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