Chapter 0 Preliminaries 1
0.1 Introduction 1
0.2 Measure Spaces 3
0.3 Integrationry 6
0.4 Absolutely Continuous Measures and Conditional Expectations 8
0.5 Function Spaces 9
0.6 Haar Measure 11
0.7 Character Theory 12
0.8 Endomorphisms of Tori 14
0.9 Perron-Frobenius Theory 16
0.10 Topology 17
Chapter 1 Measure-Preserving Transformations 19
1.l Definition and Examples 19
1.2 Problems in Ergodic Theory 23
1.3 Associated Isometrics 24
1.4 Recurrence 26
1.5 Ergodicity 26
1.6 The Ergodic Theorem 34
1.7 Mixing 39
Chapter 2 Isomorphism, Conjugacy, and Spectral Isomorphism 53
2.1 Point Maps and Set Maps 53
2.2 Isomorphism of Measure-Preserving Transformations 57
2.3 Conjugacy of Measure-Preserving Transformations 59
2.4 The Isomorphism Problem 62
2.5 Spectral Isomorphism 63
2.6 Spectral Invariants 66
Chapter 3 Measure-Preserving Transformations with Discrete Spectrum 68
3.1 Eigenvalues and Eigenfunctions 68
3.2 Discrete Spectrum 69
3.3 Group Rotations 72
Chapter 4 Entropy 75
4.1 Partitions and Subalgebras 75
4.2 Entropy of a Partition 77
4.3 Conditional Entropy 80
4.4 Entropy of a Measure-Preserving Transformation 86
4.5 Properties of h（T,？） and h（T） 89
4.6 Some Methods for Calculating h（T） 94
4.7 Examples 100
4.8 How Good an Invariant is Entropy? 103
4.9 Bernoulli Automorphisms and Kolmogorov Automorphisms 105
4.10 The Pinsker σ-Algebra of a Measure-Preserving Transformation 113
4.11 Sequence Entropy 114
4.12 Non-invertible Transformations 115
4.13 Comments 116
Chapter 5 Topological Dynamics 118
5.1 Examples 118
5.2 Minimality 120
5.3 The Non-wandering Set 123
5.4 Topological Transitivity 127
5.5 Topological Conjugacy and Discrete Spectrum 133
5.6 Expansive Homeomorphisms 137
Chapter 6 Invariant Measures for Continuous Transformations 146
6.1 Measures on Metric Spaces 146
6.2 Invariant Measures for Continuous Transformations 150
6.3 Interpretation of Ergodicity and Mixing 154
6.4 Relation of Invariant Measures to Non-wandering Sets, Periodic Points and Topological Transitivity 156
6.5 Unique Ergodicity 158
6.6 Examples 162
Chapter 7 Topological Entropy 164
7.1 Definition Using Open Covers 164
7.2 Bowen's Definition 168
7.3 Calculation of Topological Entropy 176
Chapter 8 Relationship Between Topological Entropy and Measure-Theoretic Entropy 182
8.1 The Entropy Map 182
8.2 The Variational Principle 187
8.3 Measures with Maximal Entropy 191
8.4 Entropy of Affine Transformations 196
8.5 The Distribution of Periodic Points 203
8.6 Definition of Measure-Theoretic Entropy Using the Metrics dn 205
Chapter 9 Topological Pressure and Its Relationship with Invariant Measures 207
9.1 Topological Pressure 207
9.2 Properties of Pressure 214
9.3 The Variational Principle 217
9.4 Pressure Determines M（X,T） 221
9.5 Equilibrium States 223
Chapter 10 Applications and Other Topics 229
10.1 The Qualitative Behaviour of Difleomorphisms 229
10.2 The Subadditive Ergodic Theorem and the Multiplicative Ergodic Theorem 230
10.3 Quasi-invariant Measures 236
10.4 Other Types of Isomorphism 238
10.5 Transformations of Intervals 238
10.6 Further Reading 239
References 240
Index 247