A First Course in Ergodic Theory

268 Pages·2021·6.117 MB·English

by  Karma Dajani, Charlene Kalle

A First Course in
Ergodic Theory

A First Course in
Ergodic Theory
Karma Dajani
Utrecht University
Charlene Kalle
Leiden University
First edition published 2021
by CRC Press
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Library of Congress Cataloging-in-Publication Data
ISBN: 9780367226206 (hbk)
ISBN: 9781032021843 (pbk)
ISBN: 9780429276019 (ebk)
Typeset in LMR12
by KnowledgeWorks Global Ltd.
To Annabel, Isaac, Noam, and Rafael

Contents
Preface xi
Author Bios xiii
Chapter 1(cid:4) Measure Preservingness and Basic Examples 1
1.1 WHATISERGODICTHEORY? 1
1.2 MEASUREPRESERVINGTRANSFORMATIONS 3
1.3 BASICEXAMPLES 6
Chapter 2(cid:4) Recurrence and Ergodicity 19
2.1 RECURRENCE 19
2.2 ERGODICITY 22
2.3 EXAMPLESOFERGODICTRANSFORMATIONS 27
Chapter 3(cid:4) The Pointwise Ergodic Theorem and Mixing 33
3.1 THEPOINTWISEERGODICTHEOREM 34
3.2 NORMALNUMBERS 43
3.3 IRREDUCIBLEMARKOVCHAINS 46
3.4 MIXING 49
Chapter 4(cid:4) More Ergodic Theorems 53
4.1 THEMEANERGODICTHEOREM 53
4.2 THEHUREWICZERGODICTHEOREM 56
vii
viii (cid:4) Contents
Chapter 5(cid:4) Isomorphisms and Factor Maps 65
5.1 MEASUREPRESERVINGISOMORPHISMS 65
5.2 FACTORMAPS 71
5.3 NATURALEXTENSIONS 72
Chapter 6(cid:4) The Perron-Frobenius Operator 77
6.1 ABSOLUTELYCONTINUOUSINVARIANTMEASURES 77
6.2 EXACTNESS 82
6.3 PIECEWISEMONOTONEINTERVALMAPS 91
Chapter 7(cid:4) Invariant Measures for Continuous Transforma-
tions 101
7.1 EXISTENCE 101
7.2 UNIQUEERGODICITYANDUNIFORMDISTRIBUTION 109
7.3 SOMETOPOLOGICALDYNAMICS 115
Chapter 8(cid:4) Continued Fractions 125
8.1 REGULARCONTINUEDFRACTIONS 126
8.2 ERGODICPROPERTIESOFTHEGAUSSMAP 130
8.3 THEDOEBLIN-LENSTRACONJECTURE 135
8.4 OTHERCONTINUEDFRACTIONTRANSFORMATIONS 139
Chapter 9(cid:4) Entropy 147
9.1 RANDOMNESSANDINFORMATION 147
9.2 DEFINITIONSANDPROPERTIES 150
9.3 CALCULATIONOFENTROPYANDEXAMPLES 156
9.4 THESHANNON-MCMILLAN-BREIMANTHEOREM 160
9.5 LOCHS’THEOREM 168
Contents (cid:4) ix
Chapter 10(cid:4) The Variational Principle 173
10.1 TOPOLOGICALENTROPY 173
10.2 PROOFOFTHEVARIATIONALPRINCIPLE 188
10.3 MEASURESOFMAXIMALENTROPY 194
Chapter 11(cid:4) Infinite Ergodic Theory 199
11.1 EXAMPLES 199
11.2 CONSERVATIVEANDDISSIPATIVEPART 202
11.3 INDUCEDSYSTEMS 207
11.4 JUMPTRANSFORMATIONS 216
11.5 INFINITEERGODICTHEOREMS 218
Chapter 12(cid:4) Appendix 223
12.1 TOPOLOGY 223
12.2 MEASURETHEORY 224
12.3 LEBESGUESPACES 229
12.4 LEBESGUEINTEGRATION 231
12.5 HILBERTSPACES 233
12.6 BORELMEASURESONCOMPACTMETRICSPACES 235
12.7 FUNCTIONSOFBOUNDEDVARIATION 237
Bibliography 241
Index 247