Algebraic Ideas in Ergodic Theory

This book is based on a series of ten lectures sponsored by the Conference Board on the Mathematical Sciences and presented by the author at the University of Washington in Seattle in July 1989. The main theme of the lectures, the influence of algebraic ideas on the development of ergodic theory, was so extensive that the author chose to restrict himself to two specific topics. The first topic is the influence of operator algebras on dynamics. The author concentrates on ergodic equivalence relations, their properties, and their classification, presenting occasional glimpses of the operator-algebraic context from which many of the ideas and techniques arose. In addition, he provides a large number of examples showing that equivalence relations provide a natural setting for many classical constructions and classification problems. The second topic in the book is higher dimensional Markov shifts, a difficult field of research with no indication yet of a satisfactory general theory. After discussing some elementary examples of such shifts and the surprising difficulties these examples present, the author makes the assumption that the Markov shift carries a group structure. In that context, many of the difficulties can be resolved, and one has the beginnings of a successful analysis which exhibits an intriguing interplay between commutative algebra and dynamics.