An Introduction to Ergodic Theory

By: Peter Walters

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Paperback | 1 October 2000

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264 Pages

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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.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Numerical Analysis Topology Medicine Medicine in General Probability & Statistics Geometry Algebraic Geometry

Preliminaries
Measure-Preserving Transformations
Isomorphism, Conjugacy, and Spectral Isomorphism
Measure-Preserving Transformations with Discrete Spectrum
Entropy
Topological Dynamics
Invariant Measures for Continuous Transformations
Topological Entropy
Relationship Between Topological Entropy and Mesaure-Theoretic Entropy
Topological Pressure and Its Relationship with Invariant Measures
Applications and Other Topics
Table of Contents provided by Publisher. All Rights Reserved.

An Introduction to Ergodic Theory

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