The Theory of Bernoulli Shifts

There are many measure spaces isomorphic to the unit interval with Lebesgue measure, hence there are many ways to describe measure-preserving transformations on such spaces. For example, there are translations and automorphisms of compact metric groups, shifts on sequence spaces (such as those induced by stationary processes), and flows arising from mechanical systems. It is a natural question to ask when two such transformations are isomorphic as measure-preserving transformations. Such concepts as ergodicity and mixing and the study of unitary operators induced by such transformations have provided some rather coarse answers to this isomorphism question. The first major step forward on the isomorphism quesion was the introduction by Kolmogorov in 1958-59 of the concept of entropy as an invariant for measurepreserving transformation. In 1970, D. S. Ornstein introduced some new approximation concepts which enabled him to establish that entropy was a complete invariant for a class of transformations known as Bernoulli shifts. Subsequent work by Ornstein and others has shown that a large class of transformations of physical and mathematical interest are isomorphic to Bernoulli shifts. These lecture notes grew out of my attempts to understand and use these new results about Bernoulli shifts. Most of the material in these notes is concerned with the proof that two Bernoulli shifts with the same entropy are isomorphic. This proof makes use of a number of simple ideas about partitions and approximation by periodic transformations. These are carefully presented in Chapters 2-6. The basic results about entropy are sketched in Chapters 7-8. Ornstein's Fundamental Lemma is proved in Chapter 9. This enables one to construct partitions with perfect distribution and entropy close to those which are almost perfect, and is the key to obtaining the isomorphism theorem in Chapter 10. Chapters 11-13 contain extensions of these results, while Chapter 1 contains a summary of the measure theory used in these notes. For a more complete account of recent extensions of these ideas, the reader is referred to D. S. Ornstein's forthcoming notes.