An Outline of Ergodic Theory

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Cambridge University Press, Mar 25, 2010 - Mathematics
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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.
 

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Contents

Introduction
1
1 Measuretheoretic preliminaries
5
2 Measurepreserving systems stationary processes
26
3 Martingales and coupling
55
4 Entropy
72
5 Bernoulli transformations
96
6 Ornstein isomorphism theorem
124
7 Varieties of mixing
146
Appendix
167
References
170
Index
173
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About the author (2010)

Steven Kalikow is a Visiting Professor in the Department of Mathematical Sciences at the University of Memphis.

Randall McCutcheon is Assistant Professor in the Department of Mathematical Sciences at the University of Memphis.

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