An Outline of Ergodic Theory

An engaging introduction to ergodic theory for graduate students, and a useful reference for the professional mathematician.

Steven Kalikow (Author), Randall McCutcheon (Author)

9780521194402, Cambridge University Press

Hardback, published 25 March 2010

182 pages, 305 exercises
23.5 x 15.7 x 1.5 cm, 0.38 kg

'… explains the main ideas and topics of ergodic theory for those readers who want a basic overview of it and who do not want to be overburdened with notions and details. It also gives professional mathematicians familiar with the material the option of a quick review of it.' Mathematical Reviews

This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.

Preface
Introduction
1. Measure-theoretic preliminaries
2. Measure preserving systems, stationary processes
3. Martingales and coupling
4. Entropy
5. Bernoulli transformations
6. Ornstein isomorphism theorem
7. Varieties of mixing
Appendix
References
Index.

Subject Areas: Probability & statistics [PBT], Discrete mathematics [PBD]