An Outline of Ergodic Theory (Cambridge Studies in Advanced Mathematics, Series Number 122)

An engaging introduction to ergodic theory for graduate students, and a useful reference for the professional mathematician. 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. Steven Kalikow, Randall Mccutcheon. Includes Bibliographical References And Index. "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"--Provided by publisher. Title ......Page 3 Copyright ......Page 4 Contents ......Page 5 Preface ......Page 7 Introduction......Page 9 1 Measure-theoretic preliminaries......Page 13 2 Measure-preserving systems, stationary processes......Page 34 3 Martingales and coupling......Page 63 4 Entropy......Page 80 5 Bernoulli transformations......Page 104 6 Ornstein isomorphism theorem......Page 132 7 Varieties of mixing......Page 154 Appendix: Some open problems......Page 175 References......Page 178 Index......Page 181 Machine generated contents note: 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.