An Outline of Ergodic Theory

by Steven Kalikow , Randall McCutcheon

• ISBN13: 9780521194402
• ISBN10: 0521194407
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-04-26
• Publisher: Cambridge University Press

Summary

This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon-McMillan-Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.

Table of Contents

Preface	p. vii
Introduction	p. 1
Measure-theoretic preliminaries	p. 5
Basic definitions	p. 5
Carathéodory's theorem, isomorphism, Lebesgue spaces	p. 7
Properties of Lebesgue spaces, factors	p. 11
Random variables, integration, (stationary) processes	p. 16
Conditional expectation	p. 23
Measure-preserving systems, stationary processes	p. 26
Systems and homomorphisms	p. 26
Constructing measure-preserving transformations	p. 27
Types of processes; ergodic, independent and (P, T)	p. 30
Rohlin tower theorem	p. 32
Countable generator theorem	p. 37
Birkhoff ergodic theorem and the strong law	p. 39
Measure from a monkey sequence	p. 43
Ergodic decomposition	p. 45
Ergodic theory on L2	p. 48
Conditional expectation of a measure	p. 50
Subsequential limits, extended monkey method	p. 52
Martingales and coupling	p. 55
Martingales	p. 55
Coupling; the basics	p. 59
Applications of coupling	p. 62
The dbar and variation distances	p. 67
Preparation for the Shannon-Macmillan-Breiman theorem	p. 69
Entropy	p. 72
The 3-shift is not a factor of the 2-shift	p. 72
The Shannon-McMillan-Breiman theorem	p. 74
Entropy of a stationary process	p. 78
An abstraction: partitions of 1	p. 79
Measurable partitions, entropy of measure-preserving systems	p. 82
Krieger finite generator theorem	p. 85
The induced transformation and fbar	p. 92
Bernoulli transformations	p. 96
The cast of characters	p. 96
FB ⊂ IC	p. 99
IC ⊂ EX	p. 102
FD ⊂ IC	p. 112
EX ⊂ VWB	p. 115
EX ⊂ FD	p. 121
VWB ⊂ IC	p. 122
Ornstein isomorphism theorem	p. 124
Copying in distribution	p. 124
Coding	p. 128
Capturing entropy: preparation	p. 129
Tweaking a copy to get a better copy	p. 135
Sinai's theorem	p. 139
Ornstein isomorphism theorem	p. 143
Varieties of mixing	p. 146
The varieties of mixing	p. 146
Ergodicity vs. weak mixing	p. 147
Weak mixing vs. mild mixing	p. 150
Mild mixing vs. strong mixing	p. 153
Strong mixing vs. Kolmogorov	p. 155
Kolmogorov vs. Bernoulli	p. 162
Appendix	p. 167
References	p. 170
Index	p. 173
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