An Introduction to Ergodic Theory

This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.

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.