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Chaotic Dynamics and Transport in Classical and Quantum Systems

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Preface vii

Part I : Theory

P Collet : A SHORT ERGODIC THEORY REFRESHER 1

15 35

G.M Zaslavsky, V Afraimovich: WORKING WITH COMPLEXITY

Fereydoon Family, C Miguel Arizmendi, Hilda A Larrondo: CONTROL OF

CHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS 269

Monique Combescure: PHASE-SPACE SEMICLASSICAL ANALYSIS AROUND

M Courbage: NOTES ON SPECTRAL THEORY, MIXING AND TRANSPORT

Giovanni Gallavotti: SRB DISTRIBUTION FOR ANOSOV MAPS

F Bardou: FRACTAL TIME RANDOM WALK AND SUBRECOIL LASER

COOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITE

Xavier Leoncini, Olivier Agullo, Sadruddin Benkadda, George M Zaslavsky:

ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA

Edward Ott,Paul So,Ernest Barreto,Thomas Antonsen: THE ONSET OF

SYNCHRONISM IN GLOBALLY COUPLED ENSEMBLES OF CHAOTIC

SYSTEMS WITH PHASE SPACE STRUCTURES

S.V.Prants: HAMILTONIAN CHAOS AND FRACTALS IN CAVITY QUANTUM

M Cencini,D Vergni, A Vulpiani: INERT AND REACTING TRANSPORT

Michael A Zaks: ANOMALOUS TRANSPORT IN STEADY PLANE FLOWS OF

J Le Sommer, V Zeitlin: TRACER TRANSPORT DURING THE GEOSTROPHIC ADJUSTMENT IN THE EQUATORIAL OCEAN

Antonio Ponno: THE FERMI-PASTA-ULAM PROBLEM IN THE THERMODYNAMIC

A Iomin, G.M Zaslavsky: QUANTUM BREAKING TIME FOR CHAOTIC

a traditional place to organize Theoretical Physics Summer Schools and Workshops

in a closed and well equiped place.The ASI was an International Summer School*on

"Chaotic Dynamics and Transport in Classical and Quantum Systems" The main goal of the school was to develop the mutual interaction between Physics and Mathematics concerning statistical properties of classical and quantum dynamical systems Various experimental and numerical observations have shown new phenomena of chaotic and anomalous transport, fractal structures, chaos in physics accelerators and in cooled atoms inside atom-optics billiards, space-time chaos, fluctuations far from equilibrium, quantum decoherence etc New theoretical methods have been developed in order to modelize and to understand these phenomena (volume preserving and ergodic dynamical systems, non-equilibrium statistical dynamics, fractional kinetics, coupled maps, space-time entropy, quantum dissipative processes etc) The school gathered a team of specialists from several horizons lecturing and discussing on the achievements, perspectives and open problems (both fundamental and applied) The school, aimed at the postdoctoral level scientists, non excluding PhD students and senior scientists, provided lectures devoted to the following topics :

Statistical properties of Dynamics and Ergodic Theory

Chaos in Smooth and Hamiltonian Dynamical Systems

Anomalous transport, fluctuations and strange kinetics

Quantum Chaos and Quantum decoherence

Lagrangian turbulence and fluid flows

Particle accelerators and solar systems

More than 70 lecturers and students from 17 countries have participated to the ASl The school has provided optimal conditions to stimulate contacts between young and senior scientists All of the young scientists have also received the opportunity

to present their works and to discuss them with the lecturers during two posters sessions that were organized during the School

The proceedings are divided into two parts as follows:

I Theory

This part contains the lectures given on basic concepts and tools of modern dynamical systems theory and their physical implications Concepts of ergodicity and mixing, complexity and entropy functions, SRB measures, fractal dimensions and bifurcations in hamiltonian systems have been thoroughly developed Then,

∗

http://www.ccr.jussieu.fr/lptmc/Cargese/CargeseMainPage.htm

We also provide most recent references of the other given lectures at the school These lecture notes represent, in our views, the vitality and the diversity of the research on Chaos and Physics, both on fundamental and applied levels, and we ff hope that this summer-school will be followed by similar meetings

The summer-school was mainly supported by NATO and the staff of the University Paris 7 M Courbage was a coordinator of the Summer-School, G Zaslavsky was a f director of the NATO-ASI and A Neishtadt was a co-director.

We would like to thank all the institutions who provided support and encouragements, namely, NATO ASI programm, the European Science Foundation through Prodyn programm, the Collectivités Territoriale Corse, the Laboratoire de Physique Théorique de La Matière Condensée (LPTMC) and the Présidence of the University Paris 7 Thanks also to the Centre de Physique Théorique de l'Ecole Polytechnique de Paris

The meeting was an occasion for a warm interactive atmosphere beside the scientific exchanges We want to thank those who contributed to its success: the director and the staff of the Institut d'Etudes Scientifiques de Cargèse and, the team of the Université Paris 7, especially Evelyne Authier, Secretary of the LPTMC, who provided essential help to organize this ASI.

P Collet, M Courbage, S Metens, A Neishtadt, G.W Zaslavsky

Faculty of Mathematics and Informatics

Nicholas Copernicus University, ul Chopina 12/18, 87-100 Torun, POLAND

Section of Astrophysics, Astronomy, and Mechanics-Department of Physics, Aristotle University of

ThessalonikiDepartment of Physics, Aristotle University of Thessaloniki 54124 Thessaloniki Grèce

Abstract We give a short refresher on some of the main definitions and results in

ergodic theory This is not intended to be an introduction nor a review

of the subject There are many very good texts about ergodic theory some of them are given in the references.

A system is characterized by the set Ω of all its possible states At agiven time, all the properties of the system can be recovered from the

knowledge of the instantaneous state x ∈ Ω The system is observed

us-ing the so called observables which are real valued functions on Ω Mostoften the space of states Ω is a metric space (so we can speak of nearbystates) and we will only consider below Borel measurable observables

As time goes on, the instantaneous state changes (unless the system

is in a situation of rest) The time evolution is a rule giving the change

of the state with time

Time evolutions come in different flavors

Discrete time evolution This is a map from the state Ω into itselfproducing the new state after one unit of time given the initial

state If x0 is the state of the system at time zero, the state at

time one is x1 = T (x0) and more generally the state at time n is given by x n = T (x n −1 ) This is often written x n = T n (x0) with

Chaotic Dynamics and Transport in Classical and Quantum Systems, 1–14.

© 2005 Kluwer Academic Publishers Printed in the Netherlands

P Collet et al (eds.),

The dynamics can be given by a differential equation on a manifold

associated to a vector field  F

d x

dt =  F ( x)

This is for example the case of a mechanical system in the

Hamilto-nian formalism Under regularity conditions on  F , the integration

of this equation leads to a semi-flow (and even a flow)

There are other more complicated situations like non-autonomoussystems (in particular stochastically forced systems), systems withmemory, etc

A dynamical system is a set of states Ω equipped with a time tion If a sigma-algebra is given on Ω, we will always assume that thetime evolution is measurable This will often be a Borel sigma-algebra.One would like to understand the effect of the time evolution and in

evolu-particular the behaviour at large time For example, if A is a subset

of the phase space Ω (describing the states with a given property), one

would like to know in a long time interval [0, N ] (N large) how much time the system has spent in A, namely how often the state has the property described by A.

Assume for simplicity we have a discrete time evolution IfχAdenotes

the characteristic function of the set A, the average time the system has spent in A over a time interval [0, N ] starting in the initial state x0 isgiven by

any y ∈ T −1 (x0) ={z | T (z) = x0}, and moreover

µ T (x0)(A) = µ y (A) = µ x0(A) (2)

Second, the limit also exists if A is replaced by T −1 (A) and is equal,

namely

µ x0(T −1 (A)) = µ

x0(A) (3)2

If one assumes that µ x0 does not depend on x0 at least for measurablesets, one is lead to the notion of invariant measure.

By definition a measure µ is invariant if for any measurable set A

µ(T −1 (A)) = µ(A) (4)

Similar considerations and definitions hold for continuous time less otherwise stated, when speaking below of an invariant measure we

Un-will assume it is a probability measure We Un-will denote by (Ω, T, B, µ)

the dynamical system with state space Ω, discrete time evolution T , B

is a sigma-algebra on Ω such that T is measurable with respect to B and

Theorem 1 Let (Ω, T, B, µ) be a dynamical system (recall that T is surable for the sigma algebra B and the measure µ on B is T invariant) Then for any f ∈ L1(dµ)

converges when N tends to infinity for µ almost every x.

We now make several remarks about this fundamental result

By (2), the set of points where the limit exists is invariant (and offull measure by Birkhoff’s Theorem) Moreover, if we denote by

g(x) the limiting function (which exists for µ almost every x), it is

The theorem is often remembered as saying that the time average

is equal to the space average This has to be taken with a grain

of salt As we will see below changing the measure may changedrastically the exceptional set of measure zero and this can lead

to completely different results

The set of initial conditions where the limit does not exist,

al-though small from the point of view of the measure µ may be big

from other points of views (see [1])

The most interesting case is of course when the limit in the ergodic

theorem is independent of the initial condition (except for a set of µ

measure zero) This leads to the definition of ergodicity

A measure µ invariant for a dynamical system (Ω, A, T ) is ergodic if

any invariant function (i.e any measurable function f such that f ◦T =

f , µ almost surely) is µ almost surely constant There are two often

used equivalent conditions The first one is in terms of invariant sets

An invariant (probability) measure is ergodic if and only if

µ(A∆T −1 (A)) = 0 ⇐⇒ µ(A) = 0 or µ(A) = 1 (5)

The second equivalent condition is in terms of the Birkhoff average

An invariant (probability) measure is ergodic if and only if for any f ∈

In formula (6), the state x does not appear on the right hand side,

but it is hidden in the fact that the formula is only true outside aset of measure zero

It often happens that a dynamical system (Ω, A, T ) has several

ergodic invariant (probability) measures Let µ and ν be two

dif-ferent ones It is easy to verify that they are disjoint One canfind a set of measure one which is of measure zero for the otherand vice versa This explains why the ergodic theorem applies toboth measures leading in general to different time averages.For non ergodic measures, one can use an ergodic decomposition(disintegration) We refer to [14] for more information However

in concrete cases this may lead to rather complicated sets

4

In probability theory, the ergodic theorem is usually called the law

of large numbers for stationary sequences

Birkhoff’s ergodic theorem holds for semi flows (continuous timeaverage) It also holds of course for the map obtained by samplingthe semi flow uniformly in time However non uniform samplingmay spoil the result (see [23] and references therein)

Simple cases of non ergodicity come from Hamiltonian systemswith the invariant Liouville measure First of all since the energy

is conserved the system is not ergodic if the number of degrees

of freedom is larger than one One has to restrict the ation to each energy surface More generally if there are otherindependent constants of the motion one should restrict oneself tolower dimensional manifolds For completely integrable systems,one is reduced to a constant flow on a torus which is ergodic if thefrequencies are incommensurable It is also known that genericHamiltonian systems are neither integrable nor ergodic (see [16])

It is natural to ask how fast is the convergence of the ergodic average

to its limit in Theorem 1 At this level of generality any kind of

veloc-ity above 1/n can occur Indeed Halasz and Krengel have proven the

following result (see [10] for a review)

Theorem 2 Consider a (measurable) automorphism T of the unit

in-terval Ω = [0, 1], leaving the Lebesgue measure dµ = dx invariant.

1 ) For any increasing diverging sequence a1, a2, · · ·, with a1 ≥ 2, and for any number α ∈]0, 1[, there is a measurable subset A ∈ Ω such that µ(A) = α, and





n1

n
−1 j=0

µ almost surely, for all n.

2 ) For any sequence b1, b2, · · ·, of positive numbers converging to zero, there is a measurable subset B ∈ Ω with µ(B) ∈]0, 1[ such that almost surely

In spite of this negative result, there is however an interesting andsomewhat surprising theorem by Ivanov dealing with a slightly differentquestion.

To formulate the result we first define the sequence of down-crossings

for a non-negative sequence (u n)n ∈N Let a and b be two numbers such that 0 < a < b For an integer k ≥ 0 such that u k ≤ a, we define the

first down crossing from b to a after k as the smallest integer n d > k (if

it exists) such that

1 ) u n d ≤ a,

2 ) There exists at least one integer k < j < n d such that u j ≥ b.

the number of successive down-crossings from b to a occurring before

time p for the sequence (u n), namely

N (a, b, p, (u n)) = sup{l | n l ≤ p}

Theorem 3 Let (Ω, A, T, µ) be a dynamical system Let f be a non negative observable with µ(f ) > 0 Let a and b be two positive real numbers such that 0 < a < µ(f ) < b, then for any integer r

r

.

We refer to [9], [5], [12] for proofs and extensions

In order to get some information on the rate of convergence in theergodic theorem, one has to make some hypothesis on the dynamicalsystem and on the observable

If one considers the numerator of the ergodic average, namely theergodic sum

S n (f )(x) =

n
−1 j=0

this can be considered as a sum of random variables, although in generalnot independent It is however natural to ask if there is somethingsimilar to the central limit theorem in probability theory To have such

a theorem, one has first to obtain the limiting variance Assuming for

simplicity that the average of f is zero, we are faced with the question

of convergence of the sequence



f2(x) dµ(x) + 2

n
−1 j=1

n − j n



f (x) f (T j (x)) dµ(x)

Here we restrict of course the discussion to observables which are square

integrable This sequence may diverge when n tends to infinity It may also tend to zero This is for example the case if f = u − u ◦ T with

u ∈ L2(dµ) Indeed, in that case S n = u − u ◦ T n is of order one in L2

and not of order √

n (see [10] for more details and references).

A quantity which occurs naturally from the above formula is the

auto-correlation function C f,f which is the sequence given by

A natural generalization of the auto-correlation is the cross

correla-tion between two square integrable observables f and g This funccorrela-tion

CχA ,χB (n) =



AχB ◦ T n dµ = µ(A) µ(T n (x) ∈ B|x ∈ A)

If for large time the system looses memory of its initial condition, it is

natural to expect that µ(T n (x) ∈ B|x ∈ A) converges to µ(B) This

leads to the definition of mixing We say that for a dynamical system

(Ω, A, T ) the T invariant measure µ is mixing if for any measurable

subsets A and B of Ω, we have

Often one does not have access to the points of phase space but only

to some fuzzy approximation For example if one uses a real apparatuswhich always has a finite precision There are several ways to formalizethis idea

The phase space Ω is a metric space with metric d For a given precision  > 0, two points at distance less than  are not distin-

If there is a given measure µ on the phase space it is often useful

to use partitions modulo sets of µ measure zero.

The notion of partition leads naturally to a coding of the dynamicalsystem This is a map Φ from Ω to{1, · · · , k}N given by

Φn (x) = l if T n (x) ∈ A l

If the map is invertible, one can also use a bilateral coding IfS denotes

the shift on sequences, it is easy to verify that Φ◦ T = S ◦ Φ In general

Φ(Ω) is a complicated subset of {1, · · · , k}N, i.e it is difficult to say

which codes are admissible There are however some examples of verynice codings like for Axiom A attractors (see [4] [17] and [21])

LetP and P  be two partitions, the partitionP ∨ P  is defined by

P ∨ P ={A ∩ B , A ∈ P , B ∈ P  }

IfP is a partition,

T −1 P = {T −1 (A) }

8

is also a partition Recall that (even in the non invertible case) T −1 (A) =

with  the partition into points In this case the coding is injective

(modulo sets of measure zero)

We now come to the definition of entropies There are two main tropies, the topological entropy and the so called metric or Kolmogorov-Sinai entropy Both measure how many different orbits one can observethrough a fuzzy observation

en-The topological entropy is defined independently of a measure Wewill only consider here the case of a metric phase space The topologicalentropy counts all the orbits modulo fuzziness We say that two orbits

of initial condition x and y respectively are  (the precision) different before time n (with respect to the metric d) if

The topological entropy is defined by

htop = lim

0lim supn →∞

1

n log N n ()

When an ergodic invariant measure µ is considered, the disadvantage

of the topological entropy is that it measures the total number of guishable) trajectories, including trajectories which have an anomalously

(distin-small probability to be chosen by µ It even often happens that these trajectories are much more numerous than the ones favored by µ The

metric or Kolmogorov-Sinai entropy is then more adequate

If P is a (measurable) partition, its entropy H µ(P) with respect to

the measure µ is defined by

H µ(P) = −

A ∈P

µ(A) log µ(A)

In communication theory one often uses the logarithm base 2

The entropy of the dynamical system with respect to the partition P

and the (invariant, ergodic, probability) measure µ is defined by

orbits for the measure µ.

Theorem 4 Let P be a finite generating partition For any  > 0 there

is an integer N () such that for any n > N () one can separate the atoms of the partition P n=∨ n −1

j=0 T −j P into two disjoint subsets

in G n) have almost the same measure, and of course their union givesalmost the total weight An immediate consequence is that

#G n  e n h µ (T ) ,

where #G n denotes the cardinality of the set G n This is similar to thewell known formula of Boltzmann in statistical mechanics relating theentropy to the logarithm of the (relevant) volume in phase space There

is also an obvious connection with the equivalence of ensembles Werefer to [14] and [15] for more information

As mentioned before, it often happens that #G n  # B n A simple

example is given by the Bernoulli shift on two symbols Let p ∈]0, 1[ with

p = 1/2 Consider the probability measure ν on {0, 1} with ν({0}) = p,

and the measure µ on Ω = {0, 1}N which is the infinite product of ν The distance between two elements x and y of Ω is defined by

d(x, y) = e − inf { i , x i =y i } .

It is easy to prove that the measure µ is invariant and ergodic for the

shift S on sequences Moreover, htop(S) = log 2 The partition

P =



{x0= 0}, {x0= 1}

is generating For the entropy one has h µ(S) = −p log p − q log q with

q = 1 − p However for n large we get (since p = 1/2)

#G n  e n h µ(S)  2 n  # B n

Another way to formulate the Shannon-McMillan-Breiman theorem

is to look at the measure of cylinder sets For a point x ∈ Ω, let C n (x)

be the atom of ∨ n −1

j=0 T −j P which contains x In other words, C n (x) is the set of y ∈ Ω such that for 0 ≤ j ≤ n − 1, T j (x) and T j (y) belong

to the same atom ofP (the trajectories are indistinguishable up to time

n −1 from the fuzzy observation defined by P) Then for µ almost every

x we have

h µ (T ) = − lim n →∞ 1

n log µ(C n (x))

A similar result holds using a metric instead of a partition It is due

to Brin and Katok, and uses the so called Bowen balls defined for x ∈ Ω,

the transformation T , δ > 0 and an integer n by

These are again the initial conditions leading to trajectories

indistin-guishable (at precision δ) from that of x up to time n − 1

Theorem 5 If µ is T ergodic, we have for µ almost any initial condition

h µ (T ) = lim

δ 0lim infn →∞ −1

n log µ(B(x, T, δ, n))

We refer to [3] for proofs and related results

A way to measure the entropy in coded systems was discovered byOrnstein and Weiss using return times to the first cylinder The resultwas motivated by the investigation of the asymptotic optimality of Ziv’scompression algorithms

Let q be a finite integer, and assume the phase space of a dynamical

system is a shift invariant subset of {1, · · · , q}N

As before, we denote the shift by S Let µ be an ergodic invariant

measure Let n be an integer and for x ∈ Ω, define R n (x) as the smallest integer such that the first n symbols of x and S R n (x) (x) are identical.

Theorem 6 For µ almost every x we have

lim

n →∞

1

n log R n (x) = h µ (T )

We refer to [19] for the proof

A metric version was recently obtained by Donarowicz and Weiss,using again Bowen balls, in the context of dynamical systems with a

metric phase space Let for δ > 0 and the integer n

[1] L.Barrera, J.Schmeling Sets of “non typical” points have full topological entropy

and full Hausdorff dimension Israel J Math 116, 29-70 (2000).

[2] G.D.Birkhoff Proof of the ergodic theorem Proc Nat Acad Sci USA, 17,

656-660 (1931).

[3] M.Brin, A.Katok On local entropy In Geometric dynamics Lecture Notes in

Math., 1007, 30-38 Springer, Berlin, 1983.

[4] R.Bowen Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms.

Lecture Notes in Mathematics 470 Springer-Verlag, Berlin Heidelberg New

York 1975.

[5] P.Collet, J.-P.Eckmann Oscillations of Observables in 1-Dimensional Lattice

Systems Math Phys Elec Journ 3, 1-19 (1997).

[6] M.Denker.The central limit theorem for dynamical systems In Dynamical

sys-tems and ergodic theory Banach Center Publ., 23, 33-62 PWN, Warsaw, 1989.

[7] T.Donarowicz, B.Weiss Entropy theorems along the time when x visits a set.

Illinois Journ Math To appear.

[8] P.Doukhan Mixing Properties and examples Lecture Notes in Statistics, 85.

Springer-Verlag, New York, 1994.

[9] V.Ivanov Geometric properties of monotone fluctuations and probabilities of

random fluctuations Siberian Math Journal 37, 102-129 (1996) Oscillation of means in the ergodic theorem Doklady Mathematics 53, 263-265 (1996).

[10] A.Kachurovskii The rate of convergence in the ergodic theorems Russian Math.

[15] O.Lanford Entropy and equilibrium states in classical statistical mechanics In

Statistical mechanics and Mathematical Problems Lecture Notes in Physics 20,

1-113, Springer-Verlag, Berlin 1973.

[16] L.Markus, K.Meyer Generic Hamiltonian dynamical systems are neither

inte-grable nor ergodic Memoirs of the American Mathematical Society, 144

Amer-ican Mathematical Society, Providence, R.I., 1974.

[17] M.Keane Ergodic theory and subshifts of finite type In Ergodic theory, bolic dynamics, and hyperbolic spaces, 35-70 Oxford Sci Publ., Oxford Univ.

sym-Press, New York, 1991.

[18] D.Ornstein Ergodic Theory, Randomness, and Dynamical Systems Yale

Uni-versity Press, New Haven London 1974.

[19] D.Ornstein, B.Weiss Entropy and data compression schemes stationary random

fields IEEE Trans Information Theory 39, 78-83 (1993).

[20] K.Petersen Ergodic Theory Cambridge University Press, Cambridge 1983 See

also http://www.math.unc.edu/Faculty/petersen.

[21] D.Ruelle Thermodynamic formalism Addison-Wesley, Reading, 1978.

[22] J.von-Neumann Proof of the quasi ergodic hypothesis Proc Nat Acad Sci.

AND TRANSPORT.

Maurice Courbage

Universit´ e Paris 7 - Denis Diderot L.P.T.M.C.

Tour 24-14.5` eme ´ etage 2, Place Jussieu ´

75251 Paris Cedex 05 - FRANCE

courbage@ccr.jussieu.fr

Abstract The purpose of this paper is to survey shortly some notions in the

spec-tral theory of ergodic dynamical systems and their relevance to mixing and weak mixing In addition, we present some dynamical systems of particles submitted to collisions with nondispersive obstacles and their ergodic and spectral properties Transport is formulated in terms of random walk generated by deterministic dynamical systems and their moments.

Transport in physical systems is mainly modelled by diffusion cesses For systems of noninteracting particles having ”chaotic” motion,diffusion appears in the long-time limit as a result of a random walk gen-erated by the deterministic dynamics with invariant probability measure.The machinery is an extension of the central limit theorem to determin-istic dynamical systems Such extensions goes back to works by Sinai[26] in 1960 It is generally beleived that diffusion is related to mixingproperties of the motion of the particles However, this question is notyet quite clear We shall discuss in the last section some recent results

pro-on central limit theorem and mixing properties

Spectral theory of Dynamical Systems provided powerful tools tostudy mixing and its rate In 1931, B.O Koopman pubished his paperentitled ” Hamiltonian systems and transformations in Hilbert space”

in which he showed how a measure-preserving transformation T induces

a unitary operator U on the Hilbert space of the measurable functions

15

Chaotic Dynamics and Transport in Classical and Quantum Systems, 15–33.

© 2005 Kluwer Academic Publishers Printed in the Netherlands

P Collet et al (eds.),

of L2 type [18] This led von Neumann to publish in 1932 a proof ofthe ergodic theorem in this Hilbert space (see reference in the paper of

P Collet in this volume [9]) Hopf, Koopman and von Neumann

stud-ied the relationship between mixing properties and the spectrum of U showing that for T , being weakly mixing is equivalent, for U , to hav-

ing continuous spectrum [16, 19] Later on, von Neumann and Halmosclassified completely the dynamical systems having purely discrete spec-trum All these results are presented in the Halmos book in 1956 [14]

In the sixties, many progresses were made in studying the stucture ofthe continuous spectrum of K-systems with works of Kolmogorov, Sinaiand Rokhlin [10] In the same period, Anzai introduced a new class ofdynamical systems, the class of skew products which have zero entrpoyand continuous spectrum Continuous spectrum has been found also inother examples of dynamical systems with zero entropy, namely intervalexchanges During the last 30 years, there were many spectral studies

in this kind of dynamical systems ( for a review, see [13]) On the otherhand, some results on the central limit theorem in weakly dependentprocesses and its relation to mixing have been presented, namely by I.A.Ibragimov [17] and P Billingsley [3], see also the review by A Liverani[21]

Dynamical Systems

We shall only consider DS with an invariant probability measure For

an ergodic system, as a result of the Birkhoff theorm (see [9]) applied

to the characteristic function 1A of a subset A, the invariant measuredescribes the frequency of visit of a typical trajectory to any given mea-surable set

Definition 2.1 Let X be a measurable space with a probability measure

µ and let A be the family of all measurable subsets A dynamical system

(DS) is an invertible measurable transformation T on a X , which is measure-preserving, that is, µ(T −1 A)) = µ(A) for any A ∈ A In what follows we shall denote a DS simply by (X, T, µ).

As T may have many invariant measures, the measure µ will be ified in each example The Koopman unitary operator associated to T

spec-is defined on the Hilbert space H = L2(X, µ) by :

16

In order to stress on the dependence of U with respect to T, we should

use the notation U T, but we shall omit it and use it only when necessary.The spectral properties of this operator are called the spectral prop-erties of the DS

The first important spectral property of a DS is a characterization of

its ergodicity Recall that (X, T, µ) is ergodic if any invariant measurable subset A (i.e T A = A, eventually modulo a subset of zero measure)

has either zero measure or full measure Spectral characterization ofergodicity is given by:

Theorem 2.2 (X, T, µ) is ergodic if and only if the only eigenfunction

of U for the eigenvalue 1 is the constant function.

The famous elementary example in this respect is the rotation of

the circle Let X = S1 be the circle of length 1, identified with the

interval [0, 1[, let α ∈ [0, 1[, T (x) = x + α and µ = µ L the Lebesgue

measure, then the orthonormal basis of L2(X, µ) given by: ϕ n (x) =

exp(2inπx), n ∈ Z, is a family of eigenfunctions of U for the eigenvalues

λ n = exp(2iπnα) The above theorem implies that T is ergodic if and

only if α is an irrational number.

Definition 2.3 T is said to have a discrete spectrum if U has a complete

orthonormal family of eigenfunctions.

The spectrum of U was very important since the early stage of the

Ergodic Theory when Halmos and von Neumann proved that two

er-godic systems, (X, T, µ) and (Y, S, ν), both having discrete spectrum,

are isomorphic if and only if U T and U S have the same set of eigenvalues.This is no more true when one of these systems has not a purely discretespectrum

Let us call, in this case, the discrete spectrum subspace of U ,Hd,

the subspace generated by all the eigenfunctions of U , this means that

the orthogonal complement of this subspace,H⊥

d, called the continuous

spectrum subspace, Hc, is not reduced to zero A useful quantity inorder to characterize the continuous spectrum is given by the spectral

measure of a function f One needs here to use the notion of Fourier coefficients of any measure µ on the circle S1 defined by:

se-is the case for the sequence < U n f, f > called the auto-correlation efficients of f , where < f, g > is the scalar product on H = L2(X, µ) Thus, we associate to each f of H = L2(X, µ) a unique spectral measure

co-σ f on the circle S1 such that:

< U n f, f >=

 1 0

exp(2iπnλ)dσ f (λ) = ˆ σ f (n) (2.3)

In other words, these autocorrelations coefficients of f are the Fourier coefficients of the spectral measure σ f It is possible to show that a

function f ∈ H c if and only if σ f (λ) = 0, ∀λ Then, the measure σ f is

said to be continuous.The Wiener Lemma asserts that:

Cesaro-measure of f A sufficient condition for the continuity of the spectral

measure is the decay of the auto-correlation coefficients ( for, the vergence to zero of a numerical sequence implies the convergence to zero

con-of its Cesaro-mean), but it is not necessary The relation con-of the nuity of the spectral measure to the behavior of the auto-correlations

conti-coefficients of f can be further characterized and we refer to [10] A continuous measure σ f is said to be ”absolutely continuous with re-

spect to the Lebesgue measure” if it has a density with respect to

the Lebesgue measure, dσ f (λ) = ρ(λ)dλ, where:

It follows that σ f is the Lebegue measure (i.e ρ = 1) if ˆ σ f (n) = 0, ∀n =

0 If an absolutely continuous measure has density ρ(λ) which can

van-ish only on a subset of zero Lebesgue measure, then it is said to be

equivalent to the Lebesgue measure This is the case, for instance,

if the Fourier coefficients ˆσ f (n) decay exponentially rapidly as |n| → ∞.

In fact, the density ρ(λ) given above as an entire series of z = exp(2iπλ)

will be uniformly convergent, so that it has analytic continuation in some

annulus around the unit circle Therefore ρ(λ) has only finite number of

zeros

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Singular continuous measures σ f are such that one can find a set

of zero Lebesgue measure which is of measure 1 for σ f and vice versa.They are often obtained in DS with zero entropy

Definition 2.4 A DS is mixing if for any A, B ∈ A,

We have the following hierarchy of ergodic properties:

Mixing⇒ Weak mixing ⇒ Ergodicity

Mixing is equivalent to the decay of correlations of two functions f and

is one-to-one and this sequence of random variables is independent Thebest known example of such system is the baker tranformation, for whichthe partition P is the division of the unit rectangle into left and right

halves Then, the system is mixing and the entropy of the system is equal

to the entropy of the Bernoulli shift (for this notion see [9]) Kolmogorovintroduced an intermediate class of transformations, the so-called K-systems which are in between mixing and Bernoulli systems A con-

venient definition is given in terms of the entropy H µ (T, P) Although

the entropy was introduced by Kolmogorov as an invariant quantity served by isomorphisms of DS, it can be understood as the measure thenon-predictability of the system under observations of the partititionP

pre-[22] The K-systems are characterized as those for which this entropy

is strictly positive for any finite partition P As we mentioned above,

Bernoulli systems satisfy the K-property Thus, we have the followinghierarchy:

Bernoulli ⇒ K- property ⇒ Mixing ⇒ Weak mixing ⇒ Ergodicity

The spectral properties of K-systems are all the same, so we can saythat all K-systems are spectrally isomorphic In particular, there is nospectral difference between those K-systems that are Bernoulli and thosewhich are not We have the following theorem:

Theorem 2.6 Let (X, T, µ) be a K-system. Then, U has countable Lebesgue spectrum as an operator restricted to the orthocomplement of the constant function, {1} ⊥ .

The theorem means that{1} ⊥ decomposes into a countable direct sum

of orthogonal invariant subspaces ⊕H i , i ∈ Z such that U is reduced

in each subspace Hi to shift operator acting on an orthonormal basis

{e i,j , j ∈ Z}: Ue i,j = e i,j+1 As a consequence, the spectral measure

σ e i,j is the Lebesgue measure This follows from the computation ofthe Fourier coefficients: ˆσ e i,j (k) are all zero for any i = j and they are

identical to the Fourier coefficients of the Lebesgue measure on the circle

This is a rich family of systems including nondispersive billiards andaperiodic transformations which display divergence of trajectories with

a power-law rate and some mildly unpredictabililty in the sense of thelinear prediction theory (see references in [12]) among them we shallconsider mainly examples of skew products of dynamical systems Wefirst define the general concept of skew products of DS

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Definition 3.1 Let (X, τ, µ) be a DS, (Y, ν) be a probability measure

space and for µ-almost all x ∈ X, let S x be a ν-measure-preserving transformations on Y The transformation T acting on the product space

X × Y defined by :

T (x, y) = (τ (x), S x (y)) (3.10)

preserves the product measure µ × ν The DS (T, X × Y, µ × ν) is called

a skew product of transformations.

A particularly important case which occurs in quasi-periodic grable systems, is the one where Y is a compact group G endowed with

inte-the Haar measure m ( a kind of ”Lebesgue measure” on G) and where

S x is given by a ”cocycle”, that is, a function φ : x ∈ X → G and defined

by : g ∈ G → S x (g) = φ(x).g The skew product T : X × G → X × G

defined by:

T (x, g) = (τ (x), φ(x).g) (3.11)

is called a group extension of τ

It is possible to study the ergodic properties of T in terms of the erties of τ and φ An important property proved by Abramov-Rokhlin

prop-is that the metric entropy prop-is zero if τ has zero entropy (see [23]) Here

we shall consider two examples of group extensions of the circle rotation

on X = S1, with τ (x) = x+α, and µ the Lebesgue measure on the circle.

I First Example : G = {±1}

We start in showing a relation between these transformations and theplate billiard model with slit introduced by Zaslavsky [27] A particlemoves on an infinite plane among periodically distributed plate obstacles

of length β < 1 separated by slits, the spatial period in both (x,y) directions is 1 (see fig.1, in the case β = 1/2 ).

The particle reflects elastically at each collision with the obstacle Let

θ be the absolute value of the angle between the outgoing velocity vector

and the obstacle The particle will move either always to the right oralways to the left undergoing reflections, according to the value of the

angle θ Let us suppose that 0 < θ < π2 At time t=0, the coordinates

of the particle are given by its position (x0, 0) and the direction of its

ingoing velocity The position of the particle will increase between time

t=0 and time t=1 by α along x-direction and by ±1 along y-direction

where α = 1/tg(θ) The trajectory of the particle will be specified, at times t=n, by its x-coordinate x nand the projection of the ingoing veloc-

ity vector along the y-direction,  n, which will change under a collision

Thus,  n=±1 according as the velocity is upward or downward We call

Figure 1. Trajectory of particle in the plate billiard with slits

 n the ”veolcity direction of the particle” Let us introduce the ”cocycle

The transformation T maps the state of the particle (x, ) at time

t = 0 into the state of the particle at time t=1 according to the formula:

As well known, the space H = L2(S1 × G, µ × m) is a tensor product

L2(S1, µ) ⊗ L2(G, m) It decomposes, by using the orthonormal basis

{1, ε} of L2(G, m), into a direct sum H = H1⊕ H2 where :

H1={g(x) ∈ L2(S1, µ) } (3.14)

H2 ={εh(y), h ∈ L2(S1, µ) } (3.15)

In other words, for any f ∈ L2(S1, µ) ⊗ L2(G, m), there are g, h ∈

L2(S1, µ) such that : f (x, ε) = g(x) + εh(x) The action of U restricted

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to H1 (resp H2) is given by: U g(x) = g(x + α) (resp U (εh(x)) =

εχ(x)h(x + α)) It shows thatH1 and H2 are invariant Therefore, thespectrum of U restricted to H1 is reduced to the spectrum of the theKoopman operator of the rotation onH1 The spectral measure associ-ated to any function from H2 is given through the Fourier coefficients:

< U n (εh), εh) > It is convenient to introduce the unitary operator V χ

on L2(S1, µ L) given by:

Now, using the relation U n (εh) = εV χ n (h) we obtain:

< U n (εh), εh) >=< εV χ n (h), εh) >=< V χ n (h), h) >

which means that the spectrum of U restricted to H2 is reduced to the

spectrum of V χ on L2(S1, µ) The spectrum of this operator have been

studied first in the case β = 1/2 and later in more general case In the first case, it is shown that for any irrational α, V χ has no eigenfunction

in L2(S1, µ), which means that U restricted to H2 has only continuousspectrum This immediately implies that the system is ergodic In

the general case, the same result holds for some class of α and β with

diophantine properties The spectrum is singularly continuous for every

irrational α and almost all β We refer to [10, 24] for more informations

on the ergodic and spectral properties of these transformations

This result holds also for the absolute value While the distance traveled

by the particle along the x-direction is alway x n = nα, the distance

traveled by the particle along y-direction depend on the initial condition

(x, ε) and seems as a random walk This distance, at t=n, is :

The mean value of S nover the initial conditions is zero and the problem

is to estimate the asymptotic behavior of the variance of S n We shallcome back to this question in the next section

II.Second example: G = S1

A particle moves on an infinite plane among periodically distributed

obstacles with spatial period equal to 1 along both (q1, q2)-directions In

the q1-direction, the motion of the particle is uniformly accelerated at

each regular time interval by an amount α and has uniform free motion along the q2-direction That is, define the velocity p1(n) = q1(n + 1) −

q1(n) , then the equations of the projection of the motion in the q1direction are :

q1(n) = q1(0) + np1(0) + n(n − 1)α/2 (3.23)The particle is moreover submitted at the begining of each time interval

to a deterministic ”collision” changing the direction of the motion motion

up and down along q2-direction in the following way: the projection of

the velocity of the particle p2 at time t = n is given by χ(q1(n)) where

χ(x) is a periodic discontinuous function defined by: