Ergodic Theory — Introductory Lectures

208 Pages·1975·1.639 MB·English

by  Peter Walters (auth.)

Lecture Notes ni
Mathematics
Edited yb .A Dold and .B Eckmann
Series: of Department ,scitamehtaM of University ,dnalyraM
College Park
Adviser: .L Greenberg
458
Waiters Peter
Ergodic Theory-
Introductory Lectures
I. I I
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grebledieH.nilreB weN kroY 5791
Dr. Peter Waiters
Mathematics Institute
University of Warwick
Coventry/England
Library of Congress Cataloging in Publication Data
Walters, Peter, 1943-
Ergodic theory.
(Lecture notes in mathematics &58) ;
Bibliography: p.
Includes index.
.1 Ergodic theory. I. Title. II. Series:
Lectures notes in mathematics (Berlin) ; .85.4/
QA3.L28 no. ~58 ~QA313~ 510'.8s r515'.&2~ 75-9853
ISBN 0-387-07163-6
AMS Subject Classifications (1970): 28A65
ISBN 3-540-07163-6 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-07163-6 Springer-Verlag New York • Heidelberg • Berlin
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Preface
These are notes of a one-semester introductory course on Ergodic
Theory that I gave at the University of Maryland in College Park
during the fall of 1970. I assumed the audience had no previous
knowledge of Ergodic Theory. My aim was to present some of the basic
facts in measure theoretic Ergodic Theory and Topological Dynamics and
show how they are related so that the audience would have the founda-
tions to read the research papers if they wished to pursue the subject
further.
At the beginning of Chapter 1 I give a list of examples of
measure-preserving transformations and at the end of each section of
Chapter 1 I investigate whether these examples have the properties
discussed in that section. These examples were chosen because of
their varied properties and importance in the subject. Similarly in
Chapter 5, on Topological Dynamics, a list of examples is given and
the properties discussed in that chapter are considered for these
examples.
I tried to deal with entropy as simply as possible. In the dis-
cussion of entropy I have inserted without proof some of the more
difficult theorems when I thought they wePe relevant to the discussio~
In particular I have discussed the recent deep results of D. .S
Ornstein on Bernoulli automorphisms and Kolmogorov automorphisms.
In the final chapter I have presented the new treatment of topo-
logical entropy due to R. E. Bowen. One of the beauties of this
treatment is that topological entropy can be defined for a uniformly
continuous map of any metric space and that its value remains un-
changed under certain types of covering maps. This enables one to
give an elegant calculation of the topological and (Haar) measure
VI
theoretic entropies of affine transformations of finite-dimensional
tori.
Since these notes have not been fully edited many references are
missing and it is likely that credit is often not given where it is
due. The theorems and definitions are numbered independently, but a
corollary is given the same number as the theorem to which it is a
corollary.
Thanks are due to Victor Charles Stasio and Suellen Eslinger who
took notes of the course and also to Allan Jaworski for editing and
compiling the bibliography. Special thanks are due to Betty Vander-
slice for her superb typing.
--Peter Waiters
Contents
Chapter :O Preliminaries
§i. Introduction 1
§2. Measure Theory 3
§3. Hilbert Spaces 8
§4. 'Haar Measure 9
§5. Character Theory i0
§6. Endomorphisms of Tori 12
Chapter :i Measure-Preserving Transformations 16
§i. Examples 16
§2. Problems in Measure Theoretic Ergodic Theory 19
§3. Recurrence 20
§4. Ergodicity 21
§5. The Ergodic Theorem 29
§6. Mixing 37
Chapter :2 Isomorphism and Spectral !nvariants 51
§i. Isomorphism of Measure-Preserving Transformations 51
§2. Conjugacy of Measure-Preserving Transformations 53
§3. Spectral Isomorphism 54
§4. Spectral Invariants 75
§5. Examples 59
Chapter :3 Measure-Preserving Transformations 63
with Pure Point Spectrum
§i. Eigenfunctions 63
§2. Pure Point Spectrum 64
§3. Group Rotations 67
Chapter :4 Entropy 70
§i. Partitions and Subalgebras 70
§2. Entropy 72
§3. Conditional Entropy 76
§4. Properties of h(T,A) 80
§S. Properties of h(T) 83
§6. Examples 92
§7. How good an invariant is entropy? 96
§8. Bernoulli and Kolmogorov Automorphisms 98
§9. Pinsker Algebra i07
.Ol§ Sequence Entropy 108
§ll. Comments 109
§12. Non-invertible Transformations ii0
lV
Chapter :5 Topologica! Dynamics 112
§0. Introduction 112
§i. Minimality 113
§2. Topological Transitivity i17
§3. Topological Conjugacy and Discrete Spectrum 122
§4. Invariant Measures for Homeomorphisms 128
Chapter :6 Topological Entropy 140
§i. Definition by Open Covers 140
§2. Bowen's Definition 146
§3. Connections with Measure Theoretic Entropy 155
§4. Topological Entropy of Linear Maps
and Total Affines 160
§5. Expansive Homeomorphisms 168
§6. Examples 182
Bibliography 185
Index 197
Chapter 0: Preliminaries
31. Introduction
Generally speaking, ergodic theory is the study of transforma-
tions and flows from the point of view of recurrence properties,
mixing properties, and other global, dynamical properties connected
with asymptotic behavior. Abstractly,one has a space X and a
transformation T of X (or a family {Tt: t ~ ~} of transforma-
tions of X) with some structure on X which is preserved by T
(or by {Tt}). The nature of most of the work so far can be cate-
gorized into one of the four following types:
(i) measure theoretic:
Here one deals with a measure space X and a measure pre-
serving transformation T: X ~ X.
(2) topological:
Here X is a topological space and T: X ~ X is a continuous
map.
(3) mixture of (i) and (2):
In this situation X is a topological space equipped with a
measure m on its Borel sets while T: X ~ X is continuous and
preserves m.
(4) smooth:
One considers a smooth manifold X and a smooth map
T: X ~ X.
We shall deal with some topics from (i), (2), and (3).
To see how this study arose consider, for example, a system of
k particles in 3-space moving under known forces. Suppose that the
phase of the system at a given time is completely determined by the
positions and the momenta of each of the k particles. Thus, at a
given time the system is determined by a point in 6k-dimensional
space. As time continues the phase of the system alters according to
the differential equations governing the motion, e.g., Hamilton's
equations
dqi ~H dPi ~H
dt ~Pi dt aqi
If we are given an initial condition and such equations can be unique-
ly solved then the corresponding solution gives us the entire history
of the system, which is determined by a curve in phase space.
If x is a point in phase space representing the system at a
time t0, let Tt(x) denote the point of phase space representing
the system at time t+t .0 From this we see that t T is a transforma-
tion of phase space and, moreover, O = T id., Tt+ s = TtOT .s Thus
{Tt: t E ~} is a one-parameter group of transformations of phase
space. In dynamics one is interested in the asymptotic properties of
the family {Tt}. It seems reasonable to study the system at discrete
times 0, t 2t0, 3t 0 ..... i.e., study the family {T~ }~-i ' since we
0 -
expect the properties of {T t} to be reflected in those of {T~ .}
0
For this reason, as well as the fact that it is simpler, one
studies individual transformations and their iterates. One is par-
ticularly interested in the flow on an energy surface, which is some-
times smooth (hence considerations of type (4) arise), and sometimes
is not (one then investigates along the lines of (2) .) Measure
theory enters the picture via a theorem of Liouville which tells us
that if the forces are of a certain type one can choose coordinates
in phase space so that the usual 6k-dimensional measure in these
coordinates is preserved by each transformation T t.
Around 1900 Gibbs suggested using the measure-theoretic approach
in mechanics because of the difficulty of solving the equations of
motion and also because this deterministic approach does not answer
several important questions in mechanics. In discovering statistical
mechanics Gibbs suggested looking at what happens to subsets of phase
space. For example, if A and B are subsets of phase space what
is the probability that the system is in B at the time t given
that the system is in A at the time t0? Given that the system be-
gins in A at time 0 t what is the average time the system spends
in B? Such questions motivate the type of study undertaken in
ergodic theory.
We now list some general references for the material we shall
discuss:
For topics of types (I) and (3) mentioned above see Halmos [2],
Billingsley [i], Hopf [i], Jacobs [i] [2], Parry [3], Rohlin [3][4][5],
Friedman [i], Shields [2]. In addition to the Shields notes, further
details on the results of Ornstein described in Chapter 4 may be found
in a forthcoming book by Friedman and Ornstein [2].
For material of type (2) see Gottschalk and Hedlund [I],
Nemytskii and Stepanov [I], and Ellis [i].
And material of type (4) may be found in Avez and Arnold [i],
Smale [i], Abraham [i], Abraham and Robbin [i], Nitecki [i].
Khinchin [i] provides a good sketch of the foundations of ergodic
~heory. For extensive bibliographies see Jacobs [i][2]~ Gottschalk
[i], and Smale [i]. A recent survey is Mackey [i].
~2. Measure Theory
General reference - Halmos [i].
We recall some fundamental notions from measure theory.
Let X be a set. A ~-al~ebra of subsets of X is a collection
8 of subsets of X satisfying:
(i) X E 8, (2) B E 8 = X\B 6 B,
(3) n B E B, n > i = 0 n B ~ B.
n=l
We then call (X,B) a measurable space. A measure space is a
triDle (X,B,m) where X is a set, B is a q-algebra of subsets
of X, and m is a function m: B ~ + R satisfying
m( 0 B n) = ~ m(B n)
n:l n:l
if {B n} is a pairwise disjoint sequence of elements of B. We say
that (X,B,m) is a probability space, or a normalized measure space,
if m(X) = i. We shall usually deal with such spaces.
A collection A of subsets of a set X is an alsebra if:
(i) X E A, (2) A E A = X\A 6 A,
n
(3) I A .... ,A n ~ A = ~ A. E A.
i=l l
When one is trying to equip a measurable space (X,B) with a
measure one usually knows what the measure should be on an algebra
A & B, and so, one would like to know when this function defined
on A can be extended to a measure on S. The following result
deals with this situation.
Hahn-Kolmogorov Extension Theorem:
Given a set X, an algebra A of subsets of X, let m: A ~ + R
be a function satisfying
m(X) = i, m(~JA n) : ~-m(A n)
n n
whenever n A E A V n, ~JA n E A, and the {A } n are disjoint. Then
n
there is a unique probability measure m defined on the q-algebra
generated by A such that m(A) = m(A) whenever A 6 A.