CBMS Regional Conference Series in Mathematics 1990; 94 pp; softcover Number: 76 ISBN10: 0821807277 ISBN13: 9780821807279 List Price: US$39 Member Price: US$31.20 All Individuals: US$31.20 Order Code: CBMS/76
 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 operatoralgebraic 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. Table of Contents  Operator algebras and dynamical systems
 Cohomology of equivalence relations
 Rokhlin's lemma and asymptotic invariance
 Dimension
 Markov shifts in higher dimensions
 Markov shifts and Markov groups
 The dynamics of abelian Markov groups
