Transition to Chaos in Classical and Quantum Mechanics: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) ... 13, 1991

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich F. Takens, Groningen Subseries: Fondazione C.I.M.E., Firenze Advisor: Roberto Conti

1589

J. Bellissard M. Degli Esposti G. Forni S. Graffi S. Isola J.N. Mather

Transition to Chaos In Classical and Quantum Mechanics Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, July 6-13, 1991 Editor: S. Graffi

Fondazione

C.I.M.E.

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Authors

Editor

Jean Bellissard Laboratoire de Physique Quantique Universit6 Paul Sabatier 118, route de Narbonne F-31062 Toulouse Cedex, France

Sandro Graffi Dipartimento di Matematica Universith degli Study di Bologna Piazza di Porta S. Donato, 5 1-40127 Bologna, Italy

Mirko Degli Esposti Giovanni Forni Sandro Graffi Stefano Isola Dipartimento di Matematica Universith degli di Studi di Bologna Piazza di Porta S. Donato, 5 1-40127 Bologna, Italy John N. Mather Department of Mathematics Princeton University Princeton, NJ 08544, USA

Mathematics Subject Classification (1991): 58F, 58F05, 58F15, 58F36, 81Q, 81Q05, 81Q20, 81Q50, 81S, 81S05, 81S10, 81S30

ISBN 3-540-58416-1 Springer-Verlag Berlin Heidelberg New York

CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera ready by author SPIN: 10130140 46/3140-543210 - Printed on acid-free paper

FORE'WORD

This volume collects the texts of two series of 8 lectures, and the expanded version of a seminar, given at thelC.I.M.E. Session on "Transition to Chaos in Classical and Quantum Systems", which took place at the Villa "La Querceta" in Montecatini, Italy, from July 6 to July 13, 1991. The purpose of the Session was to give a broad survey of the mathematical problems and techniques, as well as of some of the most relevant physical motivations, which arise in the study of the stochastic behaviour, if any, of deterministic dynamical systems both in classical and quantum mechanics. The transition to chaos in the most relevant and widely studied examples of classical dynamical systems, the area preserving maps, is thoroughly covered in the first series of lectures, delivered by Professor John Mather and written in collaboration with Dr. Giovanni Forni. In particular the reader can find in this text an up-to-date version of the well known Aubry-Mather theory. The lectures of Professor Jean Bellissard cover in turn, in addition to his algebraic approach to the classical limit, the behaviour of the quantum counterpart of the above systems, with particular emphasis on localization, and on qualitative as well as quantitative properties of the spectra of the relevant SchrSdinger operators in classically chaotic regions. They can be therefore considered an exhaustive introduction to the mathematical aspects of the so-called "quantum chaos". The third series of lectures, delivered by Professor Anatole Katok, covered the basic stochastic properties of classical dynamical systems and some of tlheir most recent developments. Unfortunately Professor Katok could not find the time to write up the text of his course. A very prominent role in describing tlhe chaotic behaviour of classical dynamical systems is played, as discussed also in Professor's Katok lectures, by the proliferation and equidistribution of the unstable periodic orbits of increasing period. An overview of recent results in this direction, and of their intimate connection to the problem of the classical limit of the quantized toral symplectomorphism~, is contained in an outgrowth of a seminar held by M.Degli Esposti, written in collaboration with S.Isola and the Editor.

Bologna, April 1994

Sandro Graffi

TABLE

J. BELLISSARD,

M.

G.

DEGLI

FORNI,

Non Commutative Methods in Semiclassical Analysis ...................

ESPOSTI,

J.N.

OF CONTENTS

S. G R A F F I , S. I S O L A , Equidistribution of P e r i o d i c O r b i t s : a n o v e r v i e w of classical vs quantum results ............

MATHER,

A c t i o n m i n i m i z i n g o r b i t s in H a m i l t o n i a n Systems .................................

1

65

92

Non Commutative Methods in Semiclassical Analysis Jean Bellissard Laboratoire 118, route

de Physique de Narbonne

Quantique

Universit6

F-31062 Toulouse

Paul Sabatier

Cedex, France

Contents 1

The kicked rotor problem

2

The 2.1 2.2 2.3 2.4

4

2

Rotation Algebra The Polynomial Algebra 5oi . . . . . . . . . . . . . . . . . . . . . . . . Canonical calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rotation Algebra ,4i . . . . . . . . . . . . . . . . . . . . . . . . . Smooth functions in .41 . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 7 9 10

Continuity with respect to Planck's constant 3.1 Mean values of observables . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The spectrum of observables . . . . . . . . . . . . . . . . . . . . . . . .

11 12 13 14

Structure

15

o f t h e R o t a t i o n A l g e b r a `4/

S e m i c l a s s i c a l a s y m p t o t i c s for t h e s p e c t r u m 5.1 2D lattice electrons in a magnetic field . . . . . . . . . . . . . . . . . . 5.2 Low field expansion . . . . . . . . . . . . . . . . . . . . 5.3 Qualitative analysis of the spectrum . . . . . . . . . . . . . . . . . . . .

.........

17 18 19 23

Elementary Properties of the Kicked Rotor 6.1 The Furstenberg Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calculus on BI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Representations and structure of 131 . . . . . . . . . . . . . . . . . . . . 6.4 Algebraic Properties of the Kicked Rotor . . . . . . . . . . . . . . . . .

26 26 27

Localization and Dynamical Localization 7.1 Anderson's Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Observable Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Localization Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Localization in the Kicked Rotor . . . . . . . . . . . . . . . . . . . . .

34 34 38 39 43

28

30

1

The kicked rotor problem

One considers a spinning particle submitted to rotate around a fixed axis. Let 0 E T = R/27rZ be its angle of rotation, L c R its angular momentum, I its moment of inertia, # its magnetic moment, and B a uniform magnetic field parallel to the axis of rotation. Its kinetic energy is given by : L2

~t0 ~ + ~ B L ,

(1)

We assume that this system is kicked periodically in time according to the following Hamiltonian : L2 Tl = ~-[ + p B L + k cos(0) ~ 5(t - n T ) . (2) nEZ

where T is the period of the kicks, and k is a coupling constant representing the kicks strength. Here 5 is the Dirac measure. Classically the motion is provided by the solution of the Hamilton-Jacobi equations : dO cOTl dt - cOL

dL dt -

07-I 08 "

(3)

Between two kicks, 0~-//c90 = 0, so that L is constant whereas 0 varies linearly in time. When the kick is applied, L changes suddenly according to L ( n T + O) = L ( n T - O) + k sin(0). If we set : A~=T(L(nT-O)

+#B)

O~=O(nT-O),

(4)

On + An+l mod 27r ,

(5)

the equation of motion can be expressed as : A.+I = As + K sin(0.)

0.+1

=

where K is the dimensionless coupling strength namely : K-

kT I

(6)

The phase space is the cylinder C = T x R, if A is considered as a real number. If we set

f(0, A) = (0', A')

8' = 0 + A + K sin(P)

A' = A + K sin(P) ,

(7)

the solution of the equation of motion can be written as : (0,~+l,An+l) = f(On, An) 9

(s)

f is an analytic diffeomorphism of the cylinder C, which is area preserving, namely dO~ A d A ' = dO A d A , and a twist map, namely cOO~/OA > 0, which preserves the ends (see the course of John Mather in this issue). We remark that f also commutes with the translation A ~-~ A + 2~r of the action variable A in such a way that it also defines a map of the 2-torus T 2.

The orthodox way of quantizing this model consists in choosing the Hilbert space /C = L2(T, dO/27r) as the state space, and replacing L and 0 by operators as follows : L-

hO

P = multiplication by V(0),

i 00

(9)

whenever )2 is a continuous 27r -periodic function of the variable 0 . Q u a n t u m Mechanics requires using a new parameter h, the Planck constant which gives rise to a new dimensionless parameter : h T _ 47r b'QM

"Y- I

(10)

,

//ca

where vCL = 1 / T is the kicks frequency, whereas/]QM is the eigenfrequency of the free quantum rotor in a zero magnetic field. To compute the motion, we need to solve SchrSdinger's equation, namely, we look for a path t ~ R ~ r c tC such that : L2

iliCt = H ( t ) r

H ( t ) = - ~ + # B L + k cos(0) ~

5(t - n T ) .

(11)

nEZ The &kicks may create a technical difficulty. To overcome it let us consider a smooth approximation 5~ of 5 given by a non negative Ll-function on R supported by [0, e], with integral equal to 1. The solution can be given in term of a convergent Dyson expansion. Then letting e converge to zero, we get the following result (see Appendix

1): T h e o r e m 1 The solution of (11) is given by the following evolution equation : CT-0

=

F-I'r

-

F -1

=

e-iA2/2"re-iKc~176165 i9

(12)

~)=(#B)2--~.

(13)

where A=T(L+#B)

,

Let us also introduce the dimensionless magnetic field x : X2

x = -#BT

=>

/) = ~--~.

(14)

The operators of the form )2 whenever 1~(0) is a continuous 2rr -periodic function of the variable 0, can be obtained as the norm limit of polynomials in the operator U = e/~ .

(15)

In much the same way, one can quantize the action in the torus geometry by considering the operator : V = e -iA . (16) U and V are two unitary operators satisfying the following commutation rule : U V = ei~VU .

(17)

The C*-algebra generated by these two operators is the non commutative analog of the space of continuous functions on the 2-torus. By analogy with the commutative case, this algebra will be seen as the space of continuous functions on a virtual space, the "quantal phase space". Any such function will be the norm limit of polynomials of the form : a = ~ a ( m ) U m l V ' ~ 2 e -'~'~''~2/2 , (18) mEZ2,]m[_ 0, a E ~ / , (ii) normalization : T(I) = 1, (iii) trace property : 1-(ab) = r(ba), a, b E Pl. We remark that the value of ~-(a) at V = 0 is the 0 th Fourier coefficient of rl0(a), namely the integral of its Fourier transform : dOdA

-(a)l~:o = fT~ -~-~--2ad(0, A)

.

(35)

where ad is the Fourier transform of r/0(a). The angle average, is defined by the element (a / in PI given by : (a)(m) = 5m,,oa(O, m 2 ) .

(36)

The m a p a ~-* Ca) is a module-map taking values in the commutative subalgebra Z)I generated by V as a g(I)-module. The usual Fourier transform permits to associate with any element b of Z)i a continuous function of (7, A) E I • T denoted by bay as follows : bav(3',A)= ~ b(O, m 2 ; v ) e - ~ 2 A 9 (37) m,EZ ~

The mapping b E Z)i ~-~ b~ E C(I x T), is a .-homomorphism, namely (bc)a~ = b~ca~ and (b*)~v = b**. We will say that b E 7Pi is positive whenever ba~ is positive. Using these definitions, the angle averaging satisfies : (i) positivity property : (ii) projection property: (iii) normalization : (iv) conditional expectation :

(a'a) > 0 , a E P l (Ca)) = (a) , (I) = 1 , (ab) = (a)b , (ba) = b{a) , if b E D1 , a E P l (38) A differential structure is defined on P / t h r o u g h the data of two ,-derivations 00 and OA given by : (Ooa) (m) = i m l a ( m )

(OAa) (m) = i m 2 a ( m ) .

(39)

These two derivations 0" (if # = 0, A) actually commute and satisfy : (i) (ii) (iii)

(iv)

they are C ( I ) - l i n e a r c%(a*) = (Oga)* Ot,(ab ) = (Ot,a ) b + a (Ogb) o o g = i u , o o v = o , o A u = o , OAK = - i v .

a E "~I , a, b E "PI ,

(40)

8

Moreover one can exponentiate them, namely defining by 2-parameter group of *-automorphisms given by : po,A(a)(m) = e~('~'~

{PO,A;(0, A)

C T 2} as the

(41)

,

we get : = \

0#

/#=A=0

#=0, A.

(42)

Actually PO,A is a module-*-homomorphism such that (0, A) E T 2 ~-~ po,A(a) E PZ is continuous and : PO,A o PO',A' = PO+O',A+A' , (43) If a, b E 7)z their Poisson (or Moyal [Bou]) bracket {a, b} is defined as follows : {a,b}(m;7)=

~]

a(m';7)b(m-m';7)2sin(-~m'A(m-m'))

,

(44)

mtEZ 2

where we set ( s i n x ) / x = 1 for x = 0. In particular that for "~ = 0, it coincides with the usual Poisson bracket, namely : {a, b}~l = {a~l, b~l} = OoaclOAbr -- OAaclOobr ,

(45)

From (44), the right-hand-side defines a continuous function of "y on I, so that the Poisson bracket {a, b} still belongs to T'x. The "Liouville operator" associated to w E "Pl is the module map defined by : L~(a) = {w, a} , a E T'I.

(46)

The properties of this operator are the following : (i) (ii) (iii) (iv)

L~ is C ( I ) - l i n e a r L~(a*) = gw. (a)* L~(ab) = n~.(a)b + aL~.(b) [L~,,L~,] = L{~.,~,} (Jacobi's identity)

w, a C P l , w, a, b E Pz , w,w' E Pi 9

(47)

e T 2,

(48)

We also remark that T(po,a(a))=~'(a)

r({a,b})=0

a, b E P I , ( O , A )

which is equivalent to the "integration by parts formula" : "r(O~,a. b) = - ' r ( a . O~,b)

2.3

The

Rotation

Algebra

7- (L,~(a). b) = --T (a. L~(b)) ,

(49)

Az

In order to get all continuous functions on our non commutative torus, we ought to define the non commutative analog of the uniform topology on Pz. This can be done by remarking that in the commutative case, the uniform topology is defined through a C*-norm, namely a norm on the algebra which satisfies : Ilabl] _< Ilall[Ibll

[[a'all--Ilall

2 9

(5o)

The importance of this relation comes from the fact that such a norm is actually entirely defined by the algebraic structure, namely it is given by the spectral radius of a*a. Therefore, the algebraic structure is sufficient and the uniform topology becomes natural. To construct such a norm, one uses the representations of 79I. A "representation" of PI is a pair (:r, 7-/~), where 7-/~ is a separable Hilbert space, and 7r is a ,-homomorphism from 79I into the algebra B(7-/~) of bounded linear operators on 7-/~. The formulae (17)&(18) give an example of representation for which ~ = L2(T, dO/27r). In particular 7r(U), 7r(V) will be unitary operators on 7-/~ so that if a 9 79I, one gets (if HfllI denotes the sup norm in C(I)) :

Jbr(a)ll_< ~

lla(m)Ib< oc.

(51)

mEZ 2

Two representations (Tr, 7-/~) and (C, 7-/w) are equivalent whenever there is a unitary operator S from 7-(~ into 7"/~, such that for every a 9 791 : S~(a)S -l : r'(a)

(52)

9

Up to unitary equivalence, one can always assume that 7-/~ ~2(N), so that the family of all equivalence classes of representations of 79I is a set denoted by Rep(791). We remark that tile norm 117r(a)II depends only upon the equivalence class of 7r. We then define a seminorm on 791 by : =

Ilalb = sup{ II=(a)II; ~ 9 mep(79J}.

(53)

This notation agrees with the sup-norm on C(I) if a 9 C(I). Then one has [BaBeF1] : Proposition

1 The mapping a 9 79I ~-~ ]la[ll 9 I%+ is a C*-norm.

R e m a r k : The only non trivial fact in this statement is that it is a norm, namely that Ilallz : 0 implies a : 0. D e f i n i t i o n 1 The algebra .,4i (resp. .A) is the completion of 79i (resp. 79) under the norm I1" I]x (resp. I]" liT). ,4 is called the "universal rotation algebra". P r o p o s i t i o n 2 1)-Any representation of 79i extends in a unique way to a representation of A I 2)-If B is any C*-algebra, and/9 is a *-homomorphism from 79I to B, then/9 extends in a unique way as a *-homomorphism from A~ to 13. 3)-Any pointwise continuous group of *-automorphisms of 791 extends in a unique way as a norm pointwise continuous group of *-automorphisms of,41. 4)-The trace r and the angle average (.) satisfy :

IlT(a)llI < Ilalb

II(a)lb < Ila11I

a

9 791 ,

(54)

and therefore they extend uniquely to At. 5)-The norm I1" I[I satisfies : Ila11i = supll~(a)ll

a 9 79I .

(55)

10

In practice the explicit computation of the norm does not require the knowledge of every representation. It is enough to have a faithfull family, namely a family {Trj}jeg where J is a set of indices, such that ~rj(a) = 0 for all j ' s implies a = 0. In other words njegKer(Trj) = {0}. We recall that the spectrum Sp(a) of an element a of a C*-algebra with unit A, is the set of complex numbers z such that zI - a is non invertible in A. Proposition 3

Let (Trj)je J be a faithfull family of representations of the C*-algebra

A, then : ][a[[l = sup H~rj(a)[[

Sp(a) = closure{tJj~gSp(Trj(a))} .

(56)

jEJ

In particular if 7r is faithfuU (namely if J contains only one point), [[a[[i = lit(a)[[ and Sp(a) = Sp0r(a)). 2.4

Smooth

functions

i n .AI

Beside Pl, one can define many dense subalgebras of AI playing the role of various subspaces of smooth functions. (i) For N E N, the algebra CN(AI) of N-times differentiable elements of 1)1 is the completion of .AI under the norm : 1 1

~

Ilallc"/=

,~,

n! n'! IIO;OA (a)ll' "

(57)

O~_n,n ~; n T n ~~ N

(ii) Coo(.Az) = NN>OC~V(.Ax). It coincides with the set of elements a = (a(rn))meZ2 with rapidly decreasing Fourier coefficients. It is a nuclear space, similar to the Schwartz space on the torus. Its dual space S(AI) is a space of non commutative tempered distributions which can be very useful in investigating unbounded elements. (iii) 7-/s(Ar) is the Sobolev space, namely the completion of Ps under the Sobolev norm : Ilall

.,1

(T(a*a)+r(a*(--A)8/2a)) '/2

A=

2+

2

(58)

where --A is the Laplacean on the non commutative torus. The imbedding 7-/8'(.As) ~-* 7C(Ax) is compact if s ~ > s and Coo(A1) = A~>07"/~(Ax), showing that Coo(Ai) is a nuclear space. (iv) An element of AI is holomorphic in some domain D of ( T + i R ) 2 if the continuous mapping (0, A) e T 2 ~-~ Po,A(a) C At, can be extended as a holomorphic function on D. A special interesting case consists in considering the algebra .Al(r) for r > 0, obtained by completing 7)i with the norm :

miami,,, -- sup

~

]a(m;7)[e rl'~ll ,

(59)

~'EI m E Z 2

where [m[1 = Iml[ + ]m21. Then .41(r) becomes a Banach ,-algebra of holomorphic elements in the strip D(r) = {[Im0] < r , limA[ < r}. (v) Let us consider now the case for which I is an open interval, and let ~o~ be the subalgebra of ~~ the elements of which have Fourier coefficients given by C~176 on I. Let us define the operator 07 on T'~~ by :

Ova

( Oa(m) / =

)m

(60) Z2

"

11 Then az obeys the following rules (Ito's derivative) : (i)

it is linear

(ii)

cO,~(a*) = (O,~a)*

a E "R~ ,

(61)

(iii) d T ( a ) / d ' , / = "r(O.ya) a c 7:'f ~ , (iv) O~(ab) = (c9~a)b + a(O~b) + 89(OoaCgAb - cOAaOob) a, b e 7 ~ . One can extend 0r to the dense subalgebra CN'L(AI), obtained by completing P T with respect to the norm : []a[[c~,L,1 =- MaxI 0, there is a r 7~i such that ]]a-adix < ~, and therefore by (54), supTei ]TT(a)-~'7(ae)l 0, such that if It - to[ _< 5, then E(t) N F = O.

15 (ii) it is continuous from inside, namely given any open set O in X, such that E(t0) M O ~ 0, there is 5 > 0, such that if It - t01 _< 5, then E(t) M O ~ 9. If X = R a gap of E(t) is a connected component of R - E(t). One can check that this definition is equivalent to the continuity of the gap edges of E(t) at to. For a E .AI we set E(7) = Sp(r/7(a)), whenever "y 6 I. The main result of this subsection is [BaBeF1] : T h e o r e m 3 For any normal element a E ,4i, (namely such that aa* = a'a), the family (E(7))~e I is continuous at every point of I. The proof of this theorem will not be given here. It can be found in [BaBeFl]. However, it is of very high importance in view of the numerical computation of the spectrum. For we will see in the next section that the spectrum can be easily computed on a computer for rational values of 7/27r. The continuity of the gap edges everywhere on I implies that this type of computation is sufficient to get an idea of the spectrum for irrational values of 3,/2rr. Actually for smooth self adjoint elements of .AI one gets a better result [BaBeF1], namely : T h e o r e m 4 For any self adjoint element H E C3'IAj, the gap edges of any open gap of (E(7))~el are Lipshitz continuous at every point of I. Similar but weaker results have already been obtained previously by Choi et al. [ChE1Yu], and by Avron et al. [AvSi] on the almost Mathieu model. They found HSlder continuity only. Here we get a stronger result. However the Lipshitz constant depends explicitely of the width of the gap considered, and it diverges whenever the width tends to zero. As we will see in section 4 below, there is no chance to get a better result because the gap edges have discontinuous derivative at each rational value of ~//27r. On the other hand, if a gap closes for some value of "y then generically with respect to H E C3,1jtI, we only get H61der continuity with exponent 1/2 near this point.

4

S t r u c t u r e of the R o t a t i o n Algebra .4i

In this section we will give without proofs, a description of the structure of the rotation algebra. The reader interested in the proofs will be refered to [BaBeF1]. Let us consider first the case "y = 0. The algebra 7'0 is then the convolution algebra associated to the group Z 2. Therefore by Fourier transform, one transforms it into the algebra of trigonometric polynomials with the pointwise multiplication. More precisely, if a E P0, we set :

ar

= ~

a ( m ) e ~(eml-Am2) 9

(77)

m6Z 2

This a trigonometric polynomial. The main properties of the Fourier transform are : (ab)r

A) = ac,(O, A)bcl(O, A) , a, b E Po ,

(78)

16

and (a*)r

A) : ar

A)* , a e 7)0 .

(79)

It follows t h a t for every (0, A) E T 2, the m a p a E Po ~ ar tation. Therefore : sup

la~l(0, A)l
0 and 0 < ~ 1/4. At the bifurcation value, this minimum becomes fiat namely the Hessian actually vanishes identically, giving rise to a normal form like : 72 HWBK,~ = --3 + -~- (K~ + K~) + 0(74) 9

(108)

A Bohr-Sommerfeld quantization condition gives at the lowest order in 7 : 727r . En = - 3 + 4F(1/4)4 (2n + 1) 2 + O(V 4)

for n large ,

giving parabolic Landau levels, as can be observed in (Fig. 4).

(109)

22

5.3

E x p a n s i o n n e a r a r a t i o n a l field

The method outlined in the previous subsection for a low field expansion of the spectrum, can be extended to the expansion near a rational field, or also to the case of a matrix Hamiltonian, as can happen if several different bands contribute to the conduction. We will give here the method for the rational fields, leaving to the reader the case of a matrix Hamiltonian as an exercise. We consider now an interval I = [2rp/q - ~o, 27r + r and H = H* 9 .AI. Using the matrices u, v given in section 4 (84), letting Uz, V~ be the generators of the algebra .Az for 17] N, where ERw is the "RammalWilkinson" term given by the following expression : ERw

=

-

,

(115)

The strategy used to prove this theorem is based upon the so-called "Schur complement formula". Let H = H* be a selfadjoint operator acting on a Hibert space of the form 7-I = 7) @ Q. Let P, Q be the orthogonal projections on each subspace of that decomposition and let D be a partial isometry from 7"/to P such that DD* = Iv and D*D = P. We define on P the family of operators :

He~(z) = DHD* + DHQ(zI - Q H Q ) - I Q H D * ,

(116)

whenever z is a complex number which does not belong to the spectrum of QHQ. Then it is possible to show that z E Sp(H) - Sp(QHQ) if and only if z E Sp(HefF(z)). Moreover E is an eigenvalue of H not in Sp(QHQ) if and only if E is an eigenvalue of He~(E). We then denote by P -- I - Q the projection I | P~(0) of .AI(0) @ Mq. For (k;3') ~ (0,0), it follows that there is a small neighbourhood O of ej(O) such that if z E O, z ~ Sp(QHscl(k; 3`)Q). Since the eigenvalue e~(k) is simple for k ~ 0, the projector P3(k) is one dimensional for k ~ 0, and therefore there exists a partial isometry D : C q ~ C such that /)/)* = I, and D*/) = Pj(O). If D is the partial isometry I | let us introduce the effective Hamiltonian :

hi(z) = DtI~D* + DH.~Q (zI - QH.~Q)-' QH.yD* .

(117)

By construction this is an analytic family of elements in AI(0) now. We can therefore analyze it by the method developed in 5.2, and will give rise to a bundle of Landau sublevels En,j(z) near the lower edge of the band Bj. The corresponding part of the spectrum of H~ near e3(0) = 0, will then be given by solving the implicit equation E = E,~j(E). The solution can be computed explicitely order by order in powers of 3`, thanks to the hypothesis made on e~. The Rammal-Wilkinson term comes from the first order contribution of the second term in (117). It reflects the fact that the matrices H~cl(k; 3`) do not mutually commute for various values of k in general, namely it reflects the existence of a curvature in the fiber bundle over the 2-torus defined by Pj(k). The calculation of this term can be found in [RaBe:Alg, BeKrSe].

5.4

Qualitative analysis of the spectrum

Let us now comment on the formulae (114)&(115). Due to the absolute value of 3' appearing in the first term of (114), the right and left derivatives of the band edge with respect to 3` are different, showing that the band edges eventhough continuous

24

functions of 7 by the theorem 3, have nevertheless a discontinuous first derivative at each rational point. On the other hand, even if O~H = 0 the Rammal-Wilkinson term may not vanish. This is the case for instance in the Harper model for p/q = 1/3 (see Fig. 1). We can see the effect of this term by the fact that the left and right derivative of the band edge are not symmetric around 7 = 27rp/q. The difference between them reveals the occurrence of curvature effects. On the other hand one can recognize whether the band edge is a maximum or a mini at the slope of the Landau sublevels emerging away from 7 = 27rp/q. For most values of p/q, all bands are separated by gaps. However, many non generic situation can be observed on examples. (i) Two bands may overlap without touching each other. Then, each minimum or each maximum of the corresponding band will reveal itself by the occurence of a bundle of Landau levels emerging on both sides of 7 = 27rp/q (see Fig. 5), and given by the formula (114)&(115). (ii) two bands Bj, By, with or without overlap, may touch each other. In this case, generically they will touch on a conical point (see Fig. 6). This situation leads to a different canonical form. For indeed the previous analysis can be extended by replacing the projector Pj(0) by P~(0) + Pj,(0). Then the effective Hamiltonian becomes a 2 x 2 matrix unitarily equivalent to the Dirac operator [HeSj:Har2, RaBe:Alg] : HDirac = 17[ 1/2

I g l ~iK2 -

-

0

+ 0(7) .

(118)

This case will give "Dirac levels" which are parabolic namely : E ~ = +const.lnT] 1/2 , n G N ,

(119)

which is for instance what happens in the Harper model at E = 0 and p/q = 1/2 (see Fig. 1). This formula must usually be corrected by a P~mmal-Wilkinson term, giving a slope to the sublevel n -- 0. This is what happens in the WBK-model (108), at

p/q = 1/2 (see Fig. 7). (iii) Two bands can also touch with a contact of order 2. There is another example proposed by M.Wilkinson [Wil:Cri] and studied in details by Barelli and Fleckinger [BaF1], which is the following :

Hw : T1 + T2 + t3 (T21T2e-{~ + T[2T2e {~ + TiT~e -i~ + T,T;2e {') + b.c..

(120)

At E = O,p/q = 1/2, we do get two families of Landau sublevels on either side of p/q = 1/2, corresponding to the bottom wells of the two bands. The generic parabolic touching can be seen on Fig. 8. (iv) A maximum or a minimum can also be reached on a curve. This has been observed in the W B K model at p/q = 1/2. This case has been investigated in details by Helffer and SjSstrand [HeSj:Har3], who remarked that the "subprincipal symbol" may break this degeneracy and create what they have called "miniwells", namely local extremas with deepness of order 0(7). Such an example has never been investigated numerically, but there are indications that such a phenomenon should occur on the W B K model.

25 At last we must point out the occurrence of tunneling effect. For indeed, the classical model gives a Hamiltonian on the phase space given by a 2-torus. This is equivalent to choosing R 2 instead, but requiring that the Hamiltonian be periodic in both directions. This will be called the "extended picture". In this picture, each local extremum is repeated periodically, giving rise to an exact degeneracy. Therefore a tunneling effect should occur between the corresponding wells, ending into a broadening of the Landau levels or sublevels. The width of this broadening can be computed by the WKB method, and will give rise to terms of order O(exp (-S/3`)) where S is some constant equal to the real part of the tunneling action between two neighbouring wells. This effect has been studied in great details in the Harper model by Helffer & SjSstrand [HeSj:Harl, HeSj:Har2, HeSj:Har3], and for the corresponding model on a honeycomb or triangular lattice by Kerdelhud [Ker]. By evaluating precisely the tunneling matrix representing the effective Hamiltonian restricted to each of the Landau sublevel, they could prove that it is again represented by a ttamiltonian with nearest neighbour interactions, having the symmetry of the original lattice (e.g. a Harper model for a square lattice), with a small correction. Therefore, each Landau sublevel is itself decomposed into subbands, and this explain the occurence of the fractal structure. This tunneling effect has also been exhibited in a spectacular example by Barelli & Kreft [BaKr], in the WBK model for t2 > 1/4 and 3` ~ 0. As we already said, after the bifurcation the unique minimum splits into four degenerate minima surrounding one maximum. Since these four wells are very close to each other in each unit cell of the extended phase space, compare to the distance between cells, the tunneling effect between these four wells within the unit cell is likely to dominate over the other sources of tunneling. Each well gives rise to its own bunch of Landau levels, but the splitting due to the tunneling will separate them. It turns out that the tunneling action in this case is not purely imaginary, so that the Landau levels can be represented by if n c N, t2 = 1/2 and i = 1,2,3,4 : 3 2n + 1) + 0(3 ,2) E,~,i = E,~(3`) + dE,~,i(3') , E~(3`) = - 3 + 53`(

(121)

where the splitting is given by [BaKr] :

dEn,i = 3`3e-Im(S2)/~ cos(Re(S4)/43` + 7r/4) + O(e -s'/~) , 7r

(122)

where $2 represents the action lAB kldk2 for a path A B in the complex energy surface Ha(k) = E~(3`) joining two neighbouring wells A and B, while $4 is the tunneling action for a closed path in the same energy surface going through the four wells once. Moreover, S ~ is some action larger than $2. Even though there are usually many non homotopic such paths in this complex energy surface, only the "shortest" ones (in terms of the corresponding action integral) do contribute to this order. In this formula the width of the splitting is controlled by Im(S2) which gives an exponentially small term. But the occurence of a non zero real part produces a nice braiding between these four sublevels as can be seen in (Fig. 9). In a recent work, Barelli and Fleckinger exhibited a braiding of Dirac sublevels near the half flux ( see Fig. 10)[BaF11.

26

6

Elementary

6.1

The

Properties

Furstenberg

of the

Kicked

Rotor

Algebra

As we have seen the Floquet operator for the kicked rotor cannot be seen as an element of the rotation algebra. This is because the kinetic part is not a continuous function of U and V. However, we have seen that it defines a *-automorphism of the rotation algebra. To deal with that we have two choices. The first is to ignore the Floquet operator itself and to stick with its action on the non commutative torus. This is fine as long as we are interested only in the evolution of observables. However, in many occasions do physicists need to know more on the spectrum of the Floquet operator itself, the so-called "quasi-energy" spectrum. One of its most important property is the "dynamical localization", a phenomenon similar to the Anderson localization in Solid State Physics of disordered metals [FiGrPr]. In order to deal with this latter problem, we can simply enlarge our algebra by brute force, adding the missing unitary F0 equal to the kinetic energy defined in section 1 (12) by : Fo = e -~A2 /2"~ . (123) As we have seen in section 1 (19) this operator satisfies the following commutation rules (i) F o V F o ' =: V (ii) F o U F o I = U V - l e -'~/2 . (124) As before we will denote by BI the C*-algebra generated by the polynomials in U, V, Fo with coefficients in the set of continuous functions of 7 in I. This algebra can be rigorously constructed along the line developed in section 2. However one can use the general method of C*-algebras, namely the notion of crossed-product [Ped], to construct it. One can indeed see B1 in two ways : (i)-the first one comes from the previous definition, namely F0 acts on the rotation algebra Jti by mean of the *-automorphism ~o(a) = Foa~o 1

a e A .

(125)

Therefore BI can be seen as the crossed product ,41 • Z of the rotation algebra AI by the Z-action defined by ~0. Using Weyl's operators defined in section 2 (32), we notice that /3o (W(m)) = W(Gm) , m e Z2 , (126) provided G is the element of S L ( 2 , R) given by : G=[

-

11 0

]

(127)

1j "

(ii)-the second one consists in considering first the subalgebra generated by functions in g(I), together with the operators V and F0. This is an abelian C*-algebra isomorphic to g ( I x T~). This isomorphism associates to V and Fo respectively the functions fv(7, x, y) = ei~ and f F o ( % X , y ) = e% Actually, the inner automorphism associated to U leaves this algebra invariant. This is because the commutation rules (19) can be written as (i)

U V U -1 = ei~V ,

(ii)

UFo U-1 = e~'Y/2VFo .

(128)

27 In other words, for f ~ C ( I • T2), we get : U f U -1 - f o r

(129)

where r is the "Furstenberg" map acting on I • T 2 as : r

= (%x + %y + x + 7/2) ,

( % x , y ) E I • W2 .

(130)

This m a p was used by Fhrstenberg to study the ergodic properties of diophantine approximations in number theory. Thus/3i can be seen as the crossed product C ( I • T 2) xr Z by the Fhrstenberg map. This is why we propose to call this algebra the "Furstenberg algebra". We see that r leaves each fiber {7} • T2 invariant and we will denote by r the corresponding restriction. It is well-known that whenever 7/2~ is irrational, r is a minimal diffeomorphism [CoFoSi].

6.2

Calculus

o n Bx

As for Az, a calculus can be defined on the Furstenberg algebra. Since the trace on .4i is/3o-invariant, it defines a trace on the crossed product in a natural way. It is actually defined by the formula : T (W(m)Fg)

=

(~m,0.(~/,0 ,

m E Z2 , 1 C Z.

(131)

Since we have defined originally (cf. section 1) U, V, F0 in term of action-angle variables in the classical case, one also gets an angle average (.) namely : (W(rn)F0 l)

=

~ml,oVm2F~ ,

if

m = (ml,m2) C Z ~ , 1 E Z .

(132)

Thus, if a E 131, (a) E C(I x T2), and this average satisfies the properties described in (32). In much the same way, a differential structure can be defined. The derivation COo can be extended immediately to/3i by : OoU = i U ,

COoV = 0 ,

00F0 = 0 .

(133)

We notice however that COAcannot be extended as a derivation in B/ because COAFo would be unbounded, namely outside/3i. But a new derivation COyappears defined by COuU = 0 ,

COyV = 0 ,

COuFo= iFo .

(134)

Both 0o and COyare the infinitesimal generators of the following two-parameter group of *-automorphisms : P0,u (W(m)F0 l)

ei(ml~

(135)

which leaves the trace invariant. At last, the definition of a Poisson bracket is not obvious because for 3 / = 0 the algebra B0 is no longer commutative. Even though it is in principle possible to define such an object, we will not use it, and we skip this part of the calculus.

28 6.3

Representations

and

structure

of BI

Among the representations of/3I, we will select one family of special interest in view of the original definition of the kicked rotor in the physical Hilbert space L~(T) given in section 1. It is actually simpler to work in the momentum space, namely in ~ ( Z ) where the integers of the chain Z are simply the quantum numbers for the angular momentum. This family {%,~,u; (7,x,y) E I • T 2} is indexed by points in I • T 2 and acts on g2(Z) as follows : (i)

(ii) (iii) (iv)

(Tr%x,y(f)r (r~,x,y(U)r (~r~,x,y(V)r (Tr.~,x,y(F0)r

(n) = f(7)r , (n) = r - 1) , (n) = e~(x-n*)r , = ei(y-nx+n='/2)r

f C C(I) (136) ,

if

~b E g2(Z) .

Comparing with the equation (12) & (13), 7 appears as an effective Planck constant, x as an effective magnetic field, and y as a phase factor entering in the definition of F0.

With these definitions, the following result can be easily proved by standard technics

P r o p o s i t i o n 7 1)-The family {~r~#,y; (7, x,y) E I x T 2} is faithfull. In particular, the norm of a E/3I is given by : ]lalI

--

sup

sup

~EI (z,y)~T 2

II=~,~,~(a)ll

9

(137)

2)-The map ( 7 , x , y ) E I x T 2 ~-+ %,z,y(a) is strongly continuous for all a E BI. 3)-For ~/ E I, the trace is given by : r~(a) =

fT dxdy 2 4r



(138)

Moreover if "y/27r is irrational, we get 9

1

= 1,m 27- T

L,),

uniformly in (x,y) E T a. 4)-If T is the translation operator in g2(Z), namely if (Tr = r - 1) for r E g2(Z), then :

TTr~,,.y(a)T-' = ~r~(~,~,~)(a), 5)-If N is the position operator in s have : %,x,y(Oea) = i[N, rr~,~,y(a)] ,

a ~ I3i, ('r,x,y) e I x T 2 . defined by (Nr

= nr

(139)

, r E g2(Z), we

0 7r~#,y(Oya)= ~yr%x,y(a) .

(140)

Thanks to this result the elements of BI can be described as follows. For a E 131, we set :

a('~, x, y; n) = (01rr~,~,y(a)ln) .

(141)

29 This is a continuous function on I • T 2 • Z converging to zero at infinity. In terms of such functions the product and the * in BI can be expressed as follows : ab(7, x , y ; n ) = y ~ a ( 7 , x , y ; l ) b ( 7 , x

- 17,y - lx + 127/2;n - l) ,

(142)

IEZ

a* (7, x, y; n) = a(7, x - nT, y - n x + n27/2; - n ) * ,

(143)

for a, b E BI. Moreover, the representation 7%,x,y is given by : (Try,x,y(a)r (n) = ~ a ( 7 ,

x-nT, y-nx+n27/2;l-n)r

,

r 9 g2(Z) .

(144)

IEZ

In particular, due to the faithfullness of this family, a = 0 if and only if the function a(7, x, y; n) vanishes identically. If we denote by By the algebra Bl for I = {7}, the following theorem characterizes its structure : T h e o r e m 11 1)-If 7/2~r is irrational, 13y is simple. In particular, every non zero representation is faithfull. 2)-For 7 = O, the algebra Bo is isomorphic to the universal rotation algebra A . 3 ) - I f 7 = 27rp/q where p, q are positive integers p r i m e to each others, B2,p/q is isomorphic to the sub C*-algebra of Mq(C) | Bo generated by : ~]=u|

l/=v|

F0 = w |

(145)

where Uy, Vy, Fo,y are the generators of B~, and u, v, w are three unitary q x q matrices fulfilling the following conditions : u q = v q = w 2q = I , u v u -1 = e2i'P/qv ,

(146)

u w u -1 = ei'P/qvw ,

vw = wv .

(147)

P r o o f : 1)-For 7/27r irrational, the Furstenberg m a p r : (x, y) E T 2 ~-+ (x + 7, Y + x + 3'/2) E T 2 is a minimal diffeomorphism of the torus [CoFoSi]. Therefore, the crossed p r o d u c t By = C(T 2) x r Z is simple [HiSk]. 2)-For 7 = 0 the commutation rules become : UV

= VU

,

VFo

~- F o V

UFoU -1 = VFo .

,

(148)

These rules are precisely the ones defining the universal rotation algebra .4 if we identify V with the m a p 3' c T ~-~ e ~y C C (cf. section 2). 3)-If one chooses the matrices u, v as in (84), the matrix w becomes : 1 0

0 A'

0 0

... -.-

0 0

0 0 (149)

W =

0 0 where ~'

=

e iTrp/q.

0 0

0 0

... .-.

~,(q_~)2 0 0 )r

30 It is easy to check that U, I), F0 satisfy the commutation rules for the algebra B2~p/q. Hence they define a *-homomorphism p from B2,p/q into Mq(C) | Bo. 4)-To achieve our result it is sufficient to prove that p is one-to-one. For (x, y) C T 2, let ~'~,~ be the representation of Mq(C) | Bo given by id | 7ro,~,y acting on C q | g2(Z). Any a E Bo can be seen as a function on T 2 x Z as (see (141)), and for r E C q | andAEMqweget : q-1

(150)

(n) = ~ ~ A3,j,a(x, y - nx;1 - n)r

[~'~,~(A | a)r

j=0 IEZ q-1 Let {ej}j= o be the canonical basis of C q with the convention that e~+q = ej, and let

{Sn; n E Z} be the canonical basis of g2(Z). We set : IJ, n) = ej | 5,~ .

(151)

(j, 01#,,y(A | a)IJ', l) = A j d , a(x, y, ;l) .

(152)

Then : It is not difficult to check that if now b c B2~p/q and r E C q N g2(Z) we get : [5,,y (p(b))r

(n) = ~ b(x - 2rjp/q, y - nx + j2~p/q; l)r

+ l) ,

(153)

lEZ

where j + l is defined modulo q. It is actually sufficient to check this formula on the generators Ue~p/q, V2,p/q, Fo,~p/q since #x,y and p are .-homomorphisms. In particular (0, Ol~x,y(p(b) )Il, l) = b(z, y; l ) .

(154)

Thus p(b) = 0 if and only if b(x, y; l) = 0 for any (x, y; l), namely b = 0. Hence p is one-to-one. Using the same strategy we can easily get : C o r o l l a r y 4 1)-for "7 c R the algebra B2~p/q+~ is isomorphic to the subalgebra of Mq @ B 7 generated by u | U~, v | V~, w | Fo,~.

2)-for "7, 7' C R the algebra B~+~, is isomorphic to the subalgebra of B~ | by U~ | U~,, V~ | V~,,Fo,~ | Fo,~,. 6.4

Algebraic

Properties

of the

z, generated

Rotor

Kicked

In section 1 we have expressed the Floquet operator of the kicked rotor as : -1

~ e-iA2/27e-iKc~176

if1

(155)

where 7 = l i T / I is the effective Plan& constant and x = - # B T is the effective magnetic field. Moreover in the momentum space representation, A = "/N - x if N is the position operator (see prop. 7). Using the previous algebraic framework, it follows that : FK,-r,x = 7r,y,~,o(rK) , (156) with : FK

eiK(U+U-1)/25Fo ,

(157)

31

where "~ : 3' E I ~ 3' c R. In this special case we notice the following property :

(158)

~r~,x,~(F~:) - e'~FK,~,~ ,

so that one can set y = 0 without loss of generality. It follows that FK belongs to/3i for any compact set I in the real line not containing the origin. Our first set of results concerns the spectrum as a set of this Floquet operator. Since it is unitary its spectrum is necessarily contained in the unit circle S~. Actually the following results are still valid if we replace cos(O) = (U + U-1)/2 by any real valued 27r-periodic continuous function g(O) on the real line. T h e o r e m 12 1)-For any 3" ~ O, the spectrum of FK,7 = ~ ( F K ) is the full circle.

2)-If 3"/27r is irrational, the spectrum of FK,~,~ is the full circle for any x e T. 3)-If "y/2~ is rational, but x/2~r is irrational, the spectrum of FK,~,~ is the full circle. 4)-If 3"/2~r and x/2~r are rational, FK,~,~ admits a band spectrum. P r o o f : 1)-Since the family {n~,~,~; (x, y) e T 2} of representations of/3~ is faithfull, we get : SPB,(FK,~t) = U(x,y)Ew2eiUSp(FK,.~,x) 9 (159) Taking the union over y clearly gives the full circle. 2)-If 7/27r is irrational, B~ is simple. Thus each of the ~r~,~,y's is faithfull, in particular S1

=

SpB~(Fg,~ ) = Sp(FK,~,~) ,

Vx ~ T ~ .

(160)

n C Z

(161)

Vn e Z .

(162)

3)-If 3' = 2top~q, the covarianee condition (145) gives : T"%,~,~(Fu)T-"q

= ~,~,~+.~x(FK)

,

In particular Sp (Tr~,z,y(FK)) = Sp (Tr~,~,u+~(FK)) ,

If in addition x/2~r is irrational, given any y' E T we can find a sequence (nl) of integers such that yP - y = liml~r nlqx mod 27r. By the strong continuity of ~r~,~,y with respect to y, it follows that : Sp (Tr~,,,r

C Sp (Tr~,x,y(Fg)) .

(163)

Since y, y~ are arbitrary, the same result holds after exchanging them. In particular for any y we have : Sp (Tr~,~,u(Ft()) = e~YSp (n~,~,o(Fg)) = Sp (7c~,x,o(Fg)) , showing the result. 4)-If 3' = 2rp/q and x = 2rr/s, the periodic. By the Bloch theorem we get using the corollary 4 that the algebra Mq | Ms | r generated by u | u' | u ~, v ~) are the q x q matrices (resp. s

(164)

covariance property shows that rc~,x,o(Fg) is a band spectrum. Actually one can easily see, 7r~,x,0(/3x) is isomorphic to the subalgbera of eik, v | e ~ | 1, w | v ~| 1 where u, v, w (resp. x s) defined in the theorem 12, and k is the

32 quasimomentum. Here we used the fact that r%,~,O(FK)Q = I if Q = 2(q v s). This gives the band spectrum by diagonalizing the finite dimensional matrices and varying k. The next set of results concerns the density of states. Let A be an interval in the unit circle, namely the image by w ~-+ e i~ of an interval of the real line. Let us also call gi the restriction to the finite set [-L, L] of g(O). This is a self adjoint matrix of dimension 2L + 1. Let also ~(L) 0,"/,x be the restriction of F0,~,x to the same interval. Because it is diagonal it is a unitary (2L + 1) x (2L + 1) matrix. Then we set FK(L) 1:2(L) piKgL(O)/"/ Again, this is a unitary matrix of dimension 2L + 1. Let then ,'y,x = "tO,~',xv nn(A) be the number of eigenvalues of this matrix contained in A. As L ~-~ c~, this number increases like O(L), so that we can define the Integrated Density of States (IDS) as the following limit, if it exists : nL(A)

''

2L+l

(165)

The first important property is the "Shubin formula" [Bel:Gap] P r o p o s i t i o n 8 If'y/2~r is irrational, the limit defining the IDS exists uniformly with respect to (x, y) E T 2 and is independent of (x,y). Moreover it is equal to : Aft(A) = % (x~(FK)) ,

(Shubin's Formula) .

(166)

where Xzx is the characteristic function of the interval A. The proof of this proposition can be found in [Bel:Kth, Bel:Gap, BeBoGh] for self adjoint operators. It can be easily adapt for the Floquet operator. We notice that the limit is reached uniformly with respect to (x, y). This is because the Furstenberg map is minimal and not only ergodic. Another remark is that the eigenprojection xA(FK) does not belong in general to the algebra B~. However, it belongs to the von Neumann algebra Lcc(Bz, %), namely the weak closure of B~ in the CNS representation associated to the trace. Thus the Shubin formula is meaningfull. Thanks to the Shubin formula, the IDS can be written as : Aft(A) = f a dAft(E) ,

(167)

where dA/'.r is a probability measure on the torus T (which we identify with the unit circle) called the Density of States (DOS). We can actually compute the DOS namely

P r o p o s i t i o n 9 If "y/27r is irrational, for any continuous real valued 2rr-periodic function g on the real line, the DOS of the kicked rotor is equal to the normalized Lebesgue measure on the torus, namely :

dE

KAf~(E)-- 27r

(168)

33 P r o o f : The Shubin formula implies that the DOS is the unique probability measure on the torus such that :

fw d'N"Y(E)ei'~E =

T7 (F~) ,

n E Z.

(169)

We claim that T7 (F~) = 0 unless n = 0 which will prove the result. For indeed, the trace is invariant by the,automorphism group/~0,k. On the other hand, we have :

~o,k (FK) = e~kFK.

e"~kF~

It follows that ~3o,k(F~) =

(170)

showing that

T~ (F~) (e '"k - 1) = 0 .

(171)

Our last result concerns the algebraic way of writing the kinetic energy. In order to study numerically the spectral properties of the kicked rotor, several physicists [CaChIzFo] have iatroduced the averaged kinetic energy. Giving an initial state r c t?2(Z), it is given by (see (9) & (11)) : s

=

(r162

L2

,

t e Z,

(172)

where L is the angular momentum, I is the moment of inertia and F the Floquet operator. Thanks to the definition of the position operator N (see Prop.7) and introducing the period T of the kicks, one can write it as :

gc(t) = 2~(r162

.

(173)

In order to keep only dimensionless quantities, we will redefine this kinetic energy by forgetting the prefactor I/2T 2. Moreover physicists usually choose an initial state localized on one value of the initial angular momentum. Using the covariance condition, it is always possible to choose r = 10) by changing the value of (x, y) if necessary. This why we will rather define the mean kinetic energy in the following way : s

= 72 (01F~ ~,,xN2F~ t, ,xlO)

.

(174)

We notice that varying y will not change this definition. Using now (146), it follows immediately that if IAI 2 = AA* :

C~,~(t) = 72(01~r~,~,u(10oFk,~,~1210) . The choice of the initial average over the position generic properties of the to taking the trace. This

(175)

value of the angular momentum being arbitrary, we may of the initial state in momentum space, in order to get the system. This is equivalent to averaging over (x, y), namely why we will also consider the quantity :

C~(t) = 7%~ (1OoFk,~,J 2) 9

(176)

34

7 7.1

Localization and D y n a m i c a l Localization Anderson's

Localization

The localization phenomena was predicted in 1958 by Anderson [And] for conduction electrons in a disordered metal. The main idea underlying this effect is that the electronic wave in an infinite medium is reflected by the obstacles (ions, defects,etc,...). If the medium is a perfect crystal, the total reflection coefficient may not be equal to one due to constructive interference effects and allows the wave to travel freely towards the boundary. This happens whenever a Bragg condition is fulfilled, for special values of the total energy of the traveling particle, defining a band spectrum. This is the essence of Bloch theory for perfect metals. In such a case, the conductivity is infinite, if one neglects the influence of phonons and of the electron-electron interaction. If the medium is not periodic but quasiperiodic, such as quasicrystals, one may have also free Bloch waves if the quasiperiodic potential describing the influence of the ions on the travelling particle is not too strong [DiSi, BeLiTe, ChDe, BeIoScTe, BenSir]. However, in a disordered medium, the Bragg condition is unlikely, namely destructive interferences may force the electronic wave to vanish at infinity. Thus, the electonic wave is trapped in defects : in other words it is localized in a bounded region. Anderson proposed a tight binding model of such medium and could predict that 1-dimensional disordered chains always exhibit localization [Pas, Cyc]. Later on [AbAnLiRa] it was argued that in 2D the same effect occurs. But in higher dimension, localization holds only for strong disorder or at the band edges [FrSp, FrMaScSp]. Then if the disorder is not too strong, Ohm's law holds, leading to a finite conductivity, even if we ignore the phonons and the electron-electron interaction. The Anderson model is extremely simple but contains most of the properties necessary to describe such a medium. In a tight binding representation, the electronic states can be represented as elements of the Hilbert space g2(zD), if the crystal we start from is the D-dimensional lattice Z D. If there is no disorder, in the one electron approximation, the conduction electrons are approximately described by the free Laplacean •D namely if r ~ ~2(zD) :

A~r

~

r

(177)

In-,~q-1 where t is the "hopping" parameter which measures the energy required for an electron to hop from one site to the next one. The energy spectrum is then given by the band

[-2Dt, 2Dt]. Adding one defect in the crystal can be described by adding to the previous Laplacean a local potential in the form of a sequence Vd~f~ct = (Vd~/~ct(n); n C zD), as was shown in 1949 by Slater. To get a homogeneous distribution of defects it is therefore sufficient to replace Vd~I~r by a homogeneous sequence V. To take into account the randomness of the defect distribution we will assume that the values V(n) of this potential at each site are identically distributed random variables. Even though we expect some correlation between them in realistic systems, at least at short distances, Anderson proposed to consider the simplest case for which they are independent and uniformly distributed in an interval [-W, W 1. Then W is a measure of the disorder strength. Let s be the corresponding probability space (in this example, 9t = I-W, W] zD) and let P be the corresponding probability measure (in this

35 example, P = (~,~ez D d V ( n ) / 2 W ) . The potential becomes a function of the random variable ca E f t so that the Anderson Hamiltonian can be written as :

H~ = / % + w~.

(178)

The probability space (f~, P) can be seen as the configuration space for the disorder. The translation invariance of the original lattice is not completely lost. For indeed, translating this new system is equivalent to translate the distribution of defects back. More precisely, there is a measure preserving action of the translation group on f~. For the Anderson model this action is given by Trcan = ca,~-r. If we denote by T ( r ) the translation by r c Z D in the Hilbert space, namely for r c t~(zD), T ( r ) r = r r), we get the following "covariance condition": T ( r ) H ~ T ( r ) -1 = HTr~ .

(179)

We will complete this framework by adding two conditions. The first one is the ergodicity of the probability measure P. Thanks to Birkhoff's ergodic theorem, it expresses the fact that space averages coincide with P-average. In this way, P can be constructed in practice simply by taking space averages, an unambiguous process. The second one concerns the existence of a topology on f~ which makes it a compact Hausdorff space, and such that the P-measurable sets are generated as a a-algebra by the Borel sets, namely P is a Radon measure. In the Anderson model the product topology will do it. Actually an intrinsic definition of homogeneous system has been proposed in [BehKth, Bet:Gapl leading to the definition of a canonical topology on the disorder configuration space. For this topoloKy, the mapping w E f t ~ H~ is strongly continuous (in the resolvent sense whenever H~ is unbounded self adjoint). To summarize, homogeneous media, such as crystals, quasicrystals, glasses, amorphous, aperiodic or disordered systems, may be mathematically described by the following axioms. (D1)-The disorder configuration space is a compact Hausdorff topological space ~t endowed with a probability Radon measure P (D2)-Tbe translation group is a locally compact abelian group G acting in ft by mean of a continuous group of homeomorphisms ca ~ gca. The probability P is G-invariant and ergodic. (D3)-The quantum state space is a separable Hilbert space 7-i in which G acts through a projective unitary representation {T(g); g c G}. (D4)-The Hamiltonian is a strong-resolvent continuous family H = {H~; ca ~ ft} of self adjoint operators acting on ~ with a common G-invariant domain Z). (D5)-A covariance condition is satisfied, namely: T ( g ) H , , T ( g ) - ' = Hg~ .

(180)

In general we will prefer a projective unitary representation. For indeed there are concrete examples for which the translation group does not act as a true representation. This is the case for a crystal in a uniform magnetic field [Bel:Gap]. We have restricted ourself to abelian translation groups because no concrete useful example have been studied till now with non abelian groups. However, systems living on a Cayley tree admits a non abelian translation group which is usually a free group. We

36 can also include in G other symmetries like rotations, reflections, if necessary. This has never been investigated in detail yet, even though we believe that it should be useful: classification of defects in crystal may be related to such groups. The smallest observable algebra that can be of interest for physics, is the one constructed with the energy. In more concrete systems however, other observables like spins, may be relevant. For simplicity, we will consider the simplest case for which the only relevant observable is the energy. In a homogeneous medium, the choice of the origin is arbitrary, since the systems reproduces itself under translation. So that the physics of the system is described by any of the elements of the family H = {H~o;w C l-l} representing the energy. In order to avoid choosing arbitrarily one of them, we will include all of them. We then define a non commutative C*-algebra C*(H) as the smallest one in the space of bounded operators onT-/ containing the resolvent of each of the elements of H. In general, we do not know the structure of such an algebra. However for most concrete examples construct till now, namely by using the Schr5dinger operator for one electron systems [BehKth, Bel:Gap], like the Anderson model, this algebra is nothing but the crossed product g(ft) x G defined by the topological dynamical system (fl, G) describing the disorder configurations in the original medium. This algebra must be slightly modified if a uniform magnetic field is turned on. We will ignore this latter case here. Thanks to this framework, there is a very close analogy with aperiodic media in Solid State Physics and the dynamics of a kicked rotor. Even though the physical interpretation is very different, the C*-algebra used to describe the observables is also a crossed product. However, in the kicked rotor model, the lattice G is the quantized momentum space instead, and the space ~t admits a fairly different interpretation since the variable -y plays the role of an effective Planck constant and is related to the period of the kicks, x plays the role of a magnetic field, whereas y represents a generic translation in momentum space. We also notice that the ergodicity of the measure holds only if 7/2~r is a fixed irrational number. There is also a very close analogy with 2D-dimensional lattice electrons in a uniform magnetic field. We have already seen that the observable algebra is the rotation algebra AI which can also be seen as the crossed product g(I • T) •162Z if r : (% x) E I • T ~-* (% x + ~') E I • T. Then 7 plays the role of a dimensionless magnetic flux per plaquette, whereas x is a generic position of the origin in the xdirection of the lattice. Again, the ergodicity of the measure on gt = I • T holds only if I = {-y} where V/2~r is a fixed irrational. The main question now is whether this formal analogy between so different problems will produce phenomena similar to Anderson's localization. The common belief is that if H is a selfadjoint operator belonging to this algebra, with short range interactions, namely if it is smooth enough with respect to the differential structure that will be described in the next subsection, it will exhibit such phenomena at least if the dynamical system (~, G) is "sufficiently aperiodic". The precise meaning of "sufficiently aperiodic" is not completely understood yet. Several numerical studies have investigate this question, but they are far from having given a precise criterion yet [FiHuXX]. More precisely we define a 2-point function by C(g) = {FFg) - (F} ~, where F is a continuous function on ~t and Fg(w) = F(g-lw) while (.) is the ergodic average. If any 2-point function converges to zero fast enough as g ~ oo, the localization is expected to occur. This is certainly not the case for a periodic or an almost

37 periodic dynamics, describing for instance a perfect crystal with or whithout a uniform magnetic field. And indeed we do not expect in this case localization to occur. Still, a 1D model like the Almost Mathieu Hamiltonian [AuAn, ChDe, BeLiTe], has been proved to exhibit a metal-insulator transition at large coupling. But the Furstenberg map for instance, which satisfies this criterion, should give rise to localization. This is the basis of an argument by Fishman, Grempel and Prange [FiGrPr] predicting that localization occurs in the kicked rotor problem. The next problem therefore is to describe mathematically w!lat we expect to characterize the localization. One of the first criterion used by Anderson was connected to the time evolution of quantum states : if the time-average of the probability for the initial state to come back after time t is positive, then localization do occur. We will see later on, thanks to an early result of Pastur [Pas] that this criterion is related to the existence of a point spectrum for H~, P-almost surely. This is essentially why mathematicians describe localization in term of the existence of a point spectrum. It is related to the finiteness of the so called "inverse participation ratio" (see below). Another way consists in defining the localization length: roughly speaking it gives a measure of the diameter of the region where a typical eigenstate is localized. One of the main problems in dealing with the spectral property of the Hamiltonian, is that in many situations, this requires the choice of a fixed representation of the observable algebra. While in the Anderson model, this choice is quite natural, thanks to the description of the original disorders medium, in other models for which we would like to use the localization theory, it is not necessarily so. Two inequivalent representations of the same algebra may give different type of spectral measure for the same Hamiltonian. This happens for instance in the problem of Bloch electrons in a magnetic field. Therefore if this latter point of view were correct, localization would require to distinguish physically between different representations. However, the computation of the localization length requires a space average, in order to get a quantity insensitive to the specific configuration of the disorder, and therefore as we will see, it can be interpreted in a purely algebraic way. There is therefore an apparent contradiction between the two points of view. This is actually nothing but the usual opposition between the SchrSdinger and Heisenberg point of view in Quantum Mechanics. Our main purpose in this section is to show how to reconcile them, and to show that in some sense they are equivalent. Our last comment concerns the semiclassical limit. While this limit is meaningless in the Anderson problem, since the starting point is the band theory for perfect crystals, a fairly strong quantum theory, the kicked rotor problem gives a nice example where the semiclassical limit exists indeed together with a localization effect. It is therefore natural to consider what happens to the localization phenomena in this limit. The main discovery of Chirikov, Izrailev and Shepelyansky [ChIzSh] was to relate this limit to the diffusion constant in phase space of the classical kicked rotor. Even though this relation has not been proved to hold rigorously, many numerical studies show that it is probably correct at least under some unknown "generic condition". Therefore we have reached here one point of the so-called "quantum chaos". We will give only some pieces of this puzzle here.

38 7.2

The

Observable

Algebra

To avoid useless technical difficulties, we consider now the C*-algebra C(f~) x G where G = Z D . D will be called the dimension of the lattice. However, most of what will be described here can be extended to more general groups such a s R D for instance. As for the rotation or the Furstenberg algebra, we can develop a calculus as follows. Elements of C(f~) x Z D are continuous complex functions a ( w , n ) on the space f~ x Z D vanishing at infinity. To define this algebra properly, it is more convenient to start with the dense sUbalgebra G(f~ x Z D) of continuous functions on f~ x Z D with compact support, endowed with the following operations: ab(~; n) = ~

a(~; l)b(T-Iw; n - l) ,

(181)

IEZ D

a*(w; n) = a(T-~w; - n ) .

(182)

Since the functions a and b have compact support, the sum above is finite. Remarkable elements are given by : I(w;n) = 5~,0 ,

U ( r ) ( w ; n ) = 5n,-r ,

(183)

r C ZD .

The first one I is a unit, whereas U(r) is a group of unitaries namely U ( r ) U ( r ' ) = U(r + r'), U(0) = I and U(r)* = U ( - r ) = V(r) -1. A family of representations in the Hilberts space &(Z D) indexed by w E f~ is given by: 7rw(a)r

=

~

a(T-'~w; n' - n)r

,

a E Cc(~~ X Z D) ,

/~ E g 2 ( z D ) 9

n I EZ D

(184) In particular we get a(w; n) = (0brw(a)in). Then a C*-norm is defined by: I[all = sup II~rw(a)l[ ,

a e G(f~ x z D) .

(185)

weft

Then C(f~) x T Z D is the completion of Cc(f~ x Z D) under this norm. To shorten the notations we will denote it by .4. Given an invariant probability measure P on f~, a normalized trace wp (or T for short whenever no confusion arises) is defined by: v(a) = I n dPa(w; O) ,

a c .4.

(186)

It is easy to see, by using the Birkhoff ergodic theorem, that if P is ergodic, r(a) = lira

1

ATZ D ] - ~

TrA 0rw(a))

for P - almost all w .

(187)

At last, the differential structure is related to the group action and defined as follows. If n = ( n l , . . . , n o ) E Z D, we define the *-derivation c9, by: a,a(w; n) = i n , a(~; n) ,

# = 1,..., D .

(188)

39 These derivations commute together and are the infinitesimal generators of the Dparameter group of automorphism {po; 01T D } (the so-called dual action of Takasaki [Ped]) defined by: po(a)(w; n) = e'~

n) ,

On = Olnl + . . . + ODnD 9

Moreover, denoting by N, the position operators defined by N , r g2(zD), we get: 7c,,(O~,a) - i[N~,, 7r~(a)] . 7.3

Localization

(189) = hue(n) in (190)

Criteria

In this subsection we give several criteria for the localization and discuss the relation between its finiteness and the nature of the spectrum. We will consider a self adjoint element H = H* in the algebra A = C(9t) • zD previously described. In view of the study of a Floquet operator we may consider a unitary element F = (F*) -1 of this algebra instead. This latter case reduces to the former provided we identify F with e i T H for some T > 0 and the Borel sets A are subset of the unit circle. In the physical representation 7r~ we consider the operator 7r~(H) = H~ instead. If A is some Borel subset of R we denote by PA t eigenprojection of H corresponding to energies in A namely : (191)

PA = Xzx(H) ,

where X~ is the characteristic function of the interval A. Again, we notice that in general P/, may not belong to A. However it always belongs to the so-called Borel algebra B ( A ) [Fed], formally generated by Borel functions of elements of A. The Borel functional calculus permits to extend any representation of .4 to its Borel algebra. Hence the previous definition makes sense. The price we pay for it is that the mapping w E ~t ~-~ Try(a) may not necessarily be strongly continuous any more, but it is always strongly borelian if a E B ( A ) . If H~ has a pure point spectrum in A we get the following decomposition:

7r~(Pa) = ~ HE(w),

(192)

EEA

where HE(W) is the eigenprojection of H~ corresponding to the eigenvalue E. If E is a simple eigenvalue, one gets He(w) = [r162 where eE,~ is a normalized eigenstate namely: HeE,~][2 = ~

IeE,~(n)[ ~

1 < +C~.

(193)

nEZ D

The first quantity measuring the localization is the probability of staying at the origin. It was introduced by [And] and studied by Pastur [Pas]. To define it let us first consider the time-average A,~,~,(A, w) of the probability for an initial state at n to be localized at n t after time t: A,,n,(A,w) = lira f T dt [(n[Tc~(eitr4 PA)In')]2 T~---,(~JO T-

(194)

40 If H~ has a pure point spectrum, the decomposition (192) leads to : A.,n,(A,w) = ~

(195)

[(nlIIE(W)ln')l ~ .

EEA The covariance condition implies A~,n,(A, w) = A0,,,_~(A, T-~w), so that the staying probability is entirely given by the function A0,0(A,w), provided we consider it as a function of the disorder. We remark that if the eigenvalues are simple, since the eigenstates are normalized we get:

Ao,o(A,w ) =

EEe~ICE,~(O)I 4 (E~

1r

(196) 2 '

namely A0,0(A, w) is the mean inverse participation ratio for energies in A. To get a quantity insensitive to the disorder, let us average it with respect to P defining the averaged inverse participation ratio : (197)

~A = fn dP A0,o(A, w) .

Using now the automorphism group defined in eq.(189) and the eq.(184,186), an elcmentary calculation leads to the following expression for {~: ~a =

lim[T -Tdt fT~

r~

Jo

dDO (2~) ~

~-(e"n PApo(e-"n P~)) .

(198)

So we see that the staying probability or the inverse participation ratio, admits a purely algebraic expression. The Pastur theorem [Pas] can then be established as follows: T h e o r e m 13 For almost all w E 12, the number of eigenvalues of H~ in A is either

zero or infinity. The latter is realized, namely H~ has some point spectrum in A, if and only if the averaged inverse participation ratio ~a is positive. C o m m e n t : this criterion is not sufficient to eliminate continuous spectrum. We now introduce a stronger notion of localization giving a measurement of the localization length. Whenever Try(H) has pure point spectrum, the eigenstate may decay faster at infinity on the lattice. We are led to introduce quantities like: ] 1/p

g(P)(E,u;) =

E IeE,~(n)121nlp n~Z D ]

1

(199)

for p > 1. If the eigenstates decrease exponentially fast one can also consider the quantity g(E, w) = lira sup --In[r

(200)

However such expressions are very badly behaving with w in general and they are not suited for comparison with experiments or numerical calculations. The following

41

definition will be more convenient and will give rise to a quantity independent of w. We consider the averaged fluctuation of the position in the form: AX~,,(T) 2 =

(nl(N~(t) - N)2ln) ,

(201)

where N,~(t) = @H~tNe-'u~*, and N = ( N 1 , . . . , N o ) is the position operator. The covariance property gives A X ~ , , ( T ) = AXT-,~,o(T), so that after averaging over the disorder, we get a quantity independent of n, namely AX(T) 2 = fn dP(w)AX~.o(T) 2" An elementary calculation shows that: AX(T)

--

(202)

We will generalize this expression by considering, for every Borel subset A of the real line, the corresponding quantity AXz~(T) 2 obtained in the same way if we replace e iHt by e i H t p A . The main result in this respect is the following: T h e o r e m 14 If g2(A) = lim sup AXA(T) 2 < c~ ,

(203)

T~--*~

then H~, has a pure point spectrum in A for almost all w E ~. Moreover, if]q" denotes the density of states of H, there is an N-measurable non negative function g on R such that for every Borel subset A t of A,

e~(A ,) = f,, dN'(E)e(E) ~ ,

(204)

C o m m e n t : we will see in the proof that if ~m,(w) denotes the set of eigenvalues of

H~: g2(A ') = ~ d P ( w )

~ nc=ZD

n2

I(01HE(~)In)I ~ .

~

(205)

E6app(w)oA

In particular, letting A shrink to the point E, the function g(E) 2 represents a kind of average (over the disorder and over a small spectral set around E), of the quantity ~ e Z D n21r 2. Namely it is a measure of the extension of the eigenstate corresponding to E. This is the reason for the definition below. Definition 2 The function g will be called the localization length for H. P r o o f of t h e t h e o r e m : (i)-The basic argument we will use here is due to Guarneri [Gua, Bel:Tre]. We will denote by a~(w) the set of eigenvalues of H~ (the point spectrum), whereas II~(w) will denote its spectral projection on the point spectrum and IIc(w) = I - II~(w) will be its spectral projection on the continuous part of its spectrum. Using the definition of the trace in A, we get : nZpT(~,n) ,

(206)

p~(~, ~) = f f ~-I(Ol~(e*"P~)l~)l ~ 9

(207)

AXa(T)~ = / n dP(w) ~ nEZ D

where,

42 We will set pT(n) = fn dP(w)pT(w, n). By definition, we have :

0 < pT(W, n) < 1 ,

(208)

~_, pT(W,n)= 1,

(209)

nc::ZD

whereas the Wiener criterion gives lira pT(W, n) =

T~--*oo

~

I(01IIE(w)In)]2.

(210)

EEapp(w)NA

In particular, if L is a positive integer, and Inloo = maxl_<j ~-* its kinetic energy saturates as a function of time. This numerical result allows us to write = g~(r*) ~ Dr* , (223) where D is the classical diffusion coefficient.

45 There is here a mathematical difficulty. First of all, never was a diffusion coefficient shown to exist rigorously for the standard map. Moreover, averaging it over all possible initial conditions will not give a finite quantity due to the "Pustilnikov acceleration modes" (or "islands of stability"). This means that we should not average over the full torus. It raises the question of which quantum average should be considered. However, recent works [BeVa, Vai, Cher] have shown that for the sawtooth map, a diffusion coefficient does exist. Moreover, a conjecture states that for the standard map there is a "large" set of values of K for which no island of stability occurs, and a diffusion constant does exist. To get the Chirikov-Izrailev-Shepelyansky formula, we argue as follows. Since the eigenstates of the Floquet operator are localized, only a finite number g -- g~ of eigenvalues contribute effectively to the evolution of the initial state 10). Therefore we can approximate this Floquet operator by a g • g matrix F (t). The existence of classical chaos will lead to a strong level repulsion. Hence one can consider that the mean distance between the quasienergies is A E ~ 2~r/g = 0(3') on the torus. For times short enough, the discrete spectral sum arising from the previous approximation can be approximated by an integral, which will be precisely the classical approximation. Hence for t small, g~(t) ~ gcl(t) ~ Dt. This is fine as long as t A E $

I I. ~%'~AI

llllll.~ '~i~

,

I

-..,.~

A

xnL4[o!lais.~l~l,1

&

A

.,.~..::." ff

" t.--, I

.

. ~"~' .... '"i",,.

.i,,l~,

..Ix

/ I " /':1'.."', / :', ": ",, ,. :.:: :.:.-. .~..--

.

I !l'}l ~'~1~1 ~-

I

.

&

~""

,,".': ~.~"

,,~ I ~ - ......... --. I ~.-

9. : . . . .

,'"~:"

,,,i,.,'~l."/

i

"

\-,,.

L~j:~'lll'li ~'" ,~l,d t .'..',:,7,','1~'~ ' I,[~. " ;?"t' l r ~ ; ~-':;, 9 /-:,

:.... " ,

....

.#, ~ii'~

, I,

'",,,,

,

,,,,,I,:,~1

/

..

~

i'I,4 ..,I.X{ ;.

".~ .~,*"-i

,~ ~

.,,.'k',

I I,~11'7~' ' S t

I --,1r

,wl! ,~l:l.d, ~l_P(B), except for a simple pole at P(B). Then, considering the particular case B = 0, and using the above zeta function in the same way as the Riemann zeta function is used to prove the prime number theorem (that is through Wiener-Ikehara Tauberian theorem, see, e.g., [Ko]), Parry and Pollicott [PP1] proved the following 'prime orbit theorem': e hT

7r(T),~

as

T~oe

(1.6)

where h = P(0) is the topological entropy of the flow. In the special case of the geodesic flow on a manifold of negative curvature one recovers an earlier result of Margulis on the distribution of lengths of closed geodesics. However for a precise understanding on how formulae of the type (1.6) can be established in different contexts we refer the reader to the following literature: [Hu], [SI1], [Mar], [B2], [PP1], [Po]. In chapter 2 we will obtain an analogous result for the linear hyperbolic diffeomorphism of the 2-torus. We observe that far beyond the questions formulated above there is the problem of characterizing the detailed structure of the set P and in particular to obtain corrections (i.e. error terms) to the leading asymptotic behaviour (1.6). This is in general a very di~cult problem belonging to an area of active mathematical research where very subtle questions are studied as testing grounds for techniques of analytic number theory. Concerning the second question, one generically finds that the periodic orbits of a hyperbolic system not only do proliferate exponentially but their average distribution in phase space becomes so uniform that any nonperiodic trajectory can be approximated arbitrarily closely, uniformly over arbitrary long time intervals, by some periodic orbit [B2].

67 Consider again an hyperbolic flow restricted to its basic set A. Then for B H61der continuous there exist a unique equilibrium state for B, i.e. an invariant probability measure #B which attains the supremum in the variational principle (1.5): P ( B ) = h~B (r + fA BdpB [Ru2]. In particular, if B = 0 one finds the measure of maximal entropy Po of Ct. Then using some analytic properties of the pressure and again the Wiener-Ikehara Tauberian theorem one proves the following result ([Pal, [PP]): T h e o r e m 1.1. Let Ct : A ~ A be a hyperbolic flow and let B : A ~ IR be a HSlder continuous function with equilibrium state pl?. If f E C(A) then

Z(B,

E

f d#'r' ep@) f B d,,.~

f d#B

--,

as

T---*oo

(1.7)

p(9/) 0 acting on L2(V). Then the correspondence

< Ckf,r

>=/M

f(x'r162162

(1.16)

defines a probability measure #r on M. Thus, the second part of the problem can be formulate exactly as problem C2 above Q2 Find (possibly in some suitably averaged form) the weak-, limit points of the sequence of measures {#r } as k -~ oo.

69 Using Egorov's theorem IT] one can preliminarily show that any limit of the #k's must be invariant under the geodesic flow. Hence the problem reduces to investigate which invariant measure can be recovered as a classical limit of the measures #r This problem is of course well defined regardless of the ergodic properties of the classical flow. However the results proved so far concern mainly the two opposite situations: 1) The geodesic flow is completely integrable or quasi-integrable. In this case 9one is led to study how the measures #r localize on invariant lagrangian submanifolds. Since our main interest in this paper is to understand the possible relations between the classical equidistribution problem stated above and the present one, we shall not discuss this case here, referring the reader to the monograph of Lazutkin [La]. 2) The geodesic flow is ergodic with respect to the Liouville measure (in particular is a hyperbolic flow). In this case we have the following result of Schnirelman [Sehn], Zelditch [Z] and Colin de Verdiere [CdV]: T h e o r e m 1.2. Assume that the geodesic flow on M is ergodic, then there exist a subsequence {)~k~} of density one such that lira [ ~c~

fd#r

J M

----[ fd#

(1.17)

JM

It is worth noticing that a preliminary step to prove (1.17) is a result on the average analogous to (1.8):

E

fd#r

~

fd#

as

~~ ~

(1.18)

which follows from classical symbolic calculus and the Karamata Tauberian theorem. Using the ergodicity of the classical flow and the Egorov theorem one is then able to extract a subsequence of density one converging to the r.h.s (see, e.g., [CdV]). Thus, the above theorem asserts that almost all of the measures #r become equidistributed in the classical limit with respect to the Liouville measure. One interesting problem is then to find the conditions under which the Liouville measure is the only weak* limit point ('quantum unique ergodicity' in the language of [Sa]) so that one may avoid to take the limit on subsequences. In the next chapter we shall show that this happens in the case for the linear hyperbolic diffeomorphism of the 2-torus and its quantized version. For other results and/or conjectures in this direction see [Z1], [Z2], [RuSa]. The two problems (C) and (Q) stated above will be referred to in sequel as the classical and quantum equidistribution problem, respectively. Besides their own interest as individual mathematical problems one is also interested in their possible connections. A first known fact is the following: consider the particular situation of a compact manifold V of dimension d and constant negative sectional curvature, say -1. Then, every conjugacy class of the fundamental group of V contains exactly one closed geodesic so that one can arrange the set of all closed geodesics as a countable family { ~ - k } ~ with non-decreasing lengths {gk}k~v. Moreover, there is a

70 one-to-one correspondence between the closed geodesics 7-k of V and the primitive periodic orbits 7k of the associated geodesic flow Ct : M ~ M. In particular 7j will have least period ~k, i.e. P(3'k) = fk. Now, a remarkable relation between the two sequence of real numbers {g-k} (often referred to as tile length spectrum) and {~k} is provided by the Selber9 trace formula [Hell: E

k

f(x/~-

ak) - Vol(V)

-d-~T

471"

sinh(T/2) + ~

k

~

n=l

sinh

k/2)

where f : /R ~ ~ is any C ~ function of compact support and f is its Fourier transform. A great deal of mathematical work has been made around this formula and several applications and generalisations have been proposed (see [He2], [He3], [BalVor] and references therein). On the other side, very little is known about the possible connections between the two sequences of measures {#.~ } and {PCk } defined respectively in (1.2) and (1.16). One remark is the following. It is known that for a geodesic flow on a negatively curved manifold, the Liouville measure coincides with the measure of maximal entropy when the curvature is constant (see, e.g., [K]). Thus, one may argue that in this case there should exist some direct relation between (1.8) and T h e o r e m 1.2 (in the next chapter we will examine a discrete time dynamical system where these two measures coincides as well and the connection can be established explicitly). On the other hand, for the more general case of variable curvature the problem seems much more involved.

2. A n e x a m p l e :

the hyperbolic linear automorphism

of the torus.

We now consider the discrete time dynamical system T : M ~ M where M is the 2 - t o m s T 2 = 1R2/Z 2 (points on T 2 are denoted by x = (p, q) C [0, 1] x [0, 1]) and T is the hyperbolic automorphism of T 2 generated by the matrix

such that (a, b, c, d) c Z, ad - bc = 1 and la + dl > 2. The Lebesgue measure # is invariant because det A = 1. Moreover, the condition ITrAI > 2 makes this dynamical system an Anosov one and hence, in particular, ergodic and mixing with respect to #. Now, an orthonormal basis in L2(T 2, d#) is given by the set =

e z 2}

(2.2)

and A acts on points x = (q,p) and on suitably smooth functions f(x) on M respectively as: Ax = ( (aq + bp)(mod 1), (cq + dp)(mod 1))

f(Ax) = E nEZ 2

fnT(Atn)

(2.3)

7"1

where A t is the transposed matrix of A (notice that we have used the same symbol A to denote the matrix A and the map T; this will be repeatedly done in what follows without fear of confusion). Consider then in L2(T 2, dp) the unitary Koopman operatorblA defined by

(l.r

= f(Ax)

(2.4)

and recall (see, e.g., [A.A]) that T is ergodic iff 1 is a simple eigenvalue of b/A. Otherwise stated, if there is h E L2(T 2, dp) such that blAh = h then h is constant p-almost everywhere. Moreover, it is mixing iff, for any pair f , g E L2(T 2, dp), lira

k--* oo

=
< 1,g >

(2.5)

This property makes a(UA) continuous on the unit circle, but for the eigenvalue 1. Now, if A is a continuous map of a compact metric space X then htop(A), the topological entropy, satisfies the restricted variational principle:

htop(g) =

sup h,(A) t~E2~A(X)

(2.6)

where A J A ( X ) the set of the probability measures on the Borel a-algebra of X which are A-invariant and h , ( A ) is the measure theoretic entropy (see [M], [AY]). Moreover, A is intrinsically ergodic if there exist a unique p E .A4A(X) such that htop(A) = ht,(A ). In this case p is called the intrinsic measure of A, or maximal entropy measure of A. Thus, for a linear automorphism of the torus the Haar measure, i.e. the Lebesgue one, is actually an intrinsic measure. We have the T h e o r e m 2.1 (Sinai) Let A : T d --* T d be a linear automorphism with eigenvalues )~1,... ,~d. Let # be the Haar m e a s u r e o f T d. Then

htop(A) = h,(A) = E

logjam]

(2.7)

I~l_>l

Proof. See e.g. [A.A]. Notice that this result can be easily obtained also from Pesin's formula ([M], p.265). Consider again the general situation of a continuous map A of a compact metric space X. We say that x E X is a periodic point of A, of period n, if it is a fixed point of A '~, i.e. Anx = x. We denote by Fix,~ the set of such points. It is easy to see that for linear automorphism of the torus the set of periodic points of A is dense in T 2, because it coincides with the subset of T 2 formed by all points having rational coordinates (see below). More generally we have the T h e o r e m 2.2 ( B o w e n - S i n a i ) Every topologically mixing hyperbolic homeomorphism A : X ~ X is intrinsically ergodic. If p denotes its intrinsic measure then for every continuous map f : X --~ ~ :

fdp =lim

#Fix,~

E xEFixn

f(x)

(2.8)

72

and A has topological entropy htop = lira 1 log~Fix,~

(2.9)

n--4 oo n

Proof. See [M], p.254. For the linear automorphism of the torus we have the additional result: P r o p o s i t i o n 2.1. Let )~ be the eigenvalue of A whose modulus is larger than

1. Then # F i x , -= ~ + A-'~ - 2

(2.10)

Proof. We sketch the idea of the proof referring to [If for more details. Let k be the trace of A, then Ikl > 2. Consider the numbers ),'u,~ = where D = V / ~ -

_

~-(2.11)

2D

1. These numbers satisfy the recursion

u0=0, ul=l

and

un=ku,~-l-u,~-2,

forn>l

and it can be easily checked by induction that

An=(aus-un-1 \

bus

CUrt

)

(2.12)

dun - us-1

By virtue of this formula one can easily realize that for any n > 0 the set Fix~ constitutes a regular lattice on the torus and by a simple geometrical argument ~r can be computed as the inverse of the area of an elementary cell in the above lattice (see [If), thus giving (2.10). Q.E.D. R e m a r k . Notice that from (2.9) and (2.10) one immediately recovers (2.7) for this particular case: h , ( A ) = htop(A) = log A. We now deduce some further consequences of (2.9) and (2.10). Denote again by P the set of all primitive periodic orbits (prime cycles) of A and by ~rP(n) and r ( x ) the number of them whose period is n and less or equal to x respectively, i.e. ~'(n) = ~{V e PIP(V) = n)

r(x) = #{V e PIP(V) -< x}

(2.13)

P r o p o s i t i o n 2.2.

(2.1a)

n

and

,~ ~

Xx x

where f ( t ) --, g(t) means that f(t)/g(t) ~ 1 when t ~ ~ .

(2.15)

73

Proof. We make use of the zeta function Oo

~(z)=exPE

n

z-

n=l

'D,

E

1

(2.16)

xCFixn

The strategy is to gain insight on the distribution of closed orbits out of the mewmorphy domain of ((z). In our case, by Proposition 2.1, ((z) has the simple form (1 - z ) 2 = (1 - ~ z ) ( 1 - z / ~ )

r

(2.17)

so that

~'(z__~) = _ ~ ~(z)

+ g(z)

(2.1s)

1 - Az

where g(z) is analytic in {z[Iz I < e~/A} for some e > 0. The rest of the proof proceeds exactly in the same way as in the proof for subshifts of finite type ([P.P], p.100). Q.E.D. R e m a r k . Proposition 2.2 is the analogue of the prime orbit theorem proved in the context of Axiom A flows (see [P.P]). We now prove that closed orbits exhibit a regularity in a spatial sense. In particular, we show that they are equidistributed on the average with respect to the Lebesgue measure p. Let #.y be the measure defined by p('y)-i

1 #'~ = p(3') E

8Ak(x)

x C 3'

(2.19)

k=0

and set f.~ f = fr2 fd#~. Then we have the P r o p o s i t i o n 2.3. For every continuous map f : T 2 --* ]I:l :

7r'(n)

2 fd#

f----~ p

as

n--*co

=n

Proof. We first write the number of fixed points of period n in the form # F i x . = E l. 7r'(l)

(2.21)

lIn

where lIn means that l divides n. Form Proposition 2.1 we then have

El.

~'(t) = ~," + ,~-" - 2

(2.22)

l[n

On the other hand Theorem 2.2 yields ,~-,~lim

E,l~t Ep(~)=~L f ~tl,, I. ~'(1)

f

= JT: f d #

(2.23)

74 Hence,

Let m = m a x { / c Z I lln, I < n}. If n is prime then m = 1, otherwise m > 1. One finds immediately that

p

=n

n

p(../)=/

f ~

(1 - C,~,m)

tin, l < m

3'

p(.y)=/

7

i~om which we obtain

fdp,

as

n ---* cx)

(2.25)

2

p('r) =n

where C,~,m is of order at most A"~-'~. To conclude the proof it suffices to divide (2.25) by ~-'(n) and apply Proposition 2.2. Q.E.D. Proposition 2.3 yields a result of uniform distribution on the average. We now study the behaviour of measures #~ supported on single closed orbits and we prove that they converge in measure to #. P r o p o s i t i o n 2.4. For any e > 0 and for any continuous function f lim ( # { ' ~ ] p ( ~ ' ) : n '

If'Yf-fr2fdP]>e})=O

(2.26)

~'(n)

~ - ~

Proof. For the sake of simplicity we shall use the notation fr2 f d p = f. For any k C Z+ set k--1

m~f(x) = ~ ~ I(A'x)

(2.27)

l=O

Now, given ~ > 0, introduce the sets S~,,~={Tlp(7)=n,

ff-f J-y

0, k E Z+

(2.29)

The ergodicity of A implies that, for p-almost every x E 1,2, lim mkf(x) = f

(2.30)

75 Hence, for any continuous f , by the Lebesgue dominated convergence theorem we can find a k0 > 0 such that r Jmkof(X) - lid# 0 such that for any n > no r2 Imk~

- fld#~ _< ~

(2.32)

Hence by the Chebychev inequality we obtain #('YIP(~') = n, f'r Imkof - fl >- v/~} ~'(n)

< v/~

(2.33)

and therefore, by the second of (2.28):

~R~'n'k~ ( V/~, ~'(n)

n > no

(2.34)

-

Hence, from (2.29), we find #S~,,~ < v~,

~'(n)

n > no

(2.35)

and the assertion follows by taking (5 ~ O and consequently no(5) ~ c~. Q.E.D. 2.1 K o o p m a n o p e r a t o r a n d p e r i o d i c orbits on invariant l a t t i c e s . Consider any point o n / , 2 having coordinates (r/N, rl/N), with r,r~,N C ~W and 0 _< r, r ~ < N. There are exactly N 2 points of this type and they belong to the N • N subgroup of T 2 given by:

i N = ((q,P) C T2]Nq, Np E Z )

(2.36)

It is immediate to realize that LN is invariant under the action of A, so that any point in LN is periodic with period _< N 2, the origin being the only fixed point of A. Of course, any point x E Fixn belongs to a periodic orbit whose period divides n. This means that LN splits into periodic orbits (which in general may have different periods) of A. Let #N be the normalized atomic measure supported on LN. We now characterize the spectrum of UA when acting on L2(T 2, d#N). Let MN be the number of distinct periodic orbits of A t which live on LN \ (0, 0). This number is the same as that corresponding to A (see [I]). Let ~ C LN be any one of such orbits with period

76

P(7) and x = (rl/N, r2/N) e 7. Then, associated to each orbit 7 there are P(7) linearly independent vectors in ~TN2 given by: fl(k)=

v(~)-i 2ri ~ A~-Sexp-~-

r=(rl,r2), keZ~

(2.37)

s=0

where Al = e -~iz/p('~) and l = 0 , . . . ,p(~f) - 1, and they satisfy

5tAfl(k) = )~lfl(k)

(2.38)

Thus, there are N 2 - 1 eigenvectors of/dA of the form (2.37) which, together with the constant function 1, provide a canonical basis of L2(T2,d#N). Among them, there are exactly MN non-constant functions which are invariant. This is in account of the fact the dynamical system (T 2, A, #N) is not ergodic: the invariant measure #N obviously admits a decomposition into invariant ergodic measures of the type (2.19). The case of N prime. We now specialize now to the lattices L N with N prime. In this case a very precise characterization of the structure of the periodic orbits is possible (for which we refer to [D.G.I]) and moreover strong results on their equidistribution properties can be proved. First, it has been shown by Percival and Vivaldi [P.V] that all the periodic orbits living in LN with N prime have the same period, i.e. p(~/) = p(N) for any "f C L N \ {0, 0}. The relation among p(N), M N and N is then:

p ( Y ) . MN = N 2 - 1

(2.39)

Thus, from the above argument we then have the following result on the spectrum of the Koopman operator: P r o p o s i t i o n 2.5. Let N be a prime number and let p(N) be the period o/the cycles living on LN \ {0, 0}. Then a(LtA) is given by the eigenvalues

At -~ e 27ril/p(g)

l = O, 1 , . . . , p ( N ) - 1

(2.40)

To each )~1 is associated an eigenspace Ez of non constant functions to which all the periodic orbits o / L N \ {0, 0} contribute. Accordingly, the following decomposition holds: p(N)-I

L'(T2,dpN)--1G(

E,)

(2.41)

/=0

where 1 is the one-dimensional subspace spanned by the function 1 and dim(El) = MN V l = O , . . . , p ( N ) - 1. We now turn to the equidistribution results. We first prove a sharpening of Proposition 2.4:

77 P r o p o s i t i o n 2.6. For any f E C(T 2) and any e > 0 there is No such that, if N >_ No is prime:

# { T C L N \ { O , O } , If-r f - ] i - - - V q } #{~17 C LN \ {0,0}}

_< v~

(2.42)

Proof. It is easy to realize that, up to the additive correction vanishing as N -2 for N ~ cr (arising from the fixed point at the origin), the following identity holds true: 1 ~ MN

j=l

J

f=~

fd#N

(2.43)

Then the assertion follows by the same argument as in Proposition 2.4 where now Sr

f~f--f

R .... k :={717

C LN \

{0, 0}, j~

< V~}

I m k f - f l = s=O 1 p(N)--I . p(N) E ee'J~< . . . . ~+#'~

1 =

s=o

1 rap(N) E

1 e~('~r

--

p(N)

v(N)-I E ee'Frt(~;~+#'x;*) ~=o

m--1

E

E

2~,a,+b,-l~ XJ(~)e"~-' r ~

(2.47) where a ---- ~ < n,v >, b -- ~3 < n,w >E ZN and the relation p(N) -= ( N - 1)/m has been used. The functions Xj : j = 0 , . . . , m - 1 are the m distinct multiplicative characters of order m of ZN (see the Appendix). We can now apply the estimate (A.18) and obtain, for any n ~ ( 0 , 0 ) ( m o d N ) :

f~

e2~ri < K CLN

(2.48)

--

where K > 0 is independent of n and of the particular orbit 7 C LN and is uniformly bounded in N because of the boundedness of m. On the other hand, a trivial computation shows that, if n -- (0, O)(modN) then jf

CLN

e 2~ we can write

j(N)

2

n#(0,0) (mod N)

j(N)

k#(0,0)

Now, if f 9 A(T 2) the second term of the r.h.s, vanishes as N ~ oo and the first term also vanishes by the uniform estimate (2.48) on the Kloosterman sums. If moreover f E C~176 the second term vanishes at least as N -1 as N -~ oc and therefore we can conclude that there exists C > 0 such that if N is large enough: f -

f d" < _

llfllA(r )

C

(2.51)

Finally, the case N inert can be treated in a similar way by using the techniques of [PV] and the generalized Kloosterman sums over arbitrary finite fields (see [Kal]). Q.E.D.

2.2. The quantum equidistribution problem. Wigner function and classical limit. We now consider the quantum dynamical system (7-t,..4, VA) obtained by the canonical quantization of the former one. We now limit ourselves to recall the main ingredients referring the reader to [DGI] for the detailed construction. 1) The Hilbert space is :H = ~ N = L2(SI,vN) where h -~ 1/N and VN is the normalized atomic measure given by (C2~'iq E $1): N-1

VN(q) = ~i ~

~(q-- ~l)

(2.52)

/=0

Hence T / h a s dimension N. 2) The algebra .4 is the *-algebra of the observables on 7-/generated by the q uantization of the classical functions on the torus in the following way: let T(n), n E Z 2 be the canonical quantization of the basic observables T(n). This is based on the classification of the irreducible representations of the discrete Heisenberg group/-/1 (Z), in complete analogy with the well known construction of the SchrSdinger representation out of the Heisenberg group H,~(/Rn). The quantization of any f C A(T 2) is then given by f=

~

AT(n)

(2.53)

n~Z 2

3) The unitary bijection VA is the quantum propagator, i.e. the quantization of the action of t h e symplectomorphism A on the classical observables. This means that if f ~ f as above then f(Ax) ~ VAfVA 1. Therefore the quantum discrete dynamics is defined as

f ~ V U V Z k,

k9Z

(2.54)

80 Moreover, if N is a prime number, one easily finds that V~(N) = Id so that the quantum dynamics is periodic with period given by the classical periodic orbits living on the lattice LN. 4) We denote by e 2~iA(N) and r n = 0,... N - 1, the (repeated) eigenvalues of VA and the corresponding (orthonormal) eigenvectors, respectively. The quantum equidistribution problem can now be stated as follows: given any pair of eigenvectors r wz~(g)belonging to the same eigenspace, define a distribution d~N(r r on the phase space T 2 by

r IdaN(r

(2.55)

Then, we want to know what are the weak* limit points of such distributions when N --* c~. The main result of [DGI] is the following T h e o r e m 2.4. Let F be an increasing sequence of primes as in Theorem 2.3. Then, for any sequence of eigenvectors {r and f E A(T 2) we have

lira f fdf~N(r (N)) = Iv: f d # N ~JT:

(2.56)

and, if f is smooth enough, say f E C~(T2), the limit is attained with speed given by: Cllfh (2.57)

L/r

/r fd.l

= ~

nEZ~ = ~ fnfT e2ri .g and ~ _> x ~ and one of these inequalities is strict, then d


0. Also, this condition m a y be expressed in terms of f - l , namely f - 1 is a C 1 diffeomorphism and 0x/cgy ~ < 0, where x' and y' are taken as independent variables. If, in addition to the conditions we have already imposed on f, we impose (7.2)

Oy'/Oy < 8 and

Oy/cgy' > - 8 ,

Oz'/Oy

Ox/0y'

-

-

then the associated variational principle satisfies (//6o). (In the first of these inequalities, we take x and y as independent variables and x' and y' as dependent variables; in the second, we take x' and y~ as independent variables and x and y as dependent variables).

Indeed, in this case h is C 2, and

(7.3)

a~ h(x, x') - Ox,/Oy ay'/ay

and

c911h -

Oy/ay' Ox/Oy' "

These equations follow fl'om (5.2) by a calculus exercise. In doing the exercise, it is important to keep in mind that x and x' are the independent variables on the left side of each of these equations, whereas x and y are the independent variables on the right side of the first of these equations and x' and y' are the independent variables on the right side of the second of these equations. From (7.3), it is clear that (7.4)

all h(x,x') l - k + 1. []

Corollary 10.4. There exists an h-minimal configuration which is periodic of type (p, ,~).

117

w

The R o t a t i o n N u m b e r .

Let x and y be configurations. We define the relations >~, >~, ~ y, z Yi for all i > i0, etc.). As before, we suppose that h satisfies (H0) - (//4). We have I. if x ,and y are h-minimal, then x >~ y, x = y or x ~ y , x = y orx ~, >,,, O}

and similarly A,~(x) and B,~(x) (replace >~ and ,~ and p~(x) (resp. p~(x)) implies p/q E B~(x) (resp. B,,(z)). Theorem

11.1. pc~(x) = p~o(x), if x is h-minimal.

P r o o f . Suppose the contrary, e.g. p~(x) < p,o(x). Let p/q be a rational number, expressed in lowest terms, with q > 0, such that p~(x) < p/q < p,o(x). By Corollary 10.4, there exists an h-minimal configuration y which is periodic of type (p, q). From the inequalities p~(x) < p/q < p~(x), it follows that x >~ y and x >~ y, and this holds for any translate of y. However, if r is sufficiently large, then xi > (T~,I y)i does not hold fi:~r all i. Hence, the Aubry graphs of x and y cross at least twice. Since x and y are h-minimal, this contradicts Aubry's Crossing Lemma. This contradiction shows that we do not have p~(x) < p~(x). Similarly, we do not have p~(x) < p~(x). []

118

We set p(x) = p~(x) = p,~(x). Theorem

11.2. p(x) E R , i f x is h-minimM.

P r o o f . Otherwise, we would have p(x) = q-c~. Suppose, for example, t h a t p(x) = +o0. If y is a periodic configuration, y >~ x and y < ~ x. By Corollary 10.4, there exists a periodic m i n i m a l configuration y of t y p e (p, 1), for any integer p. Thus, Y i + l = Yi -[- P. By choosing p very large, and replacing y by a t r a n s l a t e , if necessary, we m a y suppose t h a t Y0 < x0 and Yl > xa. The relations y >~ x, y < ~ x, y0 < x0 a n d yl > xi imply t h a t the A u b r y graphs of x and y cross at least three times. This contradicts the A u b r y ' s Crossing Lemma. This contradiction shows t h a t p(x) 7s +oo. Similarly, it m a y be shown t h a t p(x) # - o ~ . []

We will call p(x) the rotation number of x. We let AH --- M h C R ~ denore the set of all h - m i n i m a l configurations. We provide R ~ with the p r o d u c t topology and AH with the induced topology. Theorem

11.3. Ad is dosed in R ~ and p : Ad --+ R is continuous.

P r o o f . T h e first assertion folk)ws i m m e d i a t e l y from the definition of .h/[ and the a s s u m p t i o n (H0) t h a t h is continuous. To prove the second, we will use: Lemma.

Let x = ( . . . , x i , . . . ) E A/[. Let p,q E Z, q > 1. I f xi+,l

+ p, for some i, ti en p(x) >_ (p - a)/q.

P r o o f . To prove the f r s t assertion, we use the existence of a periodic minimM configuration y of t y p e (p + 1, q). By replacing y with a suitable t r a n s l a t e , if necessary, we m a y suppose t h a t yi < xi and yi+,~ > xi + p >_ xi+q. By the A u b r y ' s Crossing L e m m a V < - x and y >~ x, so p(x) < p(y) = ( p + l ) / ( t . T h e second assertion m a y be proved similarly. []

Now the continuity of p follows easily, since x,+,1 is obviously a continuous function of x. []

We let pri : R ~176 ---+R denote the projection on the i th factor. T h e o r e m 11.4. Let I, ~ be compact subsets of R . F o r m~y i E Z, p - l ( ~ ) A p r T l ( I ) is a compact subset c l a d . P r o o f . By the Tychonoff p r o d u c t theorem and the fact t h a t M is closed in R c~, it is enough to show t h a t for ea.ch j E Z, p r j ( p - l ( ~ ) (3 pi'~-l([)) is a bounded subset of l t . In the case t h a t j > i an u p p e r b o u n d for this set m a y be found as follows. Let p E Z be an u p p e r b o u n d for ~ and let y be a periodic m i n i m a l configuration of t y p e (p, 1) such t h a t yi is an u p p e r b o u n d for I. T h e n yj is an u p p e r b o u n d for p r j ( p - l ( f t ) A pr~-a(I)). For if x E p - l ( f ~ ) N pr~-i(I), then y >,o x since p(y) = p > p(x) E f~. Moreover, yi > :r,i, so yj > x j by the A u b r y ' s Crossing

Lemma. T h e other cases m a y be t r e a t e d siinila.rly.

[]

119

T h e o r e m 11.5. Let I be a closed interva.1 of unit length in R. p : pr~-I (I) N .h4 ~ R is suriective.

For any i E Z,

P r o o f . For any rational number p/q, there exists a. minimal configuration x = (...,xi,...) of type (p,q) by Corollary 10.4. Since I has unit length, we m a y suppose that xi E I. Since p(x) -- p/q, we obtain that p/q E p(pr~-~(I) n ~ ) . Thus p(pr~-X(I) n M/t) contains q . On the other hand, by Theorem 11.4, if ~ C R is compact then

# ( p r T ' ( I ) n M ) n ~ = p(pr;-'(I) n p - ' ( ~ ) ) is compact. It follows that p(pr~-~(I) rh M ) is closed. Since this set is closed and conta.ins Q, it contains ~,.11of R. []

120

w

Irrational Rotation

Number.

Let w be an irra.tionM r o t a t i o n number. Let It be a variational principle which satisfies the B a n g e r t conditions ( H 0 ) - ( / / 4 ) . By T h e o r e m 11.5, there exist h-minimM configurations x of r o t a t i o n n u m b e r w. We let AA~ = A41~,~ denote the set of hm i n i m a l configurations of r o t a t i o n n u m b e r ~v. By T h e o r e m 11.3, A4~ is closed in R ~176 In this section, we will descrihe several results concerning the s t r u c t u r e of A.4~, due to A u b r y and Le I)a.eron [Au-LeD] and later in a more general context to Bangert [Ba]. T h e first result concerns the order relation. Given two configurations x a n d y, we will say t h a t x < y (or y > x) if :ri < yi, for every i. If x and y are h-minimM configurations, then x < y if and only if x p~(.z)} (and simila.rly for w in place of a). F r o m the definition of A~(:r,) etc., it follows t h a t y ~ x ) if and only if y < ~ x (resp. y > ~ x). T h e n the result follows f i o m Trichotomy I. T h e second step is the definition of what A u b r y called the h u l l f u n c t i o n of a m i n i m a l configuration x of irrational r o t a t i o n n u m b e r w. Let r

= sup{(Tp,q x)0 [p - qw 0 .

This condition express the property of aJ being badly approximated by rationals. If w does not satisfy this condition, it is said to be a Liouville number. We recall that

134

the sets of Diophantine and Liouville numbers are dense in R, but the latter is of Hausdorff dimension and Lebesgue nlea.sure zero, as it is not difficult to show. 1 6 . 1 . IS-Z] Let fo E yl be an exact area-preserving monotone twist diffeomorphism of the annulus S t x R. Let 7o he a rotational invarim~t curve for f0 of rotation n u m b e r aJ E R \ Q, satisfying the Diopha.ntine condition (16.1). Suppose that fo is of class C ~ and Jo13`o is Ct+l-conjugate to the rotation R~o : S 1 ~ S 1, l > 2r + 2 and l - 2 7 - 1 , l - r - 1 are not integers. Then Ju 13`ois C~176 to the rotation R~. In addition, if fo is analytic, the co1~jugacy is anadytic. Furthermore, there exists a neighborhood ld = ld(C, r, fu, 3`0) of fo in the C I topology such that every exact area-preserving monotone twist C ~ diffeomorphism of the annulus f E ld admits a r o t a t i o n d invariant curve 7 of rotation number w. Theorem

In the analytic case, H.Rfissmann has shown in a series of papers, concluded in [Rs], that the Diophantine condition (16.1) on the rotation number co E R \ Q can be replaced in Theorem 16.1 by a weaker condition. Let (Pn/%),,eN the sequence of convergents of the contimled fraction expansion of c0. The condition (16.1) can be expressed in terms of the continued fraction expansion as (16.1 ~)

Logq,,+l _< CLogq,, , n E N .

Riissmann proves that the condition (16.2)

Z hEN

Logq,,+l

< oo,

qn

previously introduced by A.D. Btjuno in connection with problems related to classical perturbation theory in Hamiltonian mechanics and known as the Brjuno conditiou, it is sufficient for the stability result of invariant curves contained in Theorem 16.1: if 3'0 is analytically conjugate to a. rotation and its rotation number a; E R \ Q satisfies the Brjuno condition (16.2), then there exists a n e i g h b o r h o o d / . / o f the map f0 in the analytic topology, such that any exact area-preserving monotone twist diffeomorphism .f E/d adnfits a. rotational invariant curve "7 of rotation number w. Clearly the Diophantine condition (16.1) implies the Brjuno condition (16.2), thus Rfissma.ml theorem is in fact stronger than Theorem 16.1 in the analytic case. We recall the basic facts known a.bout the relations between the smoothness of the invariant cmve 3' as a curve in S 1 x R and the smoothness of the conjugacy of the diffeomorphism fl3' to the corresponding rigid rotation of the circle. It is a classical eounterexample by Arnold that there exist analytic diffeomorphisms of the circle, with irrational rotation mm~ber w, whose eonjugacy to a rigid rotation (which exists and is a homeomorphism by a classic theorem due to Denjoy [De]) is not absolutely continuous. Analogous examples have been constructed in the smooth case. Since every diffeomorphism of the circle can be embedded as rotational invariant curve of an exact area-preserving monotone twist diffeomorphism of the annulus with the same degree of smoothness, the mentioned examples give examples of analytic (resp. smooth) monotone twist maps f having an analytic (resp. smooth) rotational invariant curve 3' such that f i t is not absolutely continuously conjugate to a rigid rotation (although topologically it is, by Denjoy theorem). However, in these

135

examples the rotation number w, although irrational, is very well approximated by rationals. On the other hand, in the case of rotation numbers satisfying a Diophantine condition, Herman's theorem holds. We state the improved version due to J.C. Yoccoz [Yo]: T h e o r e m 16.2. Let r : S 1 ~ S 1 he a difI'eonmrphism ot" the circle of class k E N and k >_ 3. Suppose that the rotation nmnber co E R \ Q satisfies Diophantine condition (16.1). Then, it'k > 2r - 1, q5 is Ck-~-*-conjugate to rigid rotation R~, tbr aa~y e > 0. In addition, ii" ~) is C ~ the COl~jugacy is C ~ is r e d analytic the co11.iugacy is also analytic.

C k, the the if q~

Theorem 16.2 implies that, if 3'0 is a rotational inva.riant curve of rotation number' a3 E R \ Q, satisfying the Diophantine condition (16.1), then, if 3'0 is sufficiently smooth r ~ c~rve in S 1 x R then Theorem 16.1 applies. In the analytic case, the Diophantille condition in Theorem 16.2 can be replaced by weaker conditions. The picture of the situation, due to J.-C. Yoccoz, is described in the survey paper by R.Perez-Marco [P-M]. We just mention that there exists a condition 7"[ on the rotation number a~, weaker than the Diophantine condition (16.1) but stronger than the Brjuno condition (16.2), such that, if a; satisfies ~ , then the analytic diffeomorphism of the circle ~ is a nalyticMly conjugate to its linear part R~. On the other hand, if ~o does not satisfies the Brjuno condition, then there are examples of analytic diffeomorphisms ~ which are not analytically conjugate to R~. However, there is a significant gap between the condition 7-[ and the Brjuno condition. Thus, the situation is, at the level of optimal conditions, more delicate than in the smooth case. One of the most interesting applications of K.A.M. theorem for invariant curves of exact area-preserving monotone twist diffe,omorphisms of the a,nmllus is the proof of the .~tahilit!! of elliptic periodic points of a.rea.-preserving diff~omorphislns of surfaces. The problem can be reduced to the case of a fixed point by considering the appropriate iteration of the map. C o r o l l a r y 16.3. Suppose that P is an elliptic fixed point of a C ~ area-preservi12g mapping f of an open subset of the plane into the l)lane. Suppose filrthermore that the Birkhoff invariants (s'ee w of the raN) f at P are not ali equaJ to zero. Then P is Lyapunoff stable. P r o o f . According to the Birkhoff normal form theorem, stated in w there exists an area-preserving change of coordinates in a neighborhood of P such that the mapping f takes the form f ( ( ) = ( exp 27ri(flo + flip '~ + ... + flNp 2N) + O(p k) and k = 2 N + 2 or 2 N + 3 . Here ( = (+it! denotes the complex coordinate associated to the real coordinates (, q, and p = ((2 + q2)1/2. The rearl numbers ill, ...,fiN, ... are called the Birkhoff invariants of f at P. If an eigenvalue A of d f ( P ) is not a root

136

of unity, we m a y take N = oe. If A is a primitive qth root of unity, t h e n we m a y take k = q and N = [q 2/2]. Thus f is the sum of the n o r m a l form -

N ( ( ) = ( exp 2~ri(fl,~ + / : ] l p 2 -~ ... 71- lIND2N) a.nd a r e m i n d e r t e r m which is no bigger t h a n O(pk), where k = ec if an eigenva/ue A of elf(P) is not a. root of unity and k = q if A is a primitive qth root of unity. If at least one of the Birkhoff invariants is not zero, then the n o r m a l form N(~) and f are exact area-preserving monotone twist m a p p i n g s in a sufficiently small p u n c t u r e d n e i g h b o r h o o d of P , as explained in w F u r t h e r m o r e , this n e i g h b o r h o o d is foliated by invariant curves of the n o r m a l form N(~), since rio, ill,.., are real numbers, and the set of r o t a t i o n numbers of invariant curves of the n o r m a l form is an open interval I C R , since at least one of tile Birkhoff invariants is non-zero. Therefore, by K.A.M. theorem ( T h e o r e m 16.1), if an eigenva.lue A of dr(P) is not a root of unity or a.t least it is not a. qth root of unity for small q, there exists a sufficiently small p u n c t u r e d neighl)orhood of P , which contains a positive m e a s u r e set of inva.riant curves of the m a p f . In fa.ct, f can be considered, in a small n e i g h b o r h o o d of P , as a. sma.ll p e r t u r b a t i o n of its n o r m a l form a.nd by K.A.M. t h e o r e m invariant curves having D i o p h a n t i n e r o t a t i o n numbers (i.e. satisfying condition (16.1)) persist for small p e r t u r b a t i o n s . Since in dimension 2 invaxiant curves are separating, the existence of invariant curves surrounding P implies the stability of P . []

We conclude the section by stating the m a i n application, due to L a z u t k i n [La], of K.A.M. theory to plane convex billiards. Let R be an open convex b o u n d e d region in the plane whose b o u n d a r y is of class C 2. The billiard ball p r o b l e m in R has been briefly described in w where it is explained the basic fact, a l r e a d y known to Birkhoff, t h a t its dynamics largely reduces to the d y n a m i c s of an associated exact area-preserving m o n o t o n e twist mapping. A caustic for the billiards ball p r o b l e m in R is a closed curve C in R such t h a t any t r a j e c t o r y which s t a r t s being t a n g e n t to C stays tangent to C after bouncing onto OR, forever in tile p a s t and in the future. A caustic corresponds to a r o t a t i o n a l invariant curve for tile exact area-preserving m o n o t o n e twist m a p p i n g associated to the billiard ball problem. 16.4. [La] If R is strictb~ convex (i.e. the curvature of OR never vanishes) a.nd OR is suI~ciently difl'erentia.ble, then there exist caustics tbr the billiard bM1 problem in R (arl)itra.rily close to OR). Theorem

In the next section we will give a converse, due to the first a u t h o r [Ma2], to T h e o r e m 16.4. T h e a r g u m e n t will be based on the Birkhoff invariant curve t h e o r e m described in w

137 w

Birkhoff Invariant C u r v e T h e o r e m . A p p l i c a t i o n s

In this section we discuss converse K.A.M. results, due to the first a u t h o r [Ma2][Ma4], which a.re based on the Birkhoff invariant curve theorem described ill w Then we prove the ba.sic varia.tional p r o p e r t y of rotationa.1 invariant curves, i.e. any orbit on a rotationa.1 invariant curve fi)r a C 1 exa.ct area-preserving m o n o t o n e twist diffeomorphisnl of the annulus is a minimal orbit in the sense established at the beginning of w Tiffs p r o p e r t y ha.s m a n y of i m p o r t a n t consequences, which will be discussed in sections w and w Glancing BiIliard,~.

We present a converse [Ma.2] to the Lazutkin theorem of the last section (Theorem 16.4). Let R be a convex b o u n d e d plane open region, whose b o u n d a r y OR is C ~. In R we consider tile billiard ba.ll problem, already described in w We will say t h a t a t r a j e c t o r y is e-gla.ncing if for at least one bounce the angle of reflection (with either the positive or negative tangent of 0_R at the point of reflection) is < e. If e < ~r/2, we can distinguish between a positively e-gla.ncing t r a j e c t o r y and a negatively e-glancing t r a j e c t o r y according to whether the direction of reflection is close to the positive resp. the negative tangent to 0R.

T h e o r e m 17.1. [Ma2] I f the curvature of OR is zero at some point, then the b i l l i ~ d bMl problem in R has no caustics. As a consequence, tbr any e > O, there exist tra.iectories which are both positively and negatively e-glancing.

P r o o f . The proof depends on the formulation of the billiard ball p r o b l e m in terms of a.rea-preserving diffeonmrphisnl of the anmdus. A t r a j e c t o r y is positively and negatively e-glancing if and only if the corresponding orbit for the associated exact area-preserving m o n o t o n e twist m a p p i n g of the annulus f : OR x (O,7r) --~ OR x (0, 7r) visits e-neighourhoods of both b o u n d a r i e s of the annulus, OR x (0, e) and OR x (Tr- e, 0). It is a classica.1 consequence of Birkhoff invariant curve theorem, a l r e a d y o b t a i n e d by Birkhoff himself, t h a t the following holds:

L e m m a 17.2. L e t .f : S ~ x (0, 7r) --~ S ~ x (0, 7r) be an exact area-preserving monotone t w i s t map. A s s m n e f has no rota.tionM invariant curves. Then, for a n y e > 0, i1' V_ = S 1 x [0, e) a n d I7+ = S 1 x (Tr - e, Tr], there exists an orbit of the m a p f connecting V_ and V+, i.e. there exist P E S 1 x (0, 7r) a n d integer n _ , n+ such that I " - ( P ) e V_ and f " + ( P ) e V+. P r o o f . Suppose there exists e > 0 for which tile above s t a t e m e n t does not hold. Chose such e a n d consider the correspondin sets V_ and V+. Let V=

U J " ( i n t V _ ) U (S 1 x { 0 } ) . nEZ

Clearly we would have

VNV+ = ~ . Let B be the connected component of S ~ x [0, ~r] \ V which contains S ~ x {Tr}. Let u = S'x[O,~]\B, T h e n f ( U ) = U, V_ C U, V n V + OandS'x{O}is a deformation r e t r a c t of U. Then, by Birkhoff inva.riant curve theorem ( T h o r e m 15.1'), 0U is the g r a p h over S 1 of a Lipschitz flmction. Therefore there exists a r o t a t i o n a l invariant curve for f , contradicting the hypothesis. []

138

We continue the proof of T h e o r e m 17.1 as follows. We suppose t h a t the curvature of OR is zero at some point and t h a t there exists a r o t a t i o n a l invariant curve for f (a caustic for the billiard ball problem). This will lead to a contradiction, thereby showing t h a t no caustic can exists if the curvature vanishes at some point. L e m m a 17.2 will complete the proof. Let P 6 OR be a point where the curvature vanishes. Let Pt a o n e - p a r a m e t e r family of points Pt 6 OR in a n e i g h b o r h o o d of P and let P ~ = g"(Pt), n 6 Z, where g : OR --~ OR describes the restriction of f to the invariant curve corresponding to the caustic. We notice t h a t , since f is orient a t i o n preserving a n d also preserves the two ends of OR x [0, 7r], 9 is a orientation preserving h o m e m o r p h i s m of the circle. Let h : OR x OR + R be the variational principle associa.ted to f . We have seen in w t h a t h exists and has the following form: h(p,Q) -- -lip QII , for any P, Q e OR. -

It is a consequence of the generating equations (3.2) a a d of the existence of a caustic that (17.1)

h l ( P t , P 1) + h2(P[-1,Pt) - 0 ,

and, by differentiating with respect to t, (17.2)

h 1 2 ( P t , P : ) dPr - j ~ + hn(P~-',Pt) (IP[-~ d~-

( h n ( P , P 1) + h22(P~-',Pt)) ~dPt -

Since P ) a n d Pt -1 represent points on the invariant curve for f associated to the caustic, by Birkhoff t h e o r e m they are Lipschitz function of t, hence their derivatives with respect to t exist ahnost everywhere and can be a priori b o u n d e d from above and below in terms of the m a p p i n g f . F u r t h e r m o r e , since g is o r i e n t a t i o n preserving and ~ d t have the sa.me sign as ~d t " Since h12 < 0, by the definition of a dt m o n o t o n e twist m a p p i n g , we conclude that

(17.3)

h~(Pt,Pr + h22(W~,Pt) > 0,

for all t 6 S 1, as a consequence of the existence of a caustic. On the other hand, in the case of a billiard ball p r o b l e m it is not difficult to show t h a t (17.4)

hl~(P,P") + h22(P',P) < 0

for any P ' , P " , if P is a. point of OR where the curvature vanishes. This can be u n d e r s t o o d by considering the fa.ct that the p a t h of a billiard ball bouncing on a rectilinear sca.tterer .~tvictly minimizes the euclidean length among all possible paths. Since h was defined a.s the negative of the euclidean length, (17.4) follows, thus c o n t r a d i c t i n g (17.3) and concluding the proof of the theorem. []

139

Non-existence of invariant curves in the Chirikov standard mapping. The Chirikov mapping was introduced in w as the exact area-preserving monotone twist map fk of the infinite cylinder S 1 x R, (17.5)

k

k

h(6),y)=(8+y+~-~sin27rS, y+~-#sin2rO),

kER,

whose variational principle is hk : R 2 ~ R ,

(17.5')

hk(:~, ~-') = ~1

(x -

x') 2 -

4 -k~ cos 2~:~ .

It is in fact a one-parameter family of mappings, which is sometimes called the standard family, where k plays the role of a perturbative (stochasticity) parameter. When k = 0 we obtain the completely integrable mapping, which is the exact area.preserving monotone twist map ]0 of the infinite cylinder S 1 x R into itself, given by (17.6)

2"~(0, :,j) = (6) + y, y ) ,

whose variational principle is h0 = 8 9 x') 2. This ma.p is characterized by the property of having a rota,tional invariant curve for any rotation number, i.e. the cylinder S 1 x R is completely foliated by rotational invariant curves of the map f0. It is a consequence of the K.A.M theorem, exposed in w that, when Ikl is sufficiently small, a large measure set of invariant curves, namely those whose rotation number satisfies the Diophantine condition (16.1), persist. On the other hand, numerical results due to Greene [Gr] show that there are rotational invariant curves for [k[ _< /co and there are none for [k] > k0, where the critical threshold is estimated as k0 ~ 0.97. Here we will expose a simple rigorous result, due to [Ma4], which establishes that there are no rotational invariant curves for ]k I > 4/3. T h e o r e m 17.3. If I/c[ > 4/3, then there are no rotationaJ invariant curves in 81 x R

which are invaria~t under f k. P r o o f . The argument is similar to the one given in the proof of Theorem 17.1. Let f be an exact area-preserving monotone twist map of S1 • R. Suppose f has a rotational invariant curve. As a consequence of Birkhoff inavariant curve theorem, the invariant curve will be the graph of a Lipschitz fimction r : S 1 --* R. Clearly, there exists an orientation preserving homemorphism g : S 1 --~ S 1 such that (17.7)

f(0, r

= (g(6)), r

.

This follows fl'om the fact that the curve given by the graph of the function r is invariant under f. Suppose the variational principle h : R 2 ---* R of the lift of f to the universal cover R 2 is of the following form:

h(.T,.~") = ~1( . T - x') 2 +

u(x)

.

Consequently, in view of the generating equations and (17.7), (17.9)

dg-1 dg d---7-~ + ~

-

~"(~)

-

2=

-

d ,z:,-~(;~(~' ~(~:)) + ;"~('J-~(:~)' ~)) - 0 .

140

Here the derivatives dg/dx and dg - 1 / d x exist almost everywhere and are bounded, as a consequence of Birkhoff inw~.riant curve theorem. In fact, Birkhoff theorem assures that r is a Lipschitz flmction and, since f and f - 1 are smooth maps, the definition (17.7) of the homeomorphism g gives that g is bi-Lipschitz. Let L > 0 be the m a x i m u m between the least Lipschitz constant of g and the least Lipschitz constant of 9-1, i.e. (17.9')

L = max {sup ]g(x) - g(x')l Iz - :,:'l , sup

[~-~(x) - ~-~(~')1 }. 1.~ - x'l

In view of the definition of L, we ha.ve (17.9')

dg(x) < L L - 1 g(x) and H l ( x , x ' ) > 0 when x' < g ( x ) ,

g 2 ( x , x ' ) < 0 , when x' < g(x)

and

H2(,T,,X') > 0

when z' > g(x) .

Consequently, H(:Gx' ) > C, when x' # g(x) (a.nd H ( x , x ' ) = C, when x' = g(x)). Suppose O = {(xi, Yi)}iez represents an orbit of the diff~omorphism f on the rotational invariant curve F. By (17.13), the stationary configuration x = (x~)~ez associated to the orbit O (see w and w satisfies g(xi) = x~+1. Let m < n be integers and let y = (Ym,...,Yn) be a segment of a configuration subject to the constraint Ym = x,,, and y,~ = xn. Suppose that yi # xi for some i, m < i < n.

142

Then jfxx n

n--]

h ..... ( x) = E

g ( x i , :l;,+l ) Ji-

r

d~ = (1~ - ~)C-]-

(17.17) -}-

r rn

< Z H(yi,yi+l) + ~TIt

r

= hmn(y) ,

rn

where hmn : R ...... --* R is the flmction n -- 1

(17.1s)

hm.(v) = Z h(y,,y,+,).

Thus, we ha.ve shown that x is a. minimal configuration in the sense of w and, consequently, O is a minima.1 orbit, a.ccording to the definition given in w []

143

w

D e s t r u c t i o n o f Invariant C u r v e s .

Tile variational p r o p e r t y of rotati(mal invariant curves, described at the end of w explains the relevance in the context of area-preserving m o n o t o n e twist m a p p i n g s of a notion of a barrier flmction, originally i n t r o d u c e d in solid s t a t e physics in connection with the Prenkel-Kontorova model mentioned in w [Au-LeD]. This barrier function provides a tool which is sensitive to the existence of a r o t a t i o n a l invariant curve of given r o t a t i o n number.

The Peierls ~ barrier. T h e Peierl's barrier is a real valued fimction depending on a r o t a t i o n s y m b o l w (defined in w a n d on ~ E R , where R is seen as the universal cover of the circle S t. It measures to which extent the s t a t i o n a r y configuration (~i)iEZ, subject to the condition Y0 = ~, is not minimal. In w we i n t r o d u c e d the q u a n t i t y (13.3)

+~ /,h(v,

=

h(v,, v , + , ) - h(x,,

X,+l),

for the s t u d y of m i n i m a l configurations of r o t a t i o n symbol p/q:t:. To inchtde the case of r a t i o n a l n u m b e r s as r o t a t i o n symbols, we introduce the q u a n t i t y A~,t~.(y, x), as h~llows: (18.1)

Awh(y, x) = ~

]t(~i, ~]i+l ) - h(:ci, Ti+l ) , I

where I is equal to Z when co is an irrational r o t a t i o n symbol or is equal to p/q+ and it is equal to {0, ...,q - 1}, when co = p/q. The above q u a n t i t y can be shown to be convergent (possibly to + c o ) whenever x is a m i n i m a l configuration of r o t a t i o n symbol w and the configuration y is a s y m p t o t i c to x (i.e. lYi - xil -~ 0 as i --~ +oc). The proof of this fact is contained in T h e o r e m 13.1, in the case co = p / q + , b u t it works as well in the general case. We recall t h a t , for any r o t a t i o n s y m b o l co, we i n t r o d u c e d in w A~ as the subset of the reM line which is the union of all m i n i m a l configurations of r o t a t i o n symbol co. In T h e o r e m 12.2 a n d 13.4 we proved theft A~ is a closed subset of R , for any r o t a t i o n symbol co. T h e Peierl~'~ barrier is defined as follows. Let co be a r o t a t i o n symbol and ~ E R , then P~o(~) = 0, if ~ E A~. In the case ~ r A,,, hence it belongs to a c o m p l e m e n t a r y interval J = ( J - , J + ) to A~ (i.e. a connected c o m p o n e n t of R \ A~), P ~ ( ( ) is defined as (18.2)

P~(~) = rain {A~,h(y, z - ) l y o

= ~} ,

where the m i n i m u m is taken over the set of all configurations satisfying x ~ 0 such that Pp/,l(~) >- A(p,,I)dist(~, A,,/,~) 2 , for any r 9 R , where dist(., Ap/,l ) denotes the euclidea1~ distance timction from the dosed set Ap/q on the r e d line.

146

Then, for any rotation ~'ymbol w whose underlying l m m b e r ~r(w) is' irrational ~nd ~ a t i s ~ IW("~) - "Pl < e(p,,,),

IP~(~) - G/,,~(~)I -< c(8) exp(-~(p,~,)/Iw(~) - p l ) , + or - sign according to whether w > p/q or w < p/q, where ~(p,,l) -- C/q(1 +

A--1/2 (1,,q) )

"

The conditions a) and b) in Theorem 18.3 are generic in any smooth topology and in the analytic topology. However this result can be applied to the destruction of invariant curves only when it is possible to achieve a good lower b o u n d for the constant A, which is related to the hyperbolicity of the (unique) minimal periodic orbit of type (p, q). This ha.s been done IF] in the case of the completely integrable map (17.6) and it is conjectured to be possible also in the case of a rotational invariant curve 3' such that the restriction f ] 7 of the twist diffeomorphism f to 7 is sufficiently smoothly conjugate to a rigid rotation. In the general case the problem of constructing a perturbation which yiekls the desired lower bounds for A in Theorem 18.3 is open. In the following we will apply the reduction of a periodic (minimal) orbit to a fixed (minimal) point. This is usually done in the theory of dynamical systems by considering compositions of the map with itself. In the case of exact area-preserving monotone twist mappings, generated by a. variational principle, composition corresponds at the level of variational principles to the operation of conjunction, introduced in [MaS]. Given two variational principles hi and h2, satisfying the condition (//2), their conjunction is defined in the following way: (18.4)

(hi * h2)(x, x') = ~ i ~ ( h l (x, ~) + h2((, x'))

(the m i n i m u m exists by condition (1t2)). In [Ma8] it is proved that, if both hq and h2 satisfy the conditions (H0), then so does the conjunction hi * h2 with the same 8. Notice that, even when both hi and h2 are smooth, the conjunction hi * h2 needs not to be smooth. Given a variational principle h aml a rational number p/q (in lowest terms), we will often consider in the following the variational principle H (depending on p/q) defined by (18.5)

H(x,x') = (h,...,h)(x,,x'

+ p ) ( q - times) + constant ,

where the additive constant will be chosen so that min H ( x , x) = 0. One has (18.5')

Ah

=

H Aq~_ v and

h

P:(0

=

pH

,,~-~(0 ,

as long as w is a rotation symbol which is not a rational munber or is a rational number whose denominator is divisible by q.

Th, e destruction of invariant curve,,~. The qualita.tive principle underlying the destruction results for invaria,nt curves can be described as follows. It is known since the work of Poincar6 that rotationM

147

invariant curves of rational rotation number can be destroyed by a perturbation as small as we wish in any smooth topology or in the analytic topology. On the other hand, it follows from Birkhoff invariant curve theorem, namely from the fact that rotational invariant curves are Lipschitz graphs whose Lipschitz constant is determined a priori from the map, that the set of rotational invariant curves is closed. Therefore, the destruction of an invariant curve with rationM rotation number p/q is accompanied by the destruction of an open set of nearby curves, including those with irratioim.1 rotation munbers contained in a certain open interval I(p, q) around p/q . To obtain a destruction theorem h n invariant curves of a fixed irrational rotation number, one has to provide lower e,~timates for the size of the interval I(p,q), in terms of (p, q) (in fact in terms only of the denominator q > 0). In view of Lemma 18.1, this cml be done using the modulus of continuity for the Peierls's barrier. The destruction of invariant curves of rational rotation number p/q is described, in terms of the Peierls's barriers Pv/v(~) a.nd Pp/,~+ (~), in the following. Details can be found, in the smooth case, in [Ma9], and in the analytic case, in [F]. L e m m a 18.4. There exists positive constants C,-(O) such that the tbllowing holds. Given a smooth variationM principle hi, , satisfying the conditions ( Ho ), associated to em exact area-preserving monotone twist mapping f, tbr any r E N and e > O, there exists a periodic smooth function w on R such that:

Ilwll,-+l < e a~d

maxPh, ~:~(~) > ,~ E I:L

--

P l q

where G(:~, x') = h i ( x , x') + ",,,(:~) a~d of real-valued fm~ctlons on S 1 .

C,.(e)e"+2/q (r+')~

--

I1" I[,

'

de~otes the C'" . o r m on the space

P r o o f . Choose a nfinima.1 configuration x = (xi)iEz of type (p, q). Then the set {xi + j [ i, j E Z} intersects each interval [a, a + 1), a E R, in exactly q points. By the pidgeon hole principle, there exists a complementary interval J of length > 1/q to this set. We choose a smooth non-negative function u, satisfying u(x + 1) = u(x), supported in J + i, i E Z, whose C "+1 norm is small. If IlUllr+l ~ e/2, then, since the length of J is > 1/q, u can be chosen such that (18.6)

maxu

>_ C,.e/q"+l .

On the other hand, since the support of u does not contain any point of the minimal configuration x = (xi)iEz and u is non-negative,

(18.7)

h.! p"~ r ~ = p,/,j(r p/q\"~]

+ ,u(r

where h~ is the variational principle h~(x,x') = h f ( x , x ' ) + u ( z ) . This follows immediately fl'om the definition of the Peierls's barrier Pp/v" We will now use the reduction to the case of a (minimal) fixed point. Let H~ be the variational principle associated to h,, as in (18.5). It follows fronl (18.5) and (18.5') tha.t

(18.8)

H.({,()

= phi(()

= p ~h/ uq ( ( ) ,

148

which, in view of (18.7) implies (18.8')

H , ( ( , () > u ( ( ) .

Let J = ( J - , J + ) be the complementary interval to the set {x~ + j l i, j E Z} containing the support of u. By a result in the Aubry theory of minimal configurations (Corollary 13.6), since, by (18.5'), A H~ = A p/q h~ there exist minimal configurations x i of rotation symbol 0 + such that :c~- ~ j i as i ~ Toe and x + --~ J • as i ~ +oe. We are interested in a lower bound for the m a x i n m m , over i E Z, of Ix~+1 - x ~ l . Let J' be the middle third of the interval J. If no x~ E J', then we m a y take the length of J' as a lower bound. Suppose x~ E J'. Removing :c~ from the configuration x • we obtain a new configuration y• of rotation symbol 0 • namely y]= = x~=, for j < i , a n d y ~ = ' c9 j+l, • for j > i. By Theorem 13.1, the quantity (13.3)

+c~ •

=

•

•

H~,(xi , :Ci+l)) i=-oo

exists and it is positive. On the other hand, clearly (18.9)

A(y•

•

= H,,(:r#_a,:~:~+,)-- H , , ( x # _ ~ , z # ) - - H , , ( : c # , z h i ) .

Since H.,, satisfies the conditions (H0,), for t9r = ()+ 1, provided that e is sufficiently small, the following estimate holds [Ma9, w (i8.9')

• H , , ( x i•_ l , x i +• i ) - H ~ ( x ~ _ l , x ~ ) - g , , ( . "ci• ,xi+i) C~e(C,.(8) e/qr+~)~+l = Cr(O) e~+2/q (~+1)2 .

Since v vanishes outside the union of the intervals [x~, x#+l] + j, j E Z, and it is non-negative, (18.12)

P~'/~,,~(() >- PI'/~,~ (() + v ( ( ) .

where w = u + v and hg(x, x') = h i ( x , x') + w ( x ) = h,,(x, x') + v ( x ) . The inequalities (18.11) and (18.12) immediately give tile desired estimate. []

149

In the analytic case, the argument is slightly complicated because of the absence of compactly supported analytic functions. The appropriate substitutive tools are a version of the m a x i m u m principle for holomorphic functions (Hadamard 3-circle theorem) and the approximation theorem for real analytic functions by trigonometric polynomiMs (Jackson approximation theorem). The outline of the argument is, in any other respect, similar to tile smooth case. The result is the following [F]: L e m m a 18.5. There exist positive constaalts C, C(8) such that the following holds. Given a real analytic varia, tional princilfle h s, satisi~ving the conditions ( H e ) , / b r ally r > 0 aaad e > 0 there exists a trigonolnetric polynomial w such that:

Iwl, < ~ ~ld

,,,

(

max G/,,~(~) > (~/4) e~p -C(0)~, ~/~ e•

)

~ER

where h A x , x') = h• + ,,,(:~) , Iwl,. denotes the maxinmnl moduh, s on the iu~,~ite strip S, = {z ~ C I Ihn zl _ c ~ / , 1

(18.14)

~-~ ,

besides satisfying (18.6). To this purpose, it would be sufficient that the complementary set to {:,:~ + j I i, j E Z} contains two adjacent intervals of length >_ C/q, where C > 0 is a universal constant independent of q > 0. Since w"(xi) = u"(xi), i E Z, and 5h.q/,,,~) {.. = P~/,,(~) h.! + w(C), the constant A0,,<s) > 0 contained in Theorem 18.3 can be estimated as follows:

(18.15)

A(v,v ) >_ C ~ e / q " - '

.

h!

This holds in view of the fact that P/,~(~) --- 0 in the case f is the completely integrable map. Then Theorem 18.3, together with L e m m a 1.4, allow to replace the estimate (18.13) in the proof of Theorem 18.6 by the following: (18.16)

C~(O)e~+'21q (~+1)~ _ C,,(O, e)/q("+a)/2Log q ,

for any p, q E Z, q > 0. Therefore, in this case, we obtain a destruction result in the C ~ topology under the hypothesis that the Diophantine exponent of the rotation number w is > (r + 1)/2. This has to be compared with the condition given in the K.A.M. theorem, according to which we have stability in the C ~ topology provided that the Diophantine exponent is < (r - 3)/2. We see that there is no gap in the behaviour as r ~ +ec. In the general case the estimate (18.15) is not available, as

151

a. consequence of the fact that, if' a. (minimal) 1)eriodic orbit is not equispaced mod. Z, it is unclear whether it is possible to produce a sufficiently hyperbolic orbit (with estimates) by a perturbative construction. In the analytic case, similar arguments, relying on L e m m a 18.5 instead of L e m m a 18.4, lead to the following results contained in IF]: T h e o r e m 18.7. Let 7 be a rotational invariant curve of rotation n u m b e r P(7) = ~o E R \ Q, for an exact area-preserving monotone twist analytic diffeomorplaism f ot" the ammlus. A s s u m e ~o satistles the condition (I) where of w. exact admit

lira snl) ,.--.+~

Log Log,"q,,+ 1 qn

> 0 ,

(pn / q,, )neN is the sequm~ce of convergents of" the continued fraction expansion Then in any neighborhood DIS of f in the analytic topology there exists an area-preserving monotone twist analytic diffeomorphism g which does not any rotational invariant curve o~' rotation number w.

P r o o L Let U,.,, be the set of all real analytic periodic functions w on t t which extends to a holomorphie flmction W on the strip S~ = {z 9 c I Ilmzl < r}, in the complex plane C, and IWI, < e, where, as befl:,re, IWI,. denotes the m a x i m u m of IWl on S,.. N)r any flmction e : I t + ~ R +, let (18.17)

U~ =: U U~,~(~) . r>0

The family of sets described in (18.17), as the flmction e varies, is a basis of open sets for the analytic topology. Assume by contradiction that there exists a strictly positive function e : I t + ~ I t + such that, whenever h I - h,j E ld~, then g admits a rotational invaria,nt curve of rota,tion number w and therefore (18.18)

P2~(() = 0 .

Let (Pn/q,,),,eN be the sequence of convergents of the continued fraction expansion of 0J. Then, given any r > 0 and any e < e(r), by Lemnla 18.5 one can construct, for each n E N, an exact area-preserving monotone twist analytic diffeomorphism gn such that h I - h,,. E Ur, e C Me, where h,~ is a wu'iational principle for gn and (18.19)

h,, max P:,,/q,i (r

( -3/2 ) -> (e/4)exp - C ( 0 ) E 1 exp(C/'qn)

9

By the modulus of contimfity given in Theorem 18.2: (18.20)

(e/4)exl)(-C(8)e-13fZexl'(Crq,,))

Cel q2 e x p ( - r q) .

The estimate (18.23) should be compared with (18.15) which holds in the smooth ease. The estimate (18.20) is then ret)laced by the following: @/4) exp ( - C ( O ) e l 3/2 exp(C'r qn)) i

~k

_< e + , ~ ,

is finite and converges to zero, as { ---* +o% if @i)iEN is chosen

such that ~--~ei < + o c . i>0

158

On the other hand, since r _< r for all i, k > 1, then (r is a nondecreasing sequence, hence it has a limit ~ E A. By the lower semicontinuity and the choice of the sequence (ei)~eN, (19.4) implies: (19.5)

F('4,) _< e + l i m i n f r i = e , i~+oo

which concludes the proof, by showing that ~b is a mininmm point for F.

[]

Given $ E ~4" and t E R we define configurations x a = x,,~at by (xa)i=Ca(t+wi),

for a l l i E Z

and, if r is also weakly order preserving with respect to t E R, we define configurations x ~ = x r by (:c~)i = Ca(t +coi+) , for a,lli E Z . If r E A~ a.nd x = xr

then xi_~+j,

if t + w > j + A ( j ) .

We let 7-/,oz~t- denote the set of configurations x E R z which satisfy these conditions. If r C A~ and x = z e t a , + , then zi_~+j,

if t + w i > _ j + A ( j ) .

We let 7-/~ar denote the set of configurations x E R z which satisfy the latter conditions. Clearly the sets of configurations just defined also depend on ~ E It, but we prefer to drop this dependence, since ~ will be a fixed real number to be chosen appropriately, i.e. such that P~0(~) > 0. Notice that, if r E B~ is continuous from the left with respect to t E It, then x4,~oAt = X4,~A~-- E "]-{,~A't-- 9

A configuration x E ~ , t •

is sa.id to be "minimal relative to ~.~A,+ if:

for any pair of integers rn < n and any configuration x I E "H~.xt:t: such that ,! 2L m

~-

X m

and

I X n

~--- X n ~ .....

....

where h .... is the function defined as h)llows

L e m m a 19.7. Let r E B~ emd suppose r m i n i m i z e s F~~ over B(. Then, tbr a,ny given t E R, there exists a fiHl measure set :Doo(t) C :Doo, with respect to the

159

mea~ure #, such that the configuration xr

is minimM relative to 7"l~nt+, for a11

a 9 9~(t). Thus a minimizing element r 9 B~ for the averaged Percival's Lagrangian can be used to produce an abundance of relatively minimal configurations. Assuming that P~o(~) > O, it is plausible that the minimizer q~ can be chosen satisfying strictly the inequalities defining .Ar In fa.ct, if it were not so, we could produce relatively minimal configurations x = xr177 satisfying x0 = ( + j, which is impossible since P~o(~) > O. This fact is made precise in the following: 19.8. Suppose the generating function h satisfies the conditions (H1) ( H~o ). If w is irrationM, P~( ~) > 0 and # is a shift-invariant ergodic Borel probability measure supported on 7:)r 0 < c < b0(co,P,~(~)), where 7)r = {A 9 / ) ~ I IIAII~ _< c}, then there exists r 9 B~ minimizing F~' over .A~, which satisfies strictly the inequMities defining .A(, i.e. Lemma

{r 2')

if t _ < j + A ( j ) CA(t)>~+j,

if t > j + A ( j ) ,

tbrallj 9

fbr ahnost M1 (A, t) 9 7)~ x R, with respect to the measure # c = # x Lebesgue. In view of L e m m a 19.8, the proof of the Euler-La.grange equation is immediate and it does not deviate fl'om the usual argmnent. L e m m a 19.9. Suppose the smne conditions as in L e m m a 19.8 hold. Then r satisfies the Euler-Lagrm~ge equation, i.e. h2(r

- ~o), Ca(t)) + h~(r

Ca(t + w)) = 0 ,

for aJmost MI (A, t) 9 Z)~ x R , with respect to #c = # x Lebesgue. Let f be an exact area-preserving monotone twist map of the annulus S; x R. Let h be a variational principle associated to f. The conditions imposed on f imply that h satisfies the conditions (H1) - (H6o), for some 0 > 0. Let a~ E R \ Q and assume that there is no rotational f-invariant curve of rotation number w, hence P~(~) > 0, for some ~ E R, by L e m m a 18.1, and consequently L e m m a 19.8 and 19.9 hold. Let 0 < c < 5o(w,P~o(~)) and let # be a shift-invariant ergodic Borel probability measure on T)c. We will introduce a dynamicM system (7~, #L:), which is the suspension of (~r, #) over the rotation R~, where a is, as before, the forward shift on R z and R,~ is the rigid rotation t ~ t + w (rood. Z). Let Sc be the quotient space S~ = (7)~ x R ) / T , where T : T)~ x R ~ :Dc x R is the transformation T : ( A , t ) ~ ( a A , t + 1). The measure # c induces on S c a Borel probability measure, still denoted by the same symbol. The Euler-Lagrange equation suggests to consider on the measure space ($~,#c) the transformation ~ : (A, t) ---* ( A t + w) (mod. T), induced by R~, which preserves the measure #L. We formalize this construction in the folk~wing:

160

D e f i n i t i o n 19.10. If (X, T , #) is a dynamical system on a measure space (X, 13), leaving the probabiBty measure # invariant, its ~uspen~ion over the rotation R~ is the system ($, 74, #c ) de~ned as tbllows:

1) S = (X x R ) / T , where T ( x , t ) = (7-x,t + 1), for any ( x , t ) e X x R and the measure #z; is the probability measure induced by the product measure # x s where s is the Lebesgue measure on R ; 2) T4 is the transtbmnation induced on $ by the rotation R~o, i.e. 7"4 : ( x , t ) --* ( x , t + co) Onod.T). The minimizing element r 6 B( yielded by the previous m-guments induces a Borel m a p ~5 : $~ ~ S 1 x R which semi-conjugates f to 7-4 on S~. In fact, let r]a = - h i ( C a ( t ) , Ca(t + co)) = h 2 ( r

- co), Ca(t))

and

gS:(A,t)--,(Czx(t),qa(t))

(rood. Z x { i d } ) ,

then f o (I~ = ~ o T4 , ahnost everywhere on N~:, with respect to #c. Thelefin'e the measure #L induces by push-forward an f - i n v a r i a n t Borel proba.bility measure #~ on S 1 x N, i.e. #,o = 9 ,(#~;). The condition for the measure pc to be ergodic, with respect to the transfi)rmation 7~, and consequently for the measure #~o to be ergodic with respect to the

diffeomorphism f (since the push-forward of an ergodic measure is still ergodic), is the following: (E~)

exp (2rrik,/co) r E V ( p ) , for all k E Z \

{0},

where E V ( # ) is the eigenva.lues spectrum of (a,//,). This ca.n be seen by a spectral theory a.rgmnent which shows tha.t the only eigenfimctions corresponding to the eigenvalue 1 are the constant functions. The previous construction can be summarized as follows: T h e o r e m 19.11. Let w E R \ Q and suppose tha, t the diffeomorphism f does not a&nit a n y rotational invariant cm've of rotation n u m b e r co. Then any shift-invariant ergoclic Bore/ probability measure # on ~9~, 0 < c < 60(w, P,o(~)), satisfying the condition (E~o), induces an .f-inw, riant ergodic Borel probability measure #~ on S 1 x R , of angular rotation n u m b e r co. The resulting dynamicM system (f, #~) caa2 be described as a f~ctor of the suspension (74, #s of (or, it) over the rigid rotation

R~. The thctor m a p 9 : S~ ~ S 1 x R is described as:

@(A,t) = ( r

(rood.

Z x {id}) ,

where r : 59c x R --~ R is weakly order preserving, with respect to t E It, it satisfies the following localization property:

r

- c) _< Ca(t) < r

+ c),

161

and the M( constraints strictly:

1)

~b~,zx(t + 1) = ~bLx(t) + 1 ,

&n d

{r 2')

CA(t)>(+j,

if t<j+A(j) it" t > j + A ( j ) ,

forallj EZ,

tbr ahnost ali (A, t) E Z)~ x R , with respect to the m e a s u r e #L.

As betbre, a denotes the tbrward shift on R z and r E Y1 minimizes F~ according to Theorem 6.1 a n d it is normMized as in *). Finally ~ E R is chosen such that P~(() > O. According to Theoreni 19.11, it is possible to associate to any shift-invariant Borel p r o b a b i l i t y measure # on 7Pc a f - i n v a r i a n t Borel p r o b a b i l i t y measure #,o on S i x R . T h e question n a t u r a l l y raised by the previous construction concerns the ergodic-theoretical properties or classification of the mea.sures #~, which can be o b t a i n e d t h r o u g h it. Since #~ is given as a factor, it is not s t r a i g t h f o r w a r d to u n d e r s t a n d its n a t u r e in general. However, the factor nlap is p a r t i a l l y controlled by the information provided by the constraints. This is enough to conclude t h a t if # is an almost periodic measure, satisfying condition (E~), then the factor m a p is an isomorphisni, therehy obtaining T h e o r e m 19.1. This case includes the case when # is a.n atomic sliift-inva.riant p r o b a b i l i t y measure, which give the Denjoy f invariant p r o b a b i l i t y nleasures constructed in [Ma6]. Furtherniore, it is possible to choose (a, #) isomorphic to a Bernoulli shift on two symbol, in such a way t h a t the e n t r o p y of the associated f - i n v a r i a n t measure #~ is also positive. This is the content of T h e o r e m 19.2. T h e details of these arguments are contained in [F]. We finally r e m a r k t h a t the f l e e d o m in the choice of c > 0 in T h e o r e m 19.11 can be used to locMize the measure #~ constructed there, as claimed in T h e o r e m s 19.1 and 19.2. We will briefly sketch the basic idea underlying the construction of positive e n t r o p y measures. According to Lenima 19.8 and 19.9, there exists r : :De x R ---* R which satisfies the constraints strictly and therefore the Euler-Lagrange equation, if c > 0 is chosen sutficiently small. Let A_ and A+ be the subset of the cylinder S 1 x R defined as follows: A _ = {(0, y ) [ ( - 1/2 _< 0 < ( ( n i o d . Z ) } , A+ = {(0, y ) [ ( < 0 _< ( + 1 / 2 ( m o d . Z ) } . Consider relatively m i n i m a l configurations :cA i defined by (:c~A)i = CA(t + iw+) and the corresponding orbits ((z~)i,(~J~)i), i E Z, for the lift of f to the universal cover. If r E Z is such t h a t t + 'rw E ( - c , c) (rood. Z), i.e. there exists j~ E Z such t h a t t + r,~ - j,. E ( - c , c), then it possible to choose A(j~) such t h a t I/x(j~)l < c and either t + r w < j , . + A ( j , . ) or t + , w > j r + A ( J ~ ) is a.chieved. In the first case, since r satisfies the constraints, ((x~),.,(yg)~) belongs to the lift of A _ to \

;

the universal cover. In the second case it belongs to the lift of A+. Thus, we are able to construct, by a p p r o p r i a t e choices of the constraint A E ~Dc, orbits for the d i i ~ o m o r p h i s m f which belongs a'rbitra.rily to A _ or A+ each time the orbit R~(t),

162

i E Z, of the rigid rotation R~ : t --* t + w (rood. Z) belongs to the interval ( - c , c). By the irrationality of ~o, R~o is ergodic with respect to the Lebesgue measure. Thus, R i ( t ) E (--c, c) with frequency equal to 2c. Therefore the entropy of the m a p f is positive and proportional to c.

Chaotic orbits in a Birkhoff region of instability. It is a consequence of Birkhoff invariant curve theorem that, in a Birkhoff region of instability R, for an exact area-preserving monotone twist dili~omorphism of the annulus f , there are orbits which connect two preassigned open neighborhoods of the two connected components of OR. This fact has ah'eady been noticed in the proof of the existence, for any e > 0, of e-glancing trajectories of convex plane billiards in a domain whose b o u n d a r y has zero curvature at some point. We are refering to L e m m a 17.2. This statement can be significantly strengthened by a variational construction of orbits based on the positivity of the Peierls's barriers p h ( ( ) , corresponding to rotation symbols w E (p(r_),p(r§ where F+ are the two rotational invariant curves giving the b o u n d a r y of the Birkhoff region of instability. The main idea consists, as in the previous construction of invariant measures, in minimizing the "energy" associated to the variational principle h over configurations subject to constraints. The positivity of all Peierls's barriers in the region R will assure that, for a wide but appropriate choice of constra.ints, the resulting nlinimizing configurations will be contained in the interior of the constraint, thereby satifying the stationarity condition. We recall that stationary configurations axe in one-to-one correspondence with orbits of the diffeomorphism f associated to the varia.tiona.1 principle h. The results which we will describe are contained in [Mall]. We recall that a Birkhoff region o/instability is a compact f-invariant subset of the infinite cylinder, satisfying the following properties: (1) OR consists of two connected components F_ and P+, which are rotational f-invariant curves; (2) if F is a rotational invariant curve contained in R, tlmn F = F_ or F = F+. 19.12. If p(F_) < w_, w+ < p(F+), then there is a~ f-orbit (9 in R such that 0 is a-asymptotic (i.e. asymptotic in the past) to E*~_ and w-asymptotic (i.e. asymptotic in the ti~ture) to E* provided that if w_ (resp. w+) = p ( P _ ) (resp. p(P+)), then w_ (resp. w+) is irrationM. Here ~,*~ denotes, as be~bre, the Aubry-Mather set, of rotation symbol w, described in w Theorem

The statement that O = {(8i,y~)}~ez is a-asymptotic in R to E*_ means that dist ((0~ y i ) , E * _ ) --~ 0 as i --* -r The statement that O is w-asymptotic to E* means that (list ((0~, yi), E*+) --~ 0 as i ~ +z~. T h e o r e m 19.13. Consider for each i E Z, a real number p(F_) < w~ __ p(P+) and a positive number ei. There exists an f-orbit O = {(0j, yj ) } j e z in R and m~ increasing bi-in~nite sequence..., j ( i ) .... ot" integers such that dist ((0j(0, YJ(0), E*, ) < ei. In other words, O approaches within e~ > 0 of E~, at the j(i) *h iteration. By a. con.*traint ,7, we will take in this case a bi-infinite sequence (..., Ji, ...), where each Ji is a closed, connected, non-empty subset of R. A ,]-configuration will be a biinfinite sequence (xi)iez, with xi E Ji- A ~eg'm.ent of a J-configuration will be a finite sequence (xi , ..., xk) such that xi E J,, for each j < i < k. Let h be

163

a variational principle associated to an exact area-preserving monotone twist map f. A segment (x j, ..., Xk) of a ,Y-configuration will be said to be J - m i n i m a l (with respect to h) if h(xj, ..., x~) < h(x;, ..., x ~ ) , . . Xk). such . . that xj* = xj a n d x ~ = x k . for every segment of a ,:7-configuration (x. j,..., We will say that a J - c o n f i g u r a t i o n (xi)iez is J - m i n i m a l if for every j < k, the corresponding segment (x j, ..., xk) of it is J - m i n i m M . It is not difficult to specify conditions on the constraint J which assures the existence of J - m i n i m a l configurations. L e m m a 19.14. Let (..., Ji, ...) he a, constraint sl,ch that there exist arbitrarily smadl and arbitrarily large i tbr which Ji is bom~ded. T h e n there exists a J - m i n i m M contlguratiou, P r o o f . By properties (H1) and (H2), the function h ( x - N , X - N + l , . . . , X N ) is proper, continuous and bounded below on J - N x J-N+1 x ... x JN. It follows that there is a sequence (x(NN), X -(N) N + 1 , ..., X(NN)) which minimizes this function over J - N X J - N + I X ... • JN. Furthermore, using (H2), one nlay find, for each integer j, a compact set K j such that x(.J N) E K i for a,ll N: if Jj is bounded one takes K j = Jj. Otherwise, it follows fl'om the fact that there exist j ' < j < j " for wich Jj, and Jj,, are bounded and from property (/-/2) of the variational principle h. By a compactness argument combined with the Cantor diagonal process, it is possible to choose a sequence (Ni)iEN such that the subsequence xj.(ND ~ xj C K j , as i --* +oo. It is straigthforward to verify that the limiting configuration (..., x j, ...) is a J - m i n i m a l configuration. []

,:?'-minimal configurations are not always stationary, i.e. they do not yield always orbits of the diffeomorphism f. This will happen in case the J - c o n f i g u r a t i o n is also J-free, i.e. xi E int Ji, for each i. L e m m a 19.15. Let x = (...,xi,,..) be a J - m i n i m a d contlguration. If x is J - t r e e , then it is stationary, i.e. - a l h( xi, xi+ l ) = O~h( xi-1, :ci ). In particular, .f( xi, yi ) = (Zi-I-1, ~i-I-1), wliere ?,]i = --Ol]l(:l;i, :l;i-t-1), i.e. (Xi, ~i)ieZ iS ~n f - o r 6 i t . P r o o f . Since x is J-fi'ee, any sufficiently small variation of the configuration x will still be a J-configuration. Thus, the J - m i n i m a l i t y of x implies its stationarity by a simple standaxd argument. []

164

T h e orbits whose existence is asserted in Theorems 19.12 a n d 19.13 will be constructed as the orbits associated to f f - m i n i n m l a n d `y-fl'ee configurations. T h e m e t h o d of proof consists in using tile positivity of the Peierls's barriers, for any r o t a t i o n symbol in a given range, in such a way t h a t , if `y is a p r o p e r l y chosen constraint, each `y-ininimM configuration will be `y-free, and fllrthermore the corr e s p o n d i n g orbits ha~e the properties required in T h e o r e m s 19.13 a n d 19.14. T h e specifications on 27 which produce at least parti~,ll~1 ,Y-free configurations, i.e. i f free on some subinterval of Z, are quite complicated. T h e following holds. Let ~o be a real n u m b e r and let ~b~ E Y1 be a flmction which minimizes the Percival's Lgrangian, according to T h e o r e m 6.1. Let z be a m i n i m a l configuration of r o t a t i o n n u m b e r w, which corresponds to a rec'~Lrre'r~torbit of f , i.e. xi = r + a~i• for some t E R . We also choose a real n u m b e r a such t h a t P 2 ( ( 0 > 0, which is possible by L e m m a 18.1, since there is no r o t a t i o n a l i n w u i a n t curve of r o t a t i o n n u m b e r w. For each integer i, we let ai be the unique real n u m b e r such t h a t ai - a 6 Z and xi E (ai, ai q- 1). This is possible, since :ci - a is never an integer because P2(a) > 0 and the Peierls's barrier is a periodic flmction. F u r t h e r m o r e , we will associate to certain pairs (w, a) of real numbers an integer K ( w , a). L e r n m a 19.16. L e t `y = (..., Ji, ...) be a constraint. Let jo 2 0 / p h ( a ) in the definition of K ( w , a ) . Since q,, is chosen so that k _< q,, - 1, it follows that there exists il satisfying j < il _< j + k such that z~ - y~ < k -~. Since the projection in R / Z of [y~, zi] does not overlap that of [yj,zj], for j < i < i1, and yj < aj + 1 < zj, by definition of yi and zi, it follows that ai + 1 does not belong to [yi, zi]. Since, as we assumed, Yi < ai + 1, we then obtain zi < ai + 1, for j < i < il. Let i2 be the smallest wdue of i > j such that the interval [(i - j ) w , (i - j ) w + Ilqn_lWl[] contains an integer. We have il < i2 _~ j -~ If, in fact , i2 = j + q,, or i2 = j + q,, + q,~-l, as a consequence of the fact that q,, is defined as the smallest integer (1 > 0 such that II(lWll < I]q,,_lwll. Therefore z is a minimal configuration satisfying aj .-}- 1 < zj, ai2 + 1 < zi2 and a, < zi < ai + 1, for j < i < i2. This is because ai - a (mod. Z), for all i E Z, by definition of the configurations y and z, and by the assumption that yi < ai + 1, for j < i < j + K. Since ~ is J - m i n i m a l , we obtain (using as before the adapted version of the Aubry's Crossing Lemma) that ~i < zi for j _< i < i2. In particular ~i~ < zi~ < yi~ + k -1- To summarize, we have obtained the following. There exists j < il _< j + K such that (19.7)

(it 0. In fact, the constraint ff is completely specified in the range j0 - K < i < ji + K by the configuration x and the number a. One may say that the Y/s "follow" the recto'rent configuration :c in that range. Nevertheless, there are other recurrent minimM configurations y that the intervals Yi also follow. There is, in fact, an open interval ~ such that, for any w E ~, there is a recurrent configuration y of rotation number w such that the Ji's follow the configuration y in the range

167

J0 - K < i < jl + K. Thus it is possible to have constraints which follow recurrent minimal configurations with different rotation numbers in overlapping subintervals of Z. The definition of constraints J depend on the choice of a "barrier" a E l:t and of a bi-infinite sequence of integers ( n i ) , ~ z , by setting ai = a + hi. We would like to specify conditions on the sequence (..., hi, ...) which assures that, by appropriately chosing a so that P ~ ( a ) > 0, for any co E f~, then any J - m i n i m a l configuration is J-free. Let e > 0. A bi-infinite sequence of integers (ni)iez is said to be e-restrained, if for each j E Z, there exist real numbers coj and sj such that the following holds. Let (P.i,~/qjn)neN be the sequence of convergents of the continued fraction expansion of the real number wj. We suppose tha.t co is irrational or it i~* rational with denominator (in lowe,,t term.~) > e -1. Under this conditions, it is possible to introduce the smallest integer t~(j) such that q.i,e(j) > e-1. We let I(.i = qj,e(j)-i + qj;e(j). We require that the sequence (...,hi,..,) satisfies ni < w.ii + sj < ni + 1, for j - I ( j < i < j + K j . If, furthernlore, wj E f~, where ft is some open interval, then (rti)ieZ is said to be (e, f~)-re.,trained. Clearly, if (..., hi, ...) is e-restra.ined and a E R, the constraint J = (..., Ji, ...), given as before by Ji = (ai,ai + 1), where ai = a + ni follows a recurrent minimal configuration x (j) of rotation number co.i in the interval j - I ( i e -~, as K(co, e) = qe-~ + qe, where g is the smallest integer such that qt > e -1 a.nd P l / q l , P2/q2, ... a.re the convergents of the continued fraction expa.nsion of co. The proofs of Lemma. 19.18 and Lemma 19.18' are elementary and follow from the definitions of A,,.[j, k], Bn[j, k] and from what it means for n to be e-restrained. Details can be found in [Mall, L e m m a 7.2-7.3]. It can also be shown that every erestrained biinfinite sequence of integers has a unique Farey interval associated to it in the above sense. However, what it reaJly matters for our purposes is the possibility of constructing e-restrained sequences which permit us to achieve arbitrary sequences of rotation numbers (...,coj, ...), as long a.~ they are contained in a Farey interval fl. We skip the details of this construction which can be found in [Mall, w The solution of the munber theoretical problem lea.ds, in view of L e m m a 19.17, to the possibility of constructing appropriate constraints J (i.e. constraints for which every J - m i n i m a l configuration is J-free) "following" an arbitrary sequence of rotation numbers (..., w j, ...), as long as these rotation numbers ave contained in a Farey interval, whose heigth is determined by a lower bound for the PeierIs's barriers corresponding to rotation symbols in that interval. Furthermore, it can be proved, essentially as a consequence of the Aubry's Crossing Lemma, that, given e > 0, if the constraint J "follows" a minimal configuration x of rotation number co 9 R \ Q for a sufficiently long segment (Jio,.-., Ji~ ), then the f-orbits corresponding to x and to any J - m i n i m a l and J - f r e e configura.tion ( approach within e on a subsegment i0 + K < i < il - K 1 , i.e.

where (..., (xi, yi), ...) and (..., (~i, r/i), ...) are f-orbits corresponding to x = (..., xi, ...) and ~ = (...,~i,...). An analogous result holds in case co 9 Q. We refer to [MM1, w for details. The above arguments lead to the following partial results in the direction of Theorems 19.12 a.nd 19.13. Let a 9 R a.nd P > 0. Let f~ C R be an open interval. Let k be tile snmllest integer > 2 O / P and suppose that for any rational n u m b e r p / q 9 f~ (in lowest terms), we have q > k. The positive number P has to be chosen

169

to be a lower bound for the Peierls's barriers associated to rotation symbols in ft, i.e. P h ( a ) > P , for any w E ft.

Proposition 19.19. Consider w_, w+ E ft. There exists m~ f-orbit 0 which is a-asymptotic to E * aad w-asyn~ptotic to E*r P r o p o s i t i o n 19.20. Consider tbr each i 6 Z, a reM mm2ber coi E f t emd a, positive n m n b e r ei. There exists an f-orbit 0 = (..., (Oi, yi), ...) and aa~ increasing bi-in/~nite sequence ..., j(i), ... of integers sud~ that

dist ((O.i(i), YJ(O), E*, ) < ei . In Proposition 19.19 and 19.20 it is summarized the construction of orbits which approach Aubry-Mather sets whose rotation numbers lie in an interval ft. The limitation of these results consists in the fa,ct that the interval ft cannot contain any rational numbers p/q with q 0 ,

i.e. f coincides with the time-l-map of the Hamiltonian flow associated to H. Since H satisfies the Legendre condition, f can also be interpolated by the timel-map associated to the La.grangian flow which can be obtained fl'om the previous Hamiltonian flow by the usual Legendre transformation. Thus, periodic positive definite Lagrangian systems provide a setting which generalize at once exact area-preserving twist mappings of the annulus and the geodesic flow on Riemannian manifohts diffeomorphic to the 2-torus. For these problems (2 degrees of freedom) a series of related results are known, namely the first author's results, exposed in w (and the closely related results by Aubry-Le D ~ r o n [Au-LeD]) for area-preserving mappings and Hedlund results concerning "class A" geodesics of Riemannian metrics on the 2-torus [Hd]. The reader can consult on these subjects and their nmtual connections the survey paper by V.Bangert [Ba]. In this section we will describe a generalization of these results, due to the first author [Ma.12], to more degrees of fi-eedom: an existence theorem for minimal measures, and a regularity theorem which asserts that the support of minimal measures can be expressed as a (partially defined) Lipschitz section of the tangent bundle. The first result generalizes Theorem 6.1, while the second extends to more degrees of freedom Theorem 14.1. The set of ergodic minimal invariant measures is also (partially) described for generic Lagrangians, following Mafid [Mfi]. All his results have been obtained by a slight modification of the setting proposed by the first attthor in [Ma12], which we will now describe. Let M be a compact, connected C ~ manifoht, and T M be its tangent bundle. Let L : T M x R --+ R be a C 2 flmction, called the "Lagrangian". The typical situation is when M is a torus. In particular, when M = S 1, then T M = S 1 x R is an infinite cylinder, which is familiar in the theory of twist maps. We impose the following conditions on the Lagrangian L. We suppose that L is periodic with respect to the R coordinate, i.e. L(~,t) = L(~,t + 1), ~ 6 T M and t G R, where the period is, for convenience, normalized to be 1. We suppose that L has po.,itive

171

definite fiberwise Hessian everywhere, i.e. L I T M , has positive definite Hessian, for any x E M . We suppose t h a t L has fiberwise xaperlinear growth, i.e. L(~,t)/ll~l I ---* +oo ,

a.s I1(11 --~ + o c ,

for ~ E T M ,

t ER .

Here N" H denotes the n o r m associated to a R i e m a n n i a n metric on M. Since M is compact, this condition does not del)end on the choice of the R i e m a n n i a n metric. T h e last two conditions imply t h a t the Legendre t r a n s f o r m a t i o n 1: is defined: if x E M , v E TM:,., t E R , then (20.1)

E ( x , v , t ) = ( x , d v ( L I T M , x {t}),t) .

If L is C ~ (r > 2), then s is a C ~-1 diffeomorphism of T M x R onto T M * x R , where T M * denotes the cotangent bundle of M. T h e fourth condition regards the completeness of the Euler-Lagrange flow, associated to L. T h e Euler-Lagrange flow can be o b t a i n e d by the the first variation of the action functional in the following way. We pose the variational p r o b l e m for the functional (20.2)

A(~/) =

L(dT(t), t) dt

over C i curves 7 : [t0,ti] --~ M with the fixed e n d p o i n t s constraint. Here, d 7 denotes the differential of the m a p "7. The trajectories of the Euler-Lagrange flow correspond to the solution of the va.riational equation (20.3)

~d(7) = 0 ,

associated to the variational probleni for A(~/) (with fixed endpoints). In other words, a C 1 curve in T M x S 1 is a t r a j e c t o r y of the Euler-Lagrange flow if a n d only if it is of the i b r m (d~f(t),t(mod. 1)), where 7 is a curve on M which satisfies the variational equation (20.3). The first varia.tion of the functional A(7) over the space of curves with fixed endpoints can be c o m p u t e d as:

(20.3')

~A(~)(r) = ~

St.

'

for any C 1 m a p p i n g D : [-e,e] x [t0,tl] ---* M such t h a t F ( 0 , t ) = 7(t), for all t E [t0,tl] and F ( s , t 0 ) = "/(to), F(.s, t l ) = 7 ( t i ) , for all .~ E [-e,e]. It is well known t h a t (20.3), with resl)ect to a system of C ~ coordinates ( x i , . . . , x,,), takes the form: (20.3")

d ~ L~, = L~: .

Therefore, the Euler-Lagrange flow is associated to the vectorfield EL described by (20.4)

dx d--/, = :~ '

d ~L~

=L,:.

T h e Euler-Lagrange vectorfield corresponds, t h r o u g h the Legendre t r a n s f o r m a t i o n , to a H a m i l t o n i a n vectorfield on T M * . It is not difficult to show t h a t , if the Lagrangian L is C", the corresponding H a m i l t o n i a n flmction is also C ~, thus the

172

Halniltonian vectorfield is C r - 1 . Consequently, since the Legendre t r a n s f o r m a t i o n is C ~-1, the E u l e l - L a g r a n g e flow is C ~-1, although the Euler-Lagrange veetorfield (20.4) m a y be only C r-2. Since r >_ 2, we o b t a i n t h a t even t h o u g h the vectorfield EL m a y be only C ~ it satisfies the conclusion of the f u n d a m e n t a l existence and uniqueness t h e o r e m for o r d i n a r y differential equations. We now state the fourth condition. The Euler-Lagrange flow is coTnylete, i.e. every m a x i m a l integral curve of the vectorfield EL has all of R a.s its d o m a i n of definition. In the classical calculus of variations tile following basic result, concerning the above b o u n d a r y value problem, holds: T o n e l l i ' s T h e o r e m . Let a < b E R, and let x,,, z~ ~ M. I l L : T M x R ~ R, periodic with respect to the R coordinate, is fiberwise positive definite and has superlineaz growth, then, eanong the absolutely continuous curves 7 : [a, b] --* M such that 7 ( a ) = :~:, a n d 3,(b) = z~,, there is one M~ich minimizes the action I" h

A(~) = ]. L(~t~(t), t) ~It

m

As p o i n t e d out by Mafi~ in [Mfi], it is not necessary to assume compactness of M for the Tonelli's theorem to hold, if the superlinear growth condition is satisfied with respect to some complete R i e m a n n i a n metric on M. A curve which minimizes in the sense of Tonelli's theorem is called a Tonelli minimizer. Ball and Mizel [B-M] have constructed examples of Tonelli minimizers which are not C 1, under the hypotheses of Tonelli's Theorem. However, under the a d d i t i o n a l hypothesis of completeness of the Euler-Lagrange flow, a Tonelli minimizer 3' m u s t be C 1, and therefore it satifies the Euler-Lagrange equation. In case L is C", we have seen t h a t a t r a j e c t o r y t --* (d3"(t), t) of the Euler-Lagrange flow is C "-1, thus 7 is C". T h e role of the completeness hypothesis can be explained as follows. It is possible to prove that, under the hypotheses of Tonelli's Theorem, a minimizer 3, not only exists and belongs to the space of absolutely continuous fimctions, b u t it is C 1 on an open and (lense set of full measure in the interval in which it is defined a n d its velocity goes to the infinity on the exceptional set. Consequently, the completeness hypothesis implies tha.t a Tonelli minimizer is C 1 ( a n d hence C~). Let - / ~ L be the space of EL-invariant p r o b a b i l i t y measures on P = T M x S 1. To every # E -/~L~ w e m a y associate its average action (20.5)

A(#) = / p L

d#

p

Since L is b o u n d e d below, the integral exists although it m a y be +oo. In case A(#) < +oo, we m a y associate to # its rotation vector pot) C H ~ ( M , R ) , which can be uniquely cha.raeterized as follows. Let c E H I ( M , R ) be a cohomology class. By the de R h a m Theorem, c can be represented by a closed 1-form ~. A differential 1-form is defined as a section of the cotangent b u n d l e T M * , b u t it can be considered also as a function on T M , linear on fibers, hence as a function on P .

173

Then, the integral on tile right in tile equation below is defined and it is is finite, since A(#) < + o z a.nd L satisfies the superlinear growth condition (along fibers):

(c, p(#)) = IF A elt,

(20.6)

6

The bracket on the left denotes the canonical pairing between the cohomology group H 1(M, R ) and the homology group Hj (M, R). It is elementary to show that, since # is En-invariant, if A is an exact foml, then the integrM on the rigth in (20.6) vanishes [Mal2, w Lemnla]. Since this integral is linear with respect to c E H i ( M , R), (20.6) defines a homology class p(#) E H i ( M , R). The basic idea of rotation vector goes back to Schwartzman's a.*ymptotlc cycle., [Sw]. It is not difficult to show the existence of invariant probability measures # such that A(#) < + ~ , for which consequently the rotation vector p(#) is defined. The argument is essentially based on the Kryloff-Bogoliuboff procedure to construct probability invariant measures for continuous flows on compact spaces. However, the space P = T M x S ~ is not compact. Therefore, we will consider the one point compactification J~* = P U {oz}. The Euler-Lagrange flow easily extends to P* to a flow which fixes oz and tile Lagrangian L can be extended by L ( ~ ) = oz to a flmction L : P* --~ R. 20.1. A(#) f L d # is a lower-semicontinuous timctionM on the space of Bore1 probability mea~qure on P* with the vague (weak) topology. Furthermore, there exists it E ~ n S~lC~ that A(#) < +oz.

Lemma

-=

P r o o f . Let AK(#) = f m i n ( L , K ) d # , for K E R. Then AK is continuous, since m i n ( L , K ) is a bounded fimction, and AK(F) //, A(#), as K /2 +oz. This implies the lower semicontinuity of A(p). We now apply the Kryloff-Bogoliuboff argument. Let a , be an absoulute raininlizer (i.e. with free boundaries) defined on a time interva.1 of length n. Let 7,,.(t) = (dc~,(t),t). By the previous remarks, %, is a. trajectory of the EulerLagrange flow. Let g., be the probability nleasure evenly distributed ahmg 7n and let # be an aecumula.tion point of the set {/t,,}neN, with respect to the vague topology on the space of the Borel probability measures on P*. This exists because P* is a compact space. An elementary argument, which we will omit, shows that # is n.n invariant measure for the extended Euler-Lagrange flow. On the other hand, it clearly exists for each n E N some curve fl,~, defined on an interval of length n, such that A(fl,~) < C a . Hence, (20.7)

A ( # , ) = 'n.-lA(c~,) < A(fl,,) < C .

Therefore A(t L) ~ C, by the lower semicontinuity of the action functional on the spa,ce of probability nleasures. Finally, since L(oz) = oz and A(#) < oz, tile measure # just constructed has no atomic part supported at the fixed point oo. Hence its restriction to P is a probability measure on P and tt E M L. []

174

T h e l e m m a we just proved has the following i m m e d i a t e consequence: 6 ~ L "which minimize.~ A o v e r .MIL.

there

exi~t.~ #

A refinement of the previous Kryloff-Bogoliuboff a r g u m e n t gives the following: 2 0 . 2 . Let +oc ~ d p(#) = h.

Lemma

h 6 HI(M,R).

Then there exists

# 6

J~L satisfying A(#)
0 and, for each n 6 N , a curve fl,~ : [0, n] --~ M such t h a t fin(0) = i:0, fl,~(n) = 2n and J'o~ L(rlfl,,(t), t)tlt < Cn, where fl,~ is the p r o j e c t i o n of ~,, on ~r. Consequently, (20.9)

A ( # , ) = ',~-]A(~,,) < A(/3,,) 0 such that, I'or emy x, y E ~r(suppAd~), we have dist (~r-~ (x), 7r-~ (y)) _ q (list (da(to), d[~(to)) 2 .

Thus, if r were not injeetive on supp M c, or its inverse were not Lipschitz, it would be possible to construct a probability measure # E AdL for which A c ( # ) < A~(Mr contradicting the definition of M ~. This result would be achieved by "cutting and pasting" trajectories using the "curve shortening" lemma and the Tonelli's Theorem. Then the Kryloff-Bogoliuboffargument would provide the required measure, because of the continuity of p(#) and the lower semicontinuity of A(#). The details of the arguments sketched above can be found in [Ma12, w167 T h e first applications of Theorem 20.9 are to the description of the case M = S 1, thereby completing the picture given by Proposition 20.4 and re-obtaining the basic results found in w P r o o f o f P r o p o s i t i o n 20.4

181

Suppose 13 : H I ( S X , R ) - R --~ R is not strictly convex. Then the g r a p h of 13 intersects a line l in R 2 in a segment I not reduced to a point. Let (h0,13(h0)) and ( h i , 13(hi )) be the endpoints of I. These points are e x t r e m a l points of the e p i g r a p h of 13, hence there exist action mininfizing ergodic measures #o and #1 whose r o t a t i o n n u m b e r is h0 resp. hi. Each of t h e m is contained in M r where c E H I ( S 1, R ) -= R is the slope of the line l. By T h e o r e m 20.9, the projection zr of supp M r on S 1 x S 1 is injective. But this contradicts the fact t h a t two Birkhoff generic orbits 70, 71 in zr(supp(#o)) resp. z r ( s u p p ( F 1 ) ) m u s t cross, since they have different rotation nulnbers. On the other hand, they a.re the projections of distinct (and therefore disjoint) trajectories of the Euler-Lagrange flow on T S 1 x S 1. []

Let h E H I ( S 1 , R ) , let l C H I ( S 1 , R ) x R be a s u p p o r t i n g h y p e r p l a n e of the e p i g r a p h of the flmction 13, which pa.sses t h r o u g h (h, fl(h)) and let c E H 1(S 1, R ) be the slope of l. Let Mh = T S 1 x {0} V1s u p p M r By the strict convexity of/3 ( P r o p o s i t i o n 20.4) Mh is well defined (i.e. it is independent of the choice of l), since in this case M r = .A/[h. T h e set Mh is a closed invariant set for the time-1 Poincar6 m a p fL : T S 1 --~ T S i associated to the Euler-Lagrange flow. These m a p s inchide "twist inappings" defined ill w as a p a r t i c u l a r case. C o r o l l a r y 2 0 . 1 0 . The projection 7rl of Mh (C T S 1) on S 1 is injective mid the inverse rr~ 1 : 7rl(Mh) --+ Mh C T S 1 is Lipsdlitz. Corollary 20.10 includes T h o r e m 14.1, now r e - o b t a i n e d as an i m m e d i a t e consequence of Theoreln 20.9. Let ~r : R 2 ~ S 1 x R = T S 1 be the s t a n d a r d projection and let ~/h = ~ - l ( M h ) C R 2. Let zrl : R 2 ~ R be the projection on the first factor a n d let ))~ denote a lift of fL to the universal cover R 2. Since, by Corollary 20.10, the p r o j e c t i o n 7rl : iV/t, --~ R is injective, f4r/, inherits an order s t r u c t u r e from t h a t on R. C o r o l l a r y 2 0 . 1 1 . ]L : Ml, -~ -~/h is order preserving. Consequently, it'h is irrational, Mh supports a unique invarimlt m e a s u r e /th, which is the mlique minimM measure of rotation l m l n h e r h. P r o o f . T h e order preserving p r o p e r t y follows i m m e d i a t e l y from the injectivity of the p r o j e c t i o n of s u p p M r on S 1 x S 1, which is the content of T h e o r e m 20.9 in the case M = 8 1 . T h e unique ergodicity of the closed inw~,ria.nt set Mh is a s t a n d a r d consequence of the order preserving property. T h e proof is the same as in the case of an order preserving homemorl)hisnl of the circle of irrational r o t a t i o n number. Fina.lly, since all m i n i m a l nleasures of r o t a t i o n n u m b e r It are s u p p o r t e d in Mh, it follows t h a t there is a unique measure #h haxing such properties. []

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A different application of Theorem 20.9, to small perturbations of symplectic diffeomorphisms having an invariant torus, can be found in [Mal2, w There, it is exploited the remark that we can use the Lipschitz property asserted by Theorem 20.9 (and the a priori bound on the Lipschitz constant which can be obtained through it) to localize the invariant set supp Ad ~. The result can be stated as follows. Let f be a symplectic diffeomorphism of the symplectic manifold (N, w), i.e. f*oa = w, where N is a. 2n-dimensional manifold and w is a closed non-degenerate 2-form on N. A K.A.M invariant torus of f is an n-dimensional submanifold T of N such that f ( T ) = T and f i t is smoothly conjugate to ,an irrational translation on the n-torus T" by a vector p = (pl,-.., p,) E R " which satisfies a Diophantine condition, i.e. there exist eontants C, fl > 0 such that (20.21)

Ik0 + k~p, + ... + k , p , I > C(Ik~ I + . . . + I