An Outline of Ergodic Theory

An Outline of Ergodic Theory

by Steven Kalikow, Randall McCutcheon
     
 

ISBN-10: 0521194407

ISBN-13: 9780521194402

Pub. Date: 04/30/2010

Publisher: Cambridge University Press

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

Overview

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.

Product Details

ISBN-13:
9780521194402
Publisher:
Cambridge University Press
Publication date:
04/30/2010
Series:
Cambridge Studies in Advanced Mathematics Series, #122
Edition description:
New Edition
Pages:
182
Product dimensions:
6.10(w) x 9.00(h) x 0.60(d)

Table of Contents

Preface vii

Introduction 1

1 Measure-theoretic preliminaries 5

1.1 Basic definitions 5

1.2 Carathéodory's theorem, isomorphism, Lebesgue spaces 7

1.3 Properties of Lebesgue spaces, factors 11

1.4 Random variables, integration, (stationary) processes 16

1.5 Conditional expectation 23

2 Measure-preserving systems, stationary processes 26

2.1 Systems and homomorphisms 26

2.2 Constructing measure-preserving transformations 27

2.3 Types of processes; ergodic, independent and (P, T) 30

2.4 Rohlin tower theorem 32

2.5 Countable generator theorem 37

2.6 Birkhoff ergodic theorem and the strong law 39

2.7 Measure from a monkey sequence 43

2.8 Ergodic decomposition 45

2.9 Ergodic theory on L2 48

2.10 Conditional expectation of a measure 50

2.11 Subsequential limits, extended monkey method 52

3 Martingales and coupling 55

3.1 Martingales 55

3.2 Coupling; the basics 59

3.3 Applications of coupling 62

3.4 The dbar and variation distances 67

3.5 Preparation for the Shannon-Macmillan-Breiman theorem 69

4 Entropy 72

4.1 The 3-shift is not a factor of the 2-shift 72

4.2 The Shannon-McMillan-Breiman theorem 74

4.3 Entropy of a stationary process 78

4.4 An abstraction: partitions of 1 79

4.5 Measurable partitions, entropy of measure-preserving systems 82

4.6 Krieger finite generator theorem 85

4.7 The induced transformation and fbar 92

5 Bernoulli transformations 96

5.1 The cast of characters 96

5.2 FB ⊂ IC 99

5.3 IC ⊂ EX 102

5.4 FD ⊂ IC 112

5.5 EX ⊂ VWB 115

5.6 EX ⊂ FD 121

5.7 VWB ⊂ IC 122

6 Ornstein isomorphism theorem 124

6.1 Copying in distribution 124

6.2 Coding 128

6.3 Capturing entropy: preparation 129

6.4 Tweaking a copy to get a better copy 135

6.5 Sinai's theorem 139

6.6 Ornstein isomorphism theorem 143

7 Varieties of mixing 146

7.1 The varieties of mixing 146

7.2 Ergodicity vs. weak mixing 147

7.3 Weak mixing vs. mild mixing 150

7.4 Mild mixing vs. strong mixing 153

7.5 Strong mixing vs. Kolmogorov 155

7.6 Kolmogorov vs. Bernoulli 162

Appendix 167

References 170

Index 173

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