目錄
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
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 Transformations
§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 of h (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 o-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 Afine Transformations
§8.5 The Distribution of Periodic Points
§8.6 Deinition 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 Difeomorphisms
§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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