**CBMS Regional Conference Series in Mathematics**

Volume: 76;
1990;
94 pp;
Softcover

MSC: Primary 28; 54;
Secondary 22; 46

**Print ISBN: 978-0-8218-0727-9
Product Code: CBMS/76**

List Price: $44.00

Individual Price: $35.20

**Electronic ISBN: 978-1-4704-2436-7
Product Code: CBMS/76.E**

List Price: $41.00

Individual Price: $32.80

# Algebraic Ideas in Ergodic Theory

Share this page
*Klaus Schmidt*

A co-publication of the AMS and CBMS

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.

#### Table of Contents

# Table of Contents

## Algebraic Ideas in Ergodic Theory

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction 18 free
- 1. Operator Algebras and Dynamical Systems 310 free
- 2. Cohomology of Equivalence Relations 1825
- 3. Rokhlin's Lemma and Asymptotic Invariance 3643
- 4. Dimension 4350
- 5. Markov Shifts in Higher Dimensions 5562
- 6. Markov Shifts and Markov Groups 6471
- 7. The Dynamics of Abelian Markov Groups 6774
- References 7986
- Notation 8794
- Index 9198
- Back Cover Back Cover1102