Introduction to ergodic theory

Cover......Page 1
Title page......Page 2
Preface......Page 4
0.1 Introduction......Page 10
0.2 Measure Spaces......Page 12
0.3 Integration......Page 15
0.4 Absolutely Continuous Measures and Conditional Expectations......Page 17
0.5 Function Spaces......Page 18
0.6 Haar Measure......Page 11
0.7 Character Theory......Page 21
0.8 Endomorphisms of Tori......Page 23
0.9 Perron-Frobenius Theory......Page 25
0.10 Topology......Page 26
1.1 Definition and Examples......Page 28
1.2 Problems in Ergodic Theory......Page 32
1.3 Associated Isometrics......Page 33
1.5 Ergodicity......Page 35
1.6 The Ergodic Theorem......Page 43
1.7 Mixing......Page 48
2.1 Point Maps and Set Maps......Page 62
2.2 Isomorphism of Measure-Preserving Transformations......Page 66
2.3 Conjugacy of Measure-Preserving Transformations......Page 68
2.4 The Isomorphism Problem......Page 71
2.5 Spectral Isomorphism......Page 72
2.6 Spectral Invariants......Page 75
3.1 Eigenvalues and Eigenfunctions......Page 77
3.2 Discrete Spectrum......Page 78
3.3 Group Rotations......Page 81
4.1 Partitions and Subalgebras......Page 84
4.2 Entropy of a Partition......Page 86
4.3 Conditional Entropy......Page 89
4.4 Entropy of a Measure-Preserving Transformation......Page 95
4.5 Properties of h(T,A) and h(T)......Page 98
4.6 Some Methods for Calculating h(T)......Page 103
4.7 Examples......Page 109
4.8 How Good an Invariant is Entropy?......Page 112
4.9 Bernoulli Automorphisms and Kolmogorov Automorphisms......Page 114
4.10 The Pinsker σ-Algebra of a Measure-Preserving Transformation......Page 122
4.11 Sequence Entropy......Page 123
4.13 Comments......Page 125
5.1 Examples......Page 127
5.2 Minimality......Page 129
5.3 The Non-wandering Set......Page 132
5.4 Topological Transitivity......Page 136
5.5 Topological Conjugacy and Discrete Spectrum......Page 142
5.6 Expansive Homeomorphisms......Page 146
6.1 Measures on Metric Spaces......Page 155
6.2 Invariant Measures for Continuous Transformations......Page 159
6.3 Interpretation of Ergodicity and Mixing......Page 163
6.4 Relation of Invariant Measures to Non-wandering Sets, Periodic Points and Topological Transitivity......Page 165
6.5 Unique Ergodicity......Page 167
6.6 Examples......Page 171
7.1 Definition Using Open Covers......Page 173
7.2 Bowen's Definition......Page 177
7.3 Calculation of Topological Entropy......Page 185
8.1 The Entropy Map......Page 191
8.2 The Variational Principle......Page 196
8.3 Measures with Maximal Entropy......Page 200
8.4 Entropy of Affine Transformations......Page 205
8.5 The Distribution of Periodic Points......Page 212
8.6 Definition or Measure-Theoretic Entropy Using the Metrics d_n......Page 214
9.1 Topological Pressure......Page 216
9.2 Properties or Pressure......Page 223
9.3 The Variational Principle......Page 226
9.4 Pressure Determines M(X,T)......Page 230
9.5 Equilibrium States......Page 232
10.1 The Qualitative Behaviour of Diffeomorphisms......Page 238
10.2 The Subadditive Ergodic Theorem dnd the Multplicative Ergodic Thcorem......Page 239
10.3 Quasi-invariant Measures......Page 245
10.5 Transformations of Intervals......Page 247
10.6 Further Reading......Page 248
References......Page 249
Index......Page 256