Springer chaotic dynamics and transport in classical and quantum systems (2005) YYePG lotb

Chaotic Dynamics and Transport in Classical and
Quantum Systems


NATO Science Series
A Series presenting the results of scientific meetings supported under the NATO Science
Programme.
The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction
with the NATO Scientific Affairs Division
Sub-Series
I.
II.
III.
IV.
V.

Life and Behavioural Sciences
Mathematics, Physics and Chemistry
Computer and Systems Science
Earth and Environmental Sciences
Science and Technology Policy

IOS Press
Kluwer Academic Publishers
IOS Press
Kluwer Academic Publishers
IOS Press

The NATO Science Series continues the series of books published formerly as the NATO ASI Series.

The NATO Science Programme offers support for collaboration in civil science between scientists of
countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported
are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of
meeting are supported from time to time. The NATO Science Series collects together the results of
these meetings. The meetings are co-organized bij scientists from NATO countries and scientists from
NATO’s Partner countries – countries of the CIS and Central and Eastern Europe.
Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances
in a field.
Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and
identification of directions for future action.
As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science
Series has been re-organised and there are currently Five Sub-series as noted above. Please consult
the following web sites for information on previous volumes published in the Series, as well as details of
earlier Sub-series.
/>

/>
Series II: Mathematics, Physics and Chemistry – Vol. 182


Chaotic Dynamics and Transport in
Classical and Quantum Systems
edited by

P. Collet
Ecole Polytechnique,
Paris, France

M. Courbage
Université Paris 7-Denis Diderot,

France

S. Métens
Université Paris 7-Denis Diderot,
France

A. Neishtadt
Space Research Institute,
Moscow, Russia
and

G. Zaslavsky
New-York University,
New York, NY, U.S.A.

KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW


eBook ISBN:
Print ISBN:

1-4020-2947-0
1-4020-2946-2

©2005 Springer Science + Business Media, Inc.
Print ©2005 Kluwer Academic Publishers
Dordrecht
All rights reserved
No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,

mechanical, recording, or otherwise, without written consent from the Publisher
Created in the United States of America
Visit Springer's eBookstore at:
and the Springer Global Website Online at:





Contents
Preface

vii

Participants

ix

Part I : Theory
P. Collet : A SHORT ERGODIC THEORY REFRESHER

1

M. Courbage: NOTES ON SPECTRAL THEORY, MIXING AND TRANSPORT 15
Valentin Affraimovich, Lev Glebsky: COMPLEXITY, FRACTAL DIMENSIONS
AND TOPOLOGICAL ENTROPY IN DYNAMICAL SYSTEMS

35

G.M. Zaslavsky, V. Afraimovich: WORKING WITH COMPLEXITY

FUNCTIONS

73

Giovanni Gallavotti: SRB DISTRIBUTION FOR ANOSOV MAPS

87

P. Gaspard : DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY

107

Walter T. Strunz: ASPECTS OF OPEN QUANTUM SYSTEM DYNAMICS

159

Eli Shlizerman, Vered Rom Kedar: ENERGY SURFACES AND HIERARCHIES
OF BIFURCATIONS.
189
Monique Combescure: PHASE-SPACE SEMICLASSICAL ANALYSIS. AROUND
SEMICLASSICAL TRACE FORMULAE
225

Part II : Applications

Ariel Kaplan, Mikkel Andersen, Nir Friedman and Nir Davidson:
ATOM-OPTICS BILLIARDS

239

Fereydoon Family, C. Miguel Arizmendi, Hilda A. Larrondo: CONTROL OF
CHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS

269

v


vi
F. Bardou: FRACTAL TIME RANDOM WALK AND SUBRECOIL LASER
COOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITE
MEAN WAITING TIMES

281

Xavier Leoncini, Olivier Agullo, Sadruddin Benkadda, George M. Zaslavsky:
ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA
TURBULENCE

303

Edward Ott,Paul So,Ernest Barreto,Thomas Antonsen: THE ONSET OF
SYNCHRONISM IN GLOBALLY COUPLED ENSEMBLES OF CHAOTIC
AND PERIODIC DYNAMICAL UNITS

321

A. Iomin, G.M. Zaslavsky: QUANTUM BREAKING TIME FOR CHAOTIC
SYSTEMS WITH PHASE SPACE STRUCTURES

333

S.V.Prants: HAMILTONIAN CHAOS AND FRACTALS IN CAVITY QUANTUM
ELECTRODYNAMICS
349
M. Cencini,D. Vergni, A. Vulpiani: INERT AND REACTING TRANSPORT

365

Michael A. Zaks: ANOMALOUS TRANSPORT IN STEADY PLANE FLOWS OF
VISCOUS FLUIDS
401
J. Le Sommer, V. Zeitlin: TRACER TRANSPORT DURING THE GEOSTROPHIC
ADJUSTMENT IN THE EQUATORIAL OCEAN
413
Antonio Ponno: THE FERMI-PASTA-ULAM PROBLEM IN THE THERMODYNAMIC
LIMIT
431

Lectures

441


Preface

From the 18th to the 30th August 2003 , a NATO Advanced Study Institute
(ASI) was held in Cargèse, Corsica, France. Cargèse is a nice small village situated
by the mediterranean sea and the Institut d'Etudes Scientifiques de Cargese provides
∗

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,

∗

/>
vii


viii
models of dynamical evolutions of transport processes in classical and quantum
systems have been largly explained.
II. Applications
In this part, many specific applications in physical systems have been presented. It
concerns transport in fluids, plasmas and reacting media. On the other hand, new
experiments of cold optically trapped atoms and electrodynamics cavity have been
thoroughly presented. Finally, several papers bears on synchronism and control of
chaos.
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
f
and applied levels, and we
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 off the Summer-School, G. Zaslavsky was a
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


PARTICIPANTS
Valentin Afraimovich
IICO UASLP. Communication Optica. Universitet Autonomny de San Luis. Karakorum 1470, Iomas, 4eme section.
CP78200 San Luis Potosi. SLP Mexico

Omar Al Hammal
Institute “Carlos I” for Theoretical and Computational Physics. Universidad de Granada Avda. Fuente Nueva, s/n, 1
18071 Granada Spain

Athanasios Arvanitidis
Dept. of MathematicsAristotle University of Thessaloniki 54006 Thessaloniki, Geece

Francois Bardou
IPCMS-CNRS, 23, rue du Loess, BP 43 67037 Strasbourg Cedex 2 France

Maxim Barkov
Space Plasma Physics-Space Research InstituteProfsoyuznaya str. 84/32 117997 Moscow Russie

Jacopo Bellazzini
Universita’ di Pisa - Dipartimento Ingegneria AerospazialeVia Caruso 120 56100 Pisa Italie

Viatcheslav Belyi
Theoretical department-Russian Academy of Sciences IZMIRAN 142190 Troitsk, Russia

Cristel Chandre
CPT -- CNRS-Centre de Physique Théorique Campus de Luminy - case 907 13288 Marseille, France

Hugues Chaté
SPEC - CEA - Saclay, 91191 Gif-sur-Yvette, France

Guido Ciraolo
Centre de Physique Theorique,CNRS case 907 13288 Marseille cedex 9, France

Steliana Codreanu
Department of Theoretical Physics-Babes-Bolyai University Kogalniceanu str. 1 2400 CLUJ-NAPOCA Roumanie

Pierre Collet
Centre de physique theorique-CNRS Ecole Polytechnique, route de Saclay 91128 Palaiseau cedex France

Pieter Collins
Control and computation-CWIKruislaan 413 1098 SJ Amsterdam Nederland


ix


x

Monique Combescure
Institut de Physique Nucléaire de Lyon, CNRS-Bât Paul Dirac, 4 rue Enrico Fermi 69622 VILLEURBANNE France

Maurice Courbage
LPTMC case 7020-Université de Paris 7, 2 place Jussieu 75231 Paris Cedex 05 France

Giampaolo Cristadoro
Center for Nonlinear and Complex Systems and Dipartimento di Scienze Chimiche, Fisiche e MatematicheUniversita’ dell’Insubria (sede di Como)-via Valleggio,11 22100 Como Italie

Nir Davidson
Dept. of Physics of Complex Systems-Weizmann Institute of Science Rehovot 76100 Israel

Filippo De Lillo
Dipartimento di Fisica Generale-University of TorinoVia Giuria,1 10125 Torino Italie

Jean-Pierre Eckmann
Departement de Physique Theorique and Section de Mathematiques Universite de Geneve-32, Bld D’Yvoy 1211
Geneva 4 Suisse

Massimiliano Esposito
Service de Chimie Physique CP 231-Universite Libre de Bruxelles Boulevard du Triomphe B-1050 Bruxelles
Belgique

Fereydoon Family
Physics Department-Emory UniversityPhysics Department, Emory University GA Atlanta USA

Stefano Galatolo
Dipartimento di matematica applicata-Universita di PisaVia Bonanno Pisano 25b 56126 Pisa Italie

Govanni Gallavotti

Fisica -Univ. Roma 1P.le Moro 2 00185 Roma Italie

Pierre Gaspard
Center for Nonlinear Phenomena and Complex SystemsCampus Plaine, CP 231 B-1050 Brussels Belgique

Alessandro Giuliani
Physics department-Universita’ di Roma “La Sapienza”- Via Ivanoe Bonomi, 92, 00139 Roma, Italy

Vasiliy Govorukhin
Rostov State University - Computational mathematics Zorge str. 5 344090 Rostov-on-Don Russie



xi
Seiichiro Honjo
University of Tokyo, Graduate School of Arts and Sciences-Department of Basic Science, Kaneko LaboratoryKomaba
3-8-1, Meguro-ku 153-8902 Tokyo Japon

Alexander Iomin
Department of Physics, Technion 32000 Haifa Israel

Alexander Itin
Lab. 627 Space Research Institute-Profsoyuznaya str. 84/32 117997 Moscow Russia

Brunon Kaminski
Faculty of Mathematics and Informatics
Nicholas Copernicus University, ul Chopina 12/18, 87-100 Torun, POLAND

Janina Kotus
Warsaw Univ.of Technology-Department of MathematicsPlac Politechniki 1 00-661 Warsaw Pologne

Alexandra Landsman
Princeton University, 235 Thunder Circle PA 19020 Ben Salem USA

Jacques Laskar
Astronomie et Systemes Dynamiques-IMC77, Av. Denfert-Rochereau F-75014 PARIS France

Xavier Leoncini
PIIM-Université de Provence, Centre de St Jerome, case 321 13396 Marseille Cedex 20 France

Fabio Lepreti
Section of Astrophysics, Astronomy, and Mechanics-Department of Physics, Aristotle University of
ThessalonikiDepartment of Physics, Aristotle University of Thessaloniki 54124 Thessaloniki Grèce

Emanuel Lima
Instituto de Física de São Carlos-Universidade de São PauloMajor Júlio Salles 870, Vila Pureza, 13561-010 16 São
Carlos Brazil

Helen Makarenko
Department of Physics-V.Karazin National University4 Svobody square 61077 Kharkiv Ukraine

Amar Makhlouf
Université de Annaba-Labo de Mathématiques, 14 Rue Zighoud Youcef, DREAN 36 ELTARF, ALGERIE

Miguel Manna
Physique Mathematique et Theorique CNRS-UMR5825-Universite Montpellier 2Place Eugene Bataillon 34095
Montpellier, France

xii
Paul Manneville
Laboratoire d’Hydrodynamique-CNRS-Ecole PolytechniqueEcole polytechnique 91128 Palaiseau, France

Stéphane Métens
LPTMC, case 7020-Université Paris 7-Denis Diderot, 2 place Jussieu 75231 Paris Cedex 05, France

Stavros Muronidis
Dept. of MathematicsAristotle University of Thessaloniki 54006 Thessaloniki, Grèce

Stefano Musacchio
Dipartimento di Fisica Generalevia Pietro Giuria 1 10125 Torino, Italie

Anatoly Neishtadt
Space Research Institute -Russian Academy of Sciences, Profsoyuznaya 84/32 Moscow 117997, Russia

Tali Oliker
Technionyuvalim 56 20142 d.n. misgav, Israel

Edward Ott
I.R.E.A.P.-University of Maryland, City COLLEGE PARK 20742 MARYLAND, USA

Séverine Pache
ITP-EPFL, Ecublens 1015 Lausanne, Suisse

Antonio Politi
ISTITUTO NAZIONALE DI OTTICA APPLICATA, LARGO E. FERMI 6 50125 FIRENZE, Italy

Antonio Ponno
Dipartimento di Matematica-Universita’ di Milano, Via Saldini 50 20133 Milano, Italie

Serguei Prants
Institution Pacific Institute of the Russian Academy of Sciences 43 BALTIISKAYA St. 690041 VLADIVOSTOK
RUSSIA

Saar Rahav
Technion-Department of Physics 32000 Haifa Israel

Vered Rom-Kedar
Weizmann InstituteP.O. Box 26-Department of Computer science and applied mathematics 76100 Rehovot, Israel

Majid Saberi

LPTMC, case 7020-Université Paris 7-Denis Diderot, 2 place Jussieu 75231 Paris Cedex 05,
France
Marc Senneret
LPTMC, case 7020-Université de Paris 72 place Jussieu 75251 PARIS CEDEX 05 France



xiii
Michael Shlesinger
Office of Naval Research, 800 n°Quincy Str., Arlington, VA22217-5660, USA

Eli Shlizerman
Computer Science and Applied Mathematics - Dynamical Systems- Weizmann Institute of Science 76100 Rehovot
Israel

Dominique Simpelaere
Université Paris 6-Pierre et Marie Curie, Lab. De Probabilié, 4, Place Jussieu 75252 Paris cedex 05, France

Tom Solomon
Bucknell University Lewisburg, PA 17837, USA

George Stilogiannis
Dept. of Mathematics Aristotle University of Thessaloniki 54006 Thessaloniki, Grèce

Ion Stroe
Mechanics-POLITEHNICA University of Bucharest Splaiul Independentei 313 RO-77206 Bucharest ROMANIA

Walter Strunz
Physikalisches Institut, Universitat Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany

Vladimir Ten
Dept of Mathematics and Mechanics Moscow State University-MSU, Vorobevy gory 119899 Moscow Russia

Sandro Vaienti
University of Toulon and Centre de Physique TheoriqueCase 907 3288 Marseille Cedex 09 France

Jiri Vanicek
Jefferson Physical LaboratoryMSRI, 1000 Centennial Drive CA 94720 Berkeley USA

Alexei Vasiliev
Laboratory of Chaotic Dynamics Space Research Institute-Profsoyuznaya 84/32 117997 Moscow Russie

Sebastien Viscardy
Universite Libre de Bruxelles-Service de Chimie-PhysiqueCampus Plaine, CP 231 1050 Brussels Belgium

Thérèse Vivier
UMR5584 (CNRS) - Institut Mathématiques de Bourgogne BP47970 21078 Dijon Cedex France

Angelo Vulpiani
Dipartimento di Fisica, Universita di Roma La Sapienza P.le A. Moro 2 I-00185 Roma Italie

Piotr Waz
Center for Astronomy-Nicholaus Copernicus University, Gagarina 11 87-100 Torun Pologne



xiv
Tatiana Yelenina
Numerical Simulation of Electrodynamics and Magnetohydrodynamics-Keldysh Institute of Applied Mathematics of
RAS4, Miusskaya sq. 125047 Moscow Russie

L.S. Young
Courant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA

Michael Zaks
Humboldt University of Berlin-Dept of Stochastic Processes, Newtonstr. 15 D-12489 Berlin Allemagne

Georges Zaslavsky
Courant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA

Vladimir Zeitlin
Laboratoire de Meteorologie Dynamique, E N S, 24 Rue Lhomond 75231 PARIS CEDEX 05 France



A SHORT ERGODIC THEORY REFRESHER.
P. Collet

Centre de Physique Th´eorique
CNRS UMR 7644
Ecole Polytechnique
F-91128 Palaiseau Cedex (France)


Abstract

1.

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.

Introduction.

A system is characterized by the set Ω of all its possible states. At a
given time, all the properties of the system can be recovered from the
knowledge of the instantaneous state x ∈ Ω. The system is observed using the so called observables which are real valued functions on Ω. Most
often the space of states Ω is a metric space (so we can speak of nearby
states) 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 itself
producing 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 xn = T (xn−1 ). This is often written xn = T n (x0 ) with
T n = T ◦ T ◦ · · · ◦ T n-times.
Continuous time semi flow. This is a family (ϕt )t∈R+ of maps of
Ω satisfying
ϕ0 = Id ,
ϕs ◦ ϕt = ϕs+t .

1
P. Collet et al. (eds.),
Chaotic Dynamics and Transport in Classical and Quantum Systems, 1–14.
© 2005 Kluwer Academic Publishers. Printed in the Netherlands.


2
The dynamics can be given by a differential equation on a manifold
associated to a vector field F
dx
= F (x) .
dt
This is for example the case of a mechanical system in the Hamiltonian 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-autonomous
systems (in particular stochastically forced systems), systems with
memory, etc.
A dynamical system is a set of states Ω equipped with a time evolution. If a sigma-algebra is given on Ω, we will always assume that the
time evolution is measurable. This will often be a Borel sigma-algebra.
One would like to understand the effect of the time evolution and in
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. If χA denotes
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 is
given by
N
1
χ (T j (x0 )) .
(1)
AN (x0 , A) =
N + 1 j=0 A
It is natural to ask if this quantity has a limit when N tends to infinity.
The answer may of course depend on A and x0 , but we can already make
two important remarks. Assume the limit exists and denote it by µx0 (A).
First, it is easy to check that the limit also exists for T (x0 ) and for
any y ∈ T −1 (x0 ) = {z | T (z) = x0 }, and moreover
µT (x0 ) (A) = µy (A) = µx0 (A) .

(2)

Second, the limit also exists if A is replaced by T −1 (A) and is equal,
namely
(3)
µx0 (T −1 (A)) = µx0 (A) .


3

A short ergodic theory refresher.

If one assumes that µx0 does not depend on x0 at least for measurable
sets, 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. Unless otherwise stated, when speaking below of an invariant measure we
will assume it is a probability measure. We 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
µ is a measure on B invariant by T .
We can now come back to the question of the asymptotic limit of
the ergodic average (1). This is settled by the ergodic theorems. The
ergodic theorem of Von-Neumann [22] applies in an L2 context, while the
Birkhoff ergodic theorem [2] applies almost surely (we refer to [14], [20]
for proofs and extensions). We now state the Birkhoff ergodic theorem.
Theorem 1 Let (Ω, T, B, µ) be a dynamical system (recall that T is measurable for the sigma algebra B and the measure µ on B is T invariant).
Then for any f ∈ L1 (dµ)
AN (x, f ) =

N
1
f (T j (x)) .
N + 1 j=0

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 of

full measure by Birkhoff’s Theorem). Moreover, if we denote by
g(x) the limiting function (which exists for µ almost every x), it is
invariant, namely g(T (x)) = g(x).
The set of µ measure zero where nothing is claimed depends on f
and µ. One can often use a set independent of f (for example if
L1 (dµ) is separable). We will comment below on the dependence
on µ.
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 change
drastically the exceptional set of measure zero and this can lead
to completely different results.


4
The set of initial conditions where the limit does not exist, although 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 ∈
L1 (dµ)
N
1
f (T j (x)) = f dµ
(6)
lim
N →∞ N + 1
j=0
µ almost surely.
We now make some remarks about the definition of ergodicity and its
equivalent formulations.
In the condition (6), the limit exists µ almost surely by the Birkhoff
ergodic Theorem (1). It is enough to require that the limit is µ
almost surely constant since the constant has to be equal to the
integral.
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 a
set 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 different ones. It is easy to verify that they are disjoint. One can
find a set of measure one which is of measure zero for the other
and vice versa. This explains why the ergodic theorem applies to
both 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.

5

A short ergodic theory refresher.

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 time
average). It also holds of course for the map obtained by sampling
the semi flow uniformly in time. However non uniform sampling
may spoil the result (see [23] and references therein).
Simple cases of non ergodicity come from Hamiltonian systems
with 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 consideration to each energy surface. More generally if there are other
independent constants of the motion one should restrict oneself to
lower dimensional manifolds. For completely integrable systems,
one is reduced to a constant flow on a torus which is ergodic if the
frequencies are incommensurable. It is also known that generic
Hamiltonian systems are neither integrable nor ergodic (see [16]).

2.

Rate of convergence in the ergodic theorem.

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 velocity 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 interval 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
1 n−1
n j=0

χA ◦ T j − µ(A)

≤

an
n

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
lim

n→∞

1
nbn

n−1

χB ◦ T j − µ(B)
j=0

=∞.

6
In spite of this negative result, there is however an interesting and
somewhat surprising theorem by Ivanov dealing with a slightly different
question.
To formulate the result we first define the sequence of down-crossings
for a non-negative sequence (un )n∈N . Let a and b be two numbers such
that 0 < a < b. For an integer k ≥ 0 such that uk ≤ a, we define the
first down crossing from b to a after k as the smallest integer nd > k (if
it exists) such that
1 ) und ≤ a,
2 ) There exists at least one integer k < j < nd such that uj ≥ b.
Let now (nl ) be the sequence of successive down-crossings from a to b
(this sequence may be finite and even empty). We denote by N (a, b, p, (un ))
the number of successive down-crossings from b to a occurring before

time p for the sequence (un ), namely
N (a, b, p, (un )) = sup {l | nl ≤ 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
µ

x N a, b, ∞, f (T n (x))

>r

≤

a

b

r

.

We refer to [9], [5], [12] for proofs and extensions.
In order to get some information on the rate of convergence in the
ergodic theorem, one has to make some hypothesis on the dynamical
system and on the observable.
If one considers the numerator of the ergodic average, namely the
ergodic sum
n−1

f (T j (x))

Sn (f )(x) =

(7)

j=0

this can be considered as a sum of random variables, although in general
not independent. It is however natural to ask if there is something
similar 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
1
n

2

Sn (f )(x)

dµ(x)


7

A short ergodic theory refresher.
n−1
2

=

f (x) dµ(x) + 2
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
n
2
u ∈ L2 (dµ). Indeed,

√ in that case Sn = u − u ◦ T is of order one in L
and not of order n (see [10] for more details and references).
A quantity which occurs naturally from the above formula is the autocorrelation function Cf,f which is the sequence given by
Cf,f (j) =

f (x) f (T j (x)) dµ(x) .

(8)

If this sequence belongs to l1 , the limiting variance exists and is given
by
σf2 = Cf,f (0) + 2

∞

Cf,f (j) .
j=1

If moreover σf > 0, we say that the central limit theorem holds if
lim µ

n→∞

x

Sn (f )(x)
√
≤t
σf n

1
=√
2π

t
−∞

e−u

2 /2

du .

We emphasize that this kind of result has been only established for
certain classes of dynamical systems and observables. We refer to [6]
for a review. There are also results like Berry-Essen inequalities, law of
iterated logarithms, invariance principles, large deviations etc.
A natural generalization of the auto-correlation is the cross correlation between two square integrable observables f and g. This function
(sequence) is given by
Cf,g (j) =

f (x) g(T j (x)) dµ(x) −

f dµ

g dµ .

The second term appears in the general case when neither f nor g has
zero average. The case where f and g are characteristic functions is
particularly interesting. If f = χA and g = χB , we have

Cχ

A

χB (n) =

,

A

χB ◦ 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


8
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
lim µ(A ∩ T −n (B)) = µ(A)µ(B) .

n→∞

There are many other mixing conditions including various aspects and
velocity of convergence. We refer to [8] for details.

3.

Entropies.

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 apparatus
which always has a finite precision. There are several ways to formalize
this 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 distinguishable.
One gives a (measurable) partition P of the phase space, P =
{A1 , · · · , Ak } (k finite or not) , Aj ∩ Al = ∅ for j = l and
k

Ω=

Aj .
j=1

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 dynamical
system. This is a map Φ from Ω to {1, · · · , k}N given by
Φn (x) = l

if

T n (x) ∈ Al .

If the map is invertible, one can also use a bilateral coding. If S 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 very

nice codings like for Axiom A attractors (see [4] [17] and [21]).
Let P and P be two partitions, the partition P ∨ P is defined by
P ∨ P = {A ∩ B , A ∈ P , B ∈ P } .
If P is a partition,

T −1 P = {T −1 (A)}


9

A short ergodic theory refresher.

is also a partition. Recall that (even in the non invertible case) T −1 (A) =
{x , T (x) ∈ A}.
A partition P is said to be generating if (modulo sets of measure zero)
∞

T −n P =

n=0

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 entropies, the topological entropy and the so called metric or KolmogorovSinai entropy. Both measure how many different orbits one can observe
through a fuzzy observation.
The topological entropy is defined independently of a measure. We
will only consider here the case of a metric phase space. The topological
entropy 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

sup d(T k (x), T k (y)) > .
0≤k≤n

Let Nn ( ) be the maximum number of pairwise different orbits up to
time n. In other words this is the maximum number of pairwise different
films the dynamics can generate up to time n if two images differing at
most by are considered identical.
The topological entropy is defined by
htop = lim lim sup
0 n→∞

1
log Nn ( ) .
n

When an ergodic invariant measure µ is considered, the disadvantage
of the topological entropy is that it measures the total number of (distinguishable) trajectories, including trajectories which have an anomalously
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 Hµ (P) with respect to
the measure µ is defined by
Hµ (P) = −

µ(A) log µ(A) .
A∈P

In communication theory one often uses the logarithm base 2.