An Introduction to Ergodic Theory (Paperback, Softcover Repri)

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.- 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.- 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.- 3 Measure-Preserving Transformations with Discrete Spectrum.- 3.1 Eigenvalues and Eigenfunctions.- 3.2 Discrete Spectrum.- 3.3 Group Rotations.- 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 ?-Algebra of a Measure-Preserving Transformation.- 4.11 Sequence Entropy.- 4.12 Non-invertible Transformations.- 4.13 Comments.- 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.- 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.- 7 Topological Entropy.- 7.1 Definition Using Open Covers.- 7.2 Bowen's Definition.- 7.3 Calculation of Topological Entropy.- 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.- 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.- 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.