**University Lecture Series**

Volume: 30;
2003;
121 pp;
Softcover

MSC: Primary 37;

Print ISBN: 978-0-8218-3496-1

Product Code: ULECT/30

List Price: $34.00

Individual Member Price: $27.20

**Electronic ISBN: 978-1-4704-2175-5
Product Code: ULECT/30.E**

List Price: $34.00

Individual Member Price: $27.20

# Combinatorial Constructions in Ergodic Theory and Dynamics

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*Anatole Katok*

Ergodic theory studies measure-preserving transformations of measure spaces. These objects are intrinsically infinite, and the notion of an individual point or of an orbit makes no sense. Still there are a variety of situations when a measure-preserving transformation (and its asymptotic behavior) can be well described as a limit of certain finite objects (periodic processes).

The first part of this book develops this idea systematically. Genericity of approximation in various categories is explored, and numerous applications are presented, including spectral multiplicity and properties of the maximal spectral type. The second part of the book contains a treatment of various constructions of cohomological nature with an emphasis on obtaining interesting asymptotic behavior from approximate pictures at different time scales.

The book presents a view of ergodic theory not found in other expository sources. It is suitable for graduate students familiar with measure theory and basic functional analysis.

#### Table of Contents

# Table of Contents

## Combinatorial Constructions in Ergodic Theory and Dynamics

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Table of Contents iii4 free
- 0. Introduction 16 free
- Part I. Approximation and Genericity in Ergodic Theory 510
- 1. Periodic processes 510
- 2. Genericity of approximation 1015
- 3. Various types of approximation 1217
- 4. Spectral multiplicity of ergodic transformations 1823
- 5. Approximation and coding 2732
- 6. Invariant measures for transformation with specification 3742
- 7. Generic induced maps 4247
- 8. Combinatorial approximation by conjugation construction 4449

- Part II. Cocycles, Cohomology and Combinatorial Constructions 5358
- 9. Definitions and principal constructions 5459
- 10. Structure of equivalence classes 6267
- 11. Rigidity and stability 6873
- 11.1. Definitions 6873
- 11.2. Translations of the torus and smooth rigidity 7075
- 11.3. Stability of Holder cocycles for transformations with specification 7479
- 11.4. Livshitz theory 7883
- 11.5. Invariant distributions and stability of partially hyperbolic systems 8186
- 11.6. Stability determined by invariant distributions in parabolic systems 8590

- 12. Wild cochains with tame coboundaries 8994
- 13. Non-trivial cocycles 102107

- References 117122
- Back Cover Back Cover1127

#### Readership

Graduate students and research mathematicians interested in ergodic theory.

#### Reviews

For more advanced readers, however, this volume will be highly rewarding: they will be learning from a master of the subject, presenting some of his tools.

-- Mathematical Reviews