Introduction

These notes were the basis of a series of ten CBMS lectures at the Univer-

sity of Washington, Seattle, in July 1989, whose theme was the influence of

algebraic ideas on the development of ergodic theory. However, as anybody

familiär with the subject will realize, any comprehensive exploration of this

theme would fill a substantial book and several lecture courses, even if no

proofs are included. In view of this I had to restrict myself to two specific

topics, and even within these topics the shortage of space and time imposed

severe restrictions on the material I could hope to cover.

The first of these topics is the influence of Operator algebras on dynamics.

The construction of factors from group actions on measure Spaces introduced

by F. J. Murray and J. von Neumann in the 1930s has, in turn, influenced

ergodic theory by leading to H. A. Dye's notion of orbit equivalence, G. W.

Mackey's study of Virtual groups, and the investigation of ergodic and topo-

logical equivalence relations by W. Krieger, J. Feldman and C. C. Moore,

A. Connes, and many others. The theory of Operator algebras not only mo-

tivated the study of equivalence relations (or orbit structures), but it also

provided some of the key ideas for the development of this particular branch

of ergodic theory. The first four sections of these notes are devoted to ergodic

equivalence relations, their properties, and their Classification, and present oc-

casional glimpses of the operator-algebraic context from which many of the

ideas and techniques arose. Ergodic theorists tend to regard ergodic equiv-

alence relations as a subject set apart from the main body of their field; for

this reason I have included a large number of examples which (I hope) show

that equivalence relations provide a very natural setting for many classical

constructions and Classification problems. Many of these examples are drawn

from the context of Markov shifts; this was partly motivated by the fact that

the CBMS meeting followed on from a Workshop on dynamics with signifi-

cant emphasis on coding theory, and partly by the ease and naturalness with

which some of the most useful invariants in coding theory can be derived

and interpreted from the point of view of equivalence relations.

The last three sections of these notes are dedicated to higher dimensional

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http://dx.doi.org/10.1090/cbms/076/01