| 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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