = TVTA. are disjoint. In this section we show that the skeletons can be chosen so that the Cj are indeed disjoint. Definition 18.2 A family of skeletons Aj C XPjqj is minimally linked if any pair x £ Ai, y 6 Aj with i ^ j is transverse with I(x, y) = 1. Theorem 18.3 (Disjointness Theorem) If Aj c XPjqj is a minimally linked family of maximal skeletons, then the projected ghost circles Cj = TTT^ are disjoint. Proof. Order the Aj so that their rotation numbers pj = Pj/qj are increasing. Then we claim that
(i8.i)
rMi
1
\fi = lmo&qi, otherwise
for some judiciously chosen I. Call this a move of type 2. Clearly z^ (
£ XPiqi
and x
p(w) = p(x), and I(a,(Si\w) = 1, we must have aj_x < afli < Wj-iSj Hence the one crossing of w and a^ ^, which occured between j and j +1 is now moved to a crossing that occurs at j , with no other crossing introduced with this or any other sequence of.4 2 U---U Ak. Case 2: a^> < afct0 Since by assumption p(a a^' and thus a.j > Wj, by maximality of aj- . Now choose I = j + 1 and move w to z^: the one crossing of w and a^Sj+1\ which occured between j and j + 1 is now moved to a crossing that occurs at j + 1.
20. Proof of Theorem 18.1 C a s e 3 : a%\ =
73
afc^
Si
The equality a\ ' = a\Si' cannot be true for all i > j since otherwise w and a.(s^ would have same rotation number. Hence for some i > j , Case 1 or 2 must occur. Apply the procedure for these cases there. Concatenating moves alternating between type 1 and 2, we get a curve in Q between w and and a sequence which has zero height. Concatenate this with a move of type 0 to get a curve in i? between w and x.
•
20. Proof of Theorem 18.1 Let wi, W2) • • • be an enumeration of the rational numbers in the interval (a, b). Proposition 20.1 There is a family of GCs {i^ number uij, and where r>
X -T-
, . . . ,.T„ }, where r j n ) has rotation
i/ a>j < u)j. Each r>n
contains at least one
minimizing periodic orbit of rotation number u>i, and generically all of them. Proof.
If one assumes that the map / is such that the Morse property 17.1 holds, then,
according to Theorem 18.4, one can find a minimally lirked family of maximal skeletons {.4.1 , . . . ,An
} such that A™ has rotation number ujj and contains all the absolute (n)
minimizers of that rotation number. The corresponding GCs J!> ' = -T. M then satisfy the required conditions. In general, when the Morse property 17.1 is not satisfied, one can approximate / by smooth twist maps f£ which do satisfy 17.1 (since this condition is generic); One thus obtains ghost circles F- ™ , and by the compactness of the set of GCs with a fixed rotation number (Proposition 16.3) one can extract a convergent subsequence whose limit will then be a family { r 1 ( n ) , . . . , i i n ) } of GCs. But we need to make sure that limits of strictly ordered rational GCs stay strictly ordered. To see this, notice that the set r^J
x r^J is, when i / j ,
included in: (2ij = {(v,w)
£ PCO W i x PCO w . : v fjl w
and
I(v,w)
where PCO w is the set of periodic CO sequences of rotation number u>: P C O p / 9 = COp/g fl XPiq.
= 1}
74
3: GHOST CIRCLES
The set i?^ is, by the Sturmian lemma, positively invariant under the product gradient flow C* x £' corresponding to any twist map. In fact: (£* x ^(Clos £2^) C (Int fi^), as can easily be checked (i.e.Clos fi^ is an "attractor block" in the sense of Conley). As Hausdorff limit of compact sets in ntj, the set J1/"' x .T>n) is in Clos £2^ .But, since it is both positively and negatively invariant under £' x £', r> n ' x r j n ) must in fact be in Int fi^ where the intersection number is well defined and always equal to 1. In other words, we have shown that, whenever Wj < Wj one must have r\n' -< r^'. Finally, the set r>n' contains at least a minimizing periodic orbit, since the sets r>E' contain by construction all the minimizing periodic orbits of period uii for fs, and limits of minimizers are minimizers. • A. Rational CVs We now construct the CJs of Theorem 18.1, starting with all the rational a; £ [a, b]. Again, we use the compactness of the set of GCs: For each n, Proposition 20.1 provides us with GCs i f ' , ..., r „ with rotation numbers u>i, ..., wn. By compactness we can extract a subsequence {rij} for which the r^3' converge as j —» oo to a GC of rotation number (n')
u>i. Using compactness again, we can extract a further subsequence n' for which 7\ ' and (n')
F2 ' both converge; repetition of this argument and application of the diagonal trick then finally gives a subsequence n" for which all rk 3 converge to some limiting GC i")[ (of rotation number w^) as j —» oo. By the same argument as in the previous proposition, the limits r^ oo) satisfy j ; ( o o ) -< r^°°] whenever w* < uij. We then define CUk = Trr^00' and by the Graph Ordering Lemma 16.5, the CUk 's are disjoint. In the generic case, each r>n' contains all the periodic minimizers of rotation number u)i, and hence so must the limit r^00'. In the non generic case, r>°°' must contain at least one periodic minimizer of the energy. B. Irrational CJs To complete our family of rational GCs with irrational ones, we once again take a limit. We could proceed in a way similar to what we did in order to get all rational GCs, but we would have to appeal to the axiom of choice (no diagonal tricks on uncountable sets!). To avoid this, we first prove a proposition of monotone convergence of GCs. We shall write A ^ i~2 if either Ji -< T2 or p(A) = p(r2) and 7ri~i is ( not necessarily strictly) below 7iT2.This
20. Proof of Theorem 18.1
75
last condition is equivalent to xy (£) < x{' (f) in the notation of the proof of the Graph Ordering Lemma 16.5. Proposition 20.2 (Monotone Convergence for Ghost Circles) LetT^)
be an increasing
sequence of GCs, i.e. assume that
r(!) -< f (2) x _r(3) x . . •. Asswme ako £/ia£ the rotation numbers pj = p(r^') there is a unique GC r(°°) such that f ^ is the parametrization
of F^
are bounded from above. Then
- • r(°°) as j —> oo. Moreover, if
with XQ (£) = £, i/ien Wie a:*. (£) converge
ically and uniformly to x£° (£), where x^°°\^)
is the parametrization
x^($)
monoton-
of T1-00' with
Of course, the corresponding theorem for decreasing sequences of GCs also holds. We postpone the proof of this proposition till the end of this section. Assume now that we have constructed the rational GCs i ^
as above. For any number
w € {a,b), rational or otherwise, we can then define two GCs F^ as follows. Choose a sequence of rational numbers w n . which increases monotonically to UJ. The Monotone Convergence Theorem tells us that the limit of the corresponding GCs r„°° must exist. We denote this limit by r~. This procedure might produce an ambiguous definition of F~, since the result could depend on the choice of the sequence nj: If one has two such sequences, nj and n'3,, then the r „ ° ° ' and / „ ,
might have two different limits F and i~". However, one can
take the union of the two sequences to obtain a third sequence n'', i.e. {n'j} = {nj} U {n'j}. The u)n" then also increase to u, so that the r „ ' also must converge to some GC F". Since nj and n'j are subsequences of n", both sequences rij and n'j must produce the same limiting GC: hence r = F1 = r", and the definition of r~ is independent of the choice of the nj. We choose to define Cu = -KT~ (or -KF^, but with the same choice of + or — for all u) in order to avoid using the axiom of choice...). We now check that, for ui irrational, the unique Aubry-Mather set Mu of recurrent minimizers (see Proposition 12.9) is included in C w . We can take a sequence of periodic Aubry minimizing sequences xk e i ^
where u>k /
w ( \ if one chose Cw =
nT+).
Then xk —* cc, an Aubry minimizing sequence in F~. The orbit that CE corresponds to is
76
3: GHOST CIRCLES
recurrent and minimizing, as limit of recurrent and minimizing orbits. Its closure, which is also included in Cu, must be the Aubry-Mather set Mw . From our definition of J ^ , it is clear that:
wi < i f 5 -< rz < rj -< rj°°\ for rational u>j,Wj and irrational w. Hence the set formed by the rational GCs i ^
and
the irrational ones ru is completely ordered according to their rotation numbers. By the Graph Ordering Lemma 16.5, the C w 's (irrational and rational) that we have constructed are mutually disjoint.
•
Remark 20.2 If u is a rational number, r~ is no longer necessarily in PCOu but is certainly in CO w . It may contain the sequences corresponding to homo(hetero)clinic orbits joining hyperbolic periodic orbits of rotation number ui. Hence we may (and, probably, generically do) have three distinct Ghost Circles r~
X ru
< Fj for each rational u> where ru is
PJf° for some k. We will call their projections C~, Cu and C+ respectively. Instead of the set {Cw}u,e[a,6] °f strictly non mutually intersecting curves that we have found in Theorem 18.1, one might prefer to consider the bigger set {Cu U C+ U Cz}u>e[a,b]- K is n o t hard
to
check that this is a closed set of GCs. Proof of Proposition
20.2. It follows from the Graph Ordering Lemma 16.5 that the
2^ (£) are monotonic in j . We have assumed that the rotation numbers of the r^) bounded, say by some integer M. Since x^ x
i
are
is CO, this bound implies for I > 0 that
(0 < £ + KM + 1), and in view of the monotonicity of the x\r (£) they converge to
some x, ( o o ) (0- For negative I one finds that x[j)(£) decrease to some i ,
(£). Clearly xj
> £ + l(M + 1), so that the
x\j)(£)
(£) is a nondecreasing function of £. We shall show
that it is strictly increasing, and continuous. x
i
(0 *s strictly increasing.
Let £ < rj be given. Then t \-> C*(x
C ' O ^ f a ) ) both are on the GC r^\
(£))
an
^ * l—•
so that they must be ordered in the same way for all
t 6 M. At t — 0 we have
Z=
e(x(i)(Oh oo we findthatC'(x(oo)(£)) < 00 C* (a^ ) (7/)) holds for all t. By the strict monotonicity of (*, we must have strict inequality
21. Proof of Theorem 18.1
77
for all t, unless we have equality for all t. Equality cannot happen of course, since XQ (£) = £ < 77 = a ^ f a ) - H e n c e w e h a v e a; (oo) (0 < a(°°)(?7); in particular a^ 0 0 ^) < x^irj). x i ( 0 *s continuous. Since the i j (f) are monotonically increasing in both jf and £, their limit is increasing and lower semicontinuous in £. Thus we only have to show that 4°°> (£) = x {PkVk-i + akvk + pk+\vk+i}keT otk = -d22S(xk-i,xk)
-duS(xk,xk+i),
Pk =
-d-L2S{xk-i,xk)
is strictly positive, then the flow £' is strictly monotone. L(x(t)) is an infinite tridiagonal matrix with positive off diagonal terms —di2S(xk, xk+i) (see Formula (17.1) for afinitedimensional version of this matrix). The diagonal terms dnS(xk, xk+i) + d22d2S(xk-i, xk) are uniformaly bounded by assumption on S. Hence, for any T > 0 for which x (t) = C* (x) is defined when 0 < t < T, we can find a positive A such that: B{t) = L(x{t)) + Md is a strictly positive matrix. If u{t) is solution of the equation (22.1) then eXtu(t) is solution of: (22.2)
v(t) = B(t)v(t),
hence the strict positivity of the solution operator for (22.1) is equivalent to that of (22.2). Looking at the integral equation: v(t) = v(0) + f B(s)v(s)ds, Jo one sees that Picard's iteration will give positive solutions for a positive vector v(0). This will imply, assuming that vk(0) > 0, vi(0) > 0, for l ^ k:
22. Proofs of Monotonicity and of the Sturmian Lemma
83
vk+i(t) > vk+l(0) + / Bklk+i{s)vk(s)ds > 0 Jo The same holding for vk-i. By induction, vk(t) > 0, Vfc 6 Z and the operator solution is strictly positive. Thisfinishesthe proof of Theorem 14.2.
•
B. Proof of the Sturmian Lemma
Lemma 22.1 (Sturmian Lemma) Let x(-),y(-) e CO be different solutions of — - = -d2S(xk~i,xk)
then I (x(t),y(i))
- diS(xk, %k+i) ;
at does not increase, and decreases whenever x(t) and y(t) are not
transverse. To prove this lemma, we will examine a more general situation. Let Xi(t) (io < i < h, —T < t < T) be a solution of dec' (22.3)
—± = ai{t)xi-i + bi(t)xi(t) + Ci{t)xi+i(t)
(i0 < i < h)
where we assume that the coefficients ai(t), 6j(t), Cj(t) are continuous and satisfy (22.4)
ai(t),Ci{t) >6;
at, bua < M
for all — T ±± then yi(t) j/j(t)=JVit Proof.
il_i
il
+ o ( t i _ i o ) , if i > ^ ^
then
i
+ o(t - ).
We may assume ii — io > 2. The j/j(i) are continuous, and hence bounded as
t -> 0. Therefore it follows from (22.6) that |j/j(t)| < C |t| for |t| < T. If ii — i 0 = 2, then the only i with io < i < ii is i = io + 1 = ii — 1, and we have j/ i o + 1 (t) = / {Alo+1{0)yio{0) +Cil-1{0)yil(0) Jo = Mio+it + Nio-it + o{t),
+
o(l)}dr
as claimed. If ii — io > 2, then yio+2(t)
= o(l), and (22.6) implies
Vio+i(t) = f {A i o + 1 (0)y i o (0) + o(l)}dr Jo = Mio+lyio(0)t + o{t). Likewise (22.6) implies yi a _i(t) = N^^y^^t
+ o(t). If i\ — io = 3 this proves the
claim; if i\ — i0 > 3, then for all io + 1 < i < ii — 1 one deduces from (22.6) and the estimate |i/i±i(t)| < C\t\ that \yi(t)\ < Ct2. The general induction step in the derivation of (22.7) is as follows. Assume that it has been shown that (22.7) holds for all i with io < i < io + k, or ii — k < i < i\\ moreover assume it has been shown that \yi(t)\ < C \t\k for i 0 + k < % < i\ — k. If i0 + k = i\ - k, then (22.7) implies 2/io+fcW = / {Aio+k(0)Mio+k^Tk-1 Jo = Mio+ktk
+ Nh-ktk
+ Ch-k(0)Nil_k+1rk'1
+o
{rk-l)}dT
+ o (i f c ),
with Mio+k = A i o + f c ( 0 ) - M i o + f c _ i , iVil_fe = Ci 1 _fc(0)-JVi 1 _ fc+ i. In this case the claim is proved. Otherwise i0 + k 0; For small negative t the sequence yi0 (t), yi0+i {t),..., y^ (t) alternates signs, except in the middle, i.e. if i\ — io is odd then yio+k{t) and yio+k+i(t) (with k = [ n ^ i p ]) will have the same sign. Indeed, (22.7) says the sequence {yi0(t),..., y^ (£)} has the signs as the sequence {co,C\t,C2t
,...,Ck-lt
,Ckt
, Cfc+i* ~ ,•••
,C2k-lt,C2k)
if i\ — io = 2k is even, and {yi0 (fc=0fc+#- t (r)+01(e) Rk =Tk + 01 (e) n
with ipFk - Qfc + e ^
2pkin.
1=1
where e _1 oi(e, 6, r) and its first derivatives in r, 0 tend to 0 uniformly as e —> 0. We can rewrite this as: 71(0,r) = (0 + eBr + a + oi(e),r + So for small e, the condition det 80/dr
0l(e)).
^ 0 is given by the nondegeneracy of B = {fiki},
1
one uses the fact that TZ is C close to a completely integrable symplectic twist map to show that TZ is twist in U (the twist condition is open). The fact that it is homotopic to Id derives from Exercise 23.4. Note that the set V and therefore U are not necessarily invariant under TZ. Note also that the symmetric matrix B, even though it is generically nondegenerate, is not necessarily positive definite. Herman (1992 b) has examples of Hamiltonian systems and symplectic maps arbitrarily close to completely integrable which have elliptic fixed point with B not positive definite.
25. Generating Functions
95
Exercise 24.3 Compute the expression of the lift of a symplectic twist m a p generated by: S(q, Q) = \ (A(Q - q), (Q - q)) + c.(Q - q) +
V(q),
where A is a nondegenerate n x n symmetric matrix (This is yet a further generalization of the standard map).
25. More on Generating Functions A. Homeomorphism Between Twist Maps and Generating Functions The following proposition justifies the name "generating function". Proposition 25.1
between the set of lifts F of C1
There is a homeomorphism^ n
symplectic twist maps of T*T
2
and the set of C
real valued functions
S
onTR71
satisfying the following: (a) S(q + m,Q
+ Tn)=,S(q,Q),
(b) The maps: q —> d2S(q,Q0) for any Q0 and q0
VmeT, and Q —• diS(q0,Q)
are diffeomorphisms ofTR™
respectively,
(c) 5(0,0) = 0. This homeomorphism
(25.1) Proof.
is implicitely given by:
Ft,.,)- a(v,v),
99
Vu £ IRn
then / is an embedding (diffeomorphism on its image) of B n in IR".
26. Symplectic Twist Maps on Cotangent Bundles of Compact Manifolds A. Definition Our definition of symplectic twist maps of T " x M™ is geometric enough to allow a generalization to cotangent bundles of general compact manifolds. The main difference between our general definition and the one in the case of T™ x IR™ = T*Tn is that we do not work with the universal covering space of our manifold any more, to the cost of a less global definition.^10) In this book, the main examples of symplectic twist maps on general cotangent bundles will arise in the context of Hamiltonian systems (see Chapter 7). We also present, in the next section, a generalization of the standard map in cotangent of hyperbolic manifolds. We refer the reader to Appendix 1 for a review of the concepts of cotangent bundles and their symplectic structure. In the following, U will denote an open subset of T*M such that: (26.1)
ir-1(q)nU~interior{TBn)
where -K : T*M —> M is the canonical projection, and IB™ C H™ denotes the n-ball. Hence U is a relatively compact ball bundle over M, diffeomorphic to T*M. As in Appendix 1, we denote by A the canonical 1-form on a cotangent bundle. Definition 26.1 A symplectic twist map F is a diffeomorphism of an open ball bundle U C T*M (as in (26.1)) onto itself satisfying the following: (1) F i& homotopic to Id. (2) F is exact symplectic. F*X — A = S_ for some real valued function S_ on U. (3) Twist condition: the map ipF : U -* M x M given by IPF(Z) = {n{z), -K O F{z)) is an embedding. 10
If the manifold M is not covered (topologically) by IR", problems occur when we want to make the definition of symplectic twist maps of T*M as global as in T*T": there cannot be a global diffeomorphism from a fiber of T*M to the universal cover M.
100
4: SYMPLECTIC TWIST MAPS
The function S = 5 o ipp1 on IJJF(U) is called the generating function for F. Often, the kind of neighborhood we have in mind is of the form:
U=
{(q,p)eT*M\H(q,p)™ such that: (i) Tf is a C°° Lagrangian graph over the zero section, (ii) f\
is C°° conjugated to the rigid translation by cj(p0),
(Hi) Tf and the conjugacy depend C°° on f. Moreover the measure of the complement of the union of the tori Ty(p 0 ) goes to 0 as | | / - / o | | goes to 0. Remark 34.2 1) The diophantine condition (34.1) is shared by a large set of vectors in JRn. As an example, when n = 1, the set of real numbers p, G [0,1] such that \p, — p/q\ > K/qT, T > 2 for some K is dense in [0,1] and has measure going to 1 as K goes to 0. 2) The most common versions of KAM theorems concern Hamiltonian systems with a Legendre condition. In Chapter 7 we show the intimate relationship of such Hamiltonian systems with symplectic twist maps. It therefore comes as no surprise that KAM theorems have equivalents in both categories of systems. Note that there are isoenergetic versions of the KAM theorem for Hamiltonian systems, where the existence of many invariant tori is proven in a prescribed energy level (see Broer (1996), Delshams & Gutierrez (1996a)). 3) One important contribution in Moser (1962) was his treatment of the finitely differentiable case: he was able to show a version for n = 1 (twist maps) where /o and its perturbation
34. KAM Theory
125
are Cl, I > 333 instead of analytic. This was later improved to I > 3 by Herman (1983) and in higher dimension n, to I > 2n + 1 (at least if the original / 0 is analytic). 4) There is a version of the KAM for non symplectic perturbations of completely integrable maps of the annulus, called the Theorem of translated curves, due to Russmann (1970). It states that, around an invariant circle for /o whose rotation number u> satisfies the diophantine condition (34.1) (only one j in this case), there exists a circle invariant by ta ° / for a perturbation / of f0 and ta(x,y) = (x,y + a), which has same rotation number as the original. 5) One may wonder if, among all invariant tori of a symplectic twist map close to integrable, the KAM tori are typical. KAM theory says that in measure, they are. However Herman (1992a) (see also Yoccoz (1992)) shows that, for a generic symplectic twist map close to integrable, there is a residual set of invariant tori on which the (unique) invariant measure has a support of Hausdorff dimension 0 (and hence cannot be a KAM torus). Things get even worse when the differential Du in Theorem 34.1 is not positive definite: there may be many invariant tori that project onto, but are not graphs over the 0-section, and this for maps arbitrarily close to integrable (see Herman (1992 b)). 6) KAM theory implies the stability of orbits on the KAM tori, hence stability with high probability. But in "real situations" it is impossible to tell, for lack of infinite precision on the knowledge of initial conditions, whether motion actually takes place on a KAM torus. Nekhoroshev (1977) provides an estimate of how far a trajectory can drift in the momentum direction over long periods of time: If H(q,p) = h(p) + fe{q,p) is a real analytic Hamiltonian function on T*Tn with fe < e (a small parameter) and h(p) satisfies a certain condition (steepness) implied by convexity, then there exist constants eo,Ro, To and a such that, if e < eo, one has: 1*1 < r 0 exp[( £o /e) Q ] => \P(t) - p ( 0 ) |
0 depending only on K, p and a such that, if a is a 2TT -periodic analytic function
on a strip of width p, real on the real axis with
a(z) < e on the strip and such that the circle map defined by
/ : m i + 2-7r/i + a(x) is a diffeomorphism
with rotation number p satisfying the diophantine lM-p/?l>-2+^T,
condition:
Vp/?6Q
then f is analytically conjugate to a rotation R of angle 2irp Sketch of proof: We seek a change of coordinates H : S 1 —» S 1 such that: (34.2)
HoR
=
foH
write H(z) = z + h(z), with h(z + 2ir) = h(z). Then (34.2) is equivalent to (34.3)
h{z + 2irp) - h{z) = a(z + h(z)).
Since a(z) < s, h must be of order e as well and thus, in first approximation, (34.3) is equivalent to: (34.4)
h(z + 2-n\£) - h(z) = a{z)
Decomposing a(z) = Yl a,kei2vkz, h(z) = J2 bkelkz
in their Fourier series and equating
coefficients on both sides of (34.4) we obtain: *
pi2'nkjj, __ ^
where we see the problem of small divisors arise: the coefficients bk of h may become very big if p is not sufficiently rational. It turns out that, assuming the diophantine condition and using an infinite sequence of approximate conjugacies given by solutions of (34.4), one obtains sequences hn,an corresponding Hn,fn
= H~
l
and
o / o Hn which converge to H, R for some H. The domain
35. Properties of Invariant Tori
127
of hn and /„ is a strip that shrinks with n but in a controllable way. This iterative process of "linear" approximations to the conjugacy can be interpreted as a type of Newton's method for the implicit equation T{f, H) = H~l o / o H = R (given / , find H) and inherits the quadratic convergence of the classical Newton's method: R — T(fn, Hn) — 0(e2n) (see Hasselblat & Katok (1995) Section 2.7.b).
•
35. Properties of Invariant Tori The previous section showed the existence of many invariant tori for symplectic twist maps close to integrable. These tori are Lagrangian graphs with dynamics conjugated to quasiperiodic translations. In dimension 2, the Aubry-Mather theorem gives an answer to the question of what happens to these tori when they break down, eg. in large perturbations of integrable maps. In higher dimension, Mather's theory of minimal measure also provides an answer to that question (see Chapter 9). In this section, we look for properties that invariant tori may have whether they arise from KAM or not. We will see that certain attributes of KAM tori (eg. graphs with recurrent dynamics) imply their other attributes (eg. Lagrangian), as well as other properties not usually stated by the KAM theorems (minimality of orbits). A. Recurrent Invariant Toric Graphs Are Lagrangian
Theorem 35.1 (Herman (1990)) Let T be an invariant torus for a symplectic twist map f of T*Tn and suppose / _ is conjugated by a diffeomorphism h to a an irrational translation R on T n . Then T is Lagrangian. Proof. Since the restriction of the symplectic 2-form w| T is invariant under / L and since R = h~l o f\T o h, the 2-form h*w\T is invariant under R. Since R is recurrent, h*u>\T = J2i,j akjdxk A dxj must have constant coefficients akj. Integrating h*w\T over the Xk, Xj subtorus yields on one hand a^, on the other hand 0 by Stokes' theorem since h*ui\T = dh*\\T is exact. Hence h*uj\T = 0 = ui\T and the torus T is Lagrangian. •
128
6: INVARIANT MANIFOLDS
B. Orbits on Lagrangian Invariant Tori Are Minimizers The following theorem is attributed to Herman by MacKay & al. (1989), whose proof we reproduce here. Theorem 35.2 Let T be a Lagrangian torus, C1 graph over the zero section
ofT*Tn
which is invariant for a symplectic twist map f whose generating function S satisfies the following superlinearity
(35.1)
condition:
lim
§
^
- +oo
IIQ- oo. Since R has all its
critical points on T, it must attain its minimum Rmin there. It is now easy to see that the
35. Properties of Invariant Tori
q coordinates qn,...,qkof
129
any orbit segment on T must minimize the action. Indeed, let
,rk be another sequence of points of T n with qn = r „ , qk = rk. Then:
r„,...
fc-i
W(ri,.
..,rk)
= Y^ R{rj,rj+i)
+ g{qk) - g(qn) + P{qk -
qn)
j=n
>(k-
n)Rmin
+ g(qk) - g(qn) + (3(qk - qn) = W{qx,.
..,qk) D
Remark 35.3 Arnaud (1989) (see also Herman (1990)) has interesting examples which show that the condition that the graph be Lagrangian is essential in Theorem 35.2. Consider the Hamiltonians on T*T2 is given by: Hs(qi,q2,Pi,P2)
= - ( p i -Ecos(27r<j 2 )) 2 + -p%.
The torus {(^i, q2, e cos(27rg2), 0)} is made of fixed points for the corresponding Hamiltonian system, but it is not Lagrangian (exercise). A further perturbation Ge^{q,p) Hs(q,p)
=
+ S sin(27r IR. (Hint. Show that the integral of A over any loop on Ws,u is 0). b) Show that if and W is an exact Lagrangian manifold invariant under the exact symplectic map .F, then: S(z) + constant = L(F(z)) - L(z), c) Conclude that L " ( 0 = J^ [S(Fk(z"))
- S(z*)} ,
Vp e W
Ls(zs) = - J2 [S(Fk(zs))
-
S(z*)].
For more on this approach, see Delshams & Ramirez-Ros (1997).
B. Variational Approach to Heteroclinic Orbits As a consequence of Proposition 36.1, we obtain a variational approach to heteroclinic orbits. Let z* = (q*, p*) be a hyperbolic fixed point. Let oo, ^ ( q j y ) —* ^(g*) = 0 and thus the sum converges to —(9fc,«Ifc+i)-S(g*,",s = d$™'s for some functions $ " ' s of the base variable q. Clearly, the manifolds W" , s split for e small enough whenever the following Poincare-Melnikov function: M(q) =
(#(«)-#(«))
| £=0
is not constantly zero, and their intersection is transverse if the differential DM is invertible at the zeros. We will now show that:
where L(q) is the function defined in (36.2) , expressed in our new coordinates. Formula (36.1) gives us expressions for
