The Theory of Bernoulli Shifts

Description

There are many measure spaces isomorphic to the unit interval with Lebesguemeasure, hence there are many ways to describe measure-preserving transformationson such spaces. For example, there are translations and automorphisms ofcompact metric groups, shifts on sequence spaces (such as those induced by stationaryprocesses), and flows arising from mechanical systems. It is a natural questionto ask when two such transformations are isomorphic as measure-preservingtransformations. Such concepts as ergodicity and mixing and the study of unitaryoperators induced by such transformations have provided some rather coarseanswers to this isomorphism question.The first major step forward on the isomorphism quesion was the introductionby Kolmogorov in 1958-59 of the concept of entropy as an invariant for measurepreservingtransformation. In 1970, D. S. Ornstein introduced some new approximationconcepts which enabled him to establish that entropy was a complete invariantfor a class of transformations known as Bernoulli shifts. Subsequent workby Ornstein and others has shown that a large class of transformations of physicaland mathematical interest are isomorphic to Bernoulli shifts.These lecture notes grew out of my attempts to understand and use these newresults about Bernoulli shifts. Most of the material in these notes is concerned withthe proof that two Bernoulli shifts with the same entropy are isomorphic. Thisproof makes use of a number of simple ideas about partitions and approximationby periodic transformations. These are carefully presented in Chapters 2-6. Thebasic results about entropy are sketched in Chapters 7-8. Ornstein's FundamentalLemma is proved in Chapter 9. This enables one to construct partitions withperfect distribution and entropy close to those which are almost perfect, and is thekey to obtaining the isomorphism theorem in Chapter 10. Chapters 11-13 containextensions of these results, while Chapter 1 contains a summary of the measuretheory used in these notes. For a more complete account of recent extensions ofthese ideas, the reader is referred to D. S. Ornstein's forthcoming notes.