Chapter 0 Preliminaries
0.1 Introduction
0.2 Measure Spaces
0.3 Integration
0.4 Absolutely Continuous Measures and Conditional Expectations
0.5 Function Spaces
0.6 Haar Measure
0.7 Character Theory
0.8 Endomorphisms of Tori
0.9 Perron-Frobenius Theory
0.10 Topology
Chapter 1 Measure-Preserving Transformations
1.1 Definition and Examples
1.2 Problems in Ergodic Theory
1.3 Associated Isometries
1.4 Recurrence
1.5 Ergodicity
1.6 The Ergodic Theorem
1.7 Mixing
Chapter 2 Isomorphism, Conjugacy, and Spectral Isomorphism
2.1 Point Maps and Set Maps
2.2 Isomorphism of Measure-Preserving Transformations
2.3 Conjugacy of Measure-preserving Transformhtions
2.4 The Isomorphism Problem
2.5 Spectral Isomorphism
2.6 Spectral Invariants
Chapter 3 Measure-Preserving Transformations with Discrete Spectrum
3.1 Eigenvalues and Eigenfunctions
3.2 Discrete Spectrum
3.3 Group Rotations
Chapter 4 Entropy
4.1 Partitions and Subalgebras
4.2 Entropy of a Partition
4.3 Conditional Entropy
4.4 Entropy of a Measure-Preserving Transformation
4.5 Properties orb T,A and h T
4.6 Some Methods for Calculating h T
4.7 Examples
4.8 How Good an Invariant is Entropy
4.9 Bernoulli Automorphisms and Kolmogorov Automorphisms
4.10 The Pinsker -Algebra of a Measure-Preserving Transformation
4.11 Sequence Entropy
4.12 Non-invertible Transformations
4.13 Comments
Chapter 5 Topological Dynamics
5.1 Examples
5.2 Minimality
5.3 The Non-wandering Set
5.4 Topological Transitivity
5.5 Topological Conjugacy and Discrete Spectrum
5.6 Expansive Homeomorphisms
Chapter 6 Invariant Measures for Continuous Transformations
6.1 Measures on Metric Spaces
6.2 Invariant Measures for Continuous Transformations
6.3 Interpretation of Ergodicity and Mixing
6.4 Relation of Invariant Measures to Non-wandering Sets, Periodic Points and Topological Transitivity
6.5 Unique Ergodicity
6.6 Examples
Chapter 7 Topological Entropy
7.1 Definition Using Open Covers
7.2 Bowen's Definition
7.3 Calculation of Topological Entropy
Chapter 8 Relationship Between Topological Entropy and Measure-Theoretic Entropy
8.1 The Entropy Map
8.2 The Variational Principle
8.3 Measures with Maximal Entropy
8.4 Entropy of Affine Transformations
8.5 The Distribution of Periodic Points
8.6 Definition of Measure-Theoretic Entropy Using the Metrics dn
Chapter 9 Topological Pressure and Its Relationship with Invariant Measures
9.1 Topological Pressure
9.2 Properties of Pressure
9.3 The Variational Principle
9.4 Pressure Determines M X, T
9.5 Equilibrium States
Chapter 10 Applications and Other Topics
10.1 The Qualitative Behaviour of Diffeomorphisms
10.2 The Subadditive Ergodic Theorem and the Multiplicative Ergodic Theorem
10.3 Quasi-invariant Measures
10.4 Other Types of Isomorphism
10.5 Transformations of Intervals
10.6 Further Reading
References
Index
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