**Mathematical Surveys and Monographs**

Volume: 50;
1997;
284 pp;
Hardcover

MSC: Primary 28;
Secondary 30; 58; 60

Print ISBN: 978-0-8218-0494-0

Product Code: SURV/50

List Price: $101.00

Individual Member Price: $80.80

**Electronic ISBN: 978-1-4704-1281-4
Product Code: SURV/50.E**

List Price: $101.00

Individual Member Price: $80.80

# An Introduction to Infinite Ergodic Theory

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*Jon Aaronson*

Infinite ergodic theory is the study of measure preserving
transformations of infinite measure spaces. The book focuses on
properties specific to infinite measure preserving
transformations.

The work begins with an introduction to basic nonsingular ergodic
theory, including recurrence behavior, existence of invariant
measures, ergodic theorems, and spectral theory. A wide range of
possible “ergodic behavior” is catalogued in the third
chapter mainly according to the yardsticks of intrinsic normalizing
constants, laws of large numbers, and return sequences. The rest of
the book consists of illustrations of these phenomena, including
Markov maps, inner functions, and cocycles and skew products. One
chapter presents a start on the classification theory.

#### Table of Contents

# Table of Contents

## An Introduction to Infinite Ergodic Theory

- Contents ix10 free
- Preface xi12 free
- Chapter 1. Non-singular transformations 114 free
- Chapter 2. General ergodic and spectral theorems 5366
- Chapter 3. Transformations with infinite invariant measures 8598
- §3.1 Isomorphism, factors, and similarity 8699
- §3.2 Intrinsic normalising constants and laws of large numbers 93106
- §3.3 Rational ergodicity 98111
- §3.4 Maharam transformations 102115
- §3.5 Category theorems 108121
- §3.6 Asymptotic distributional behaviour 112125
- §3.7 Pointwise dual ergodicity 118131
- §3.8 Wandering rates 130143

- Chapter 4. Markov maps 139152
- §4.1 Markov partitions 139152
- §4.2 Graph shifts 140153
- §4.3 Distortion properties 143156
- §4.4 Ergodic properties of Markov maps with distortion properties 149162
- §4.5 Markov shifts 156169
- §4.6 Schweiger's jump transformation 161174
- §4.7 Smooth Probenius-Perron operators and the Gibbs property 164177
- §4.8 Non-expanding interval maps 172185
- §4.9 Additional reading 180193

- Chapter 5. Recurrent events and similarity of Markov shifts 181194
- Chapter 6. Inner functions 201214
- Chapter 7. Hyperbolic geodesic flows 223236
- Chapter 8. Cocycles and skew products 247260
- Bibliography 275288
- Index 281294

#### Readership

Graduate students and research mathematicians interested in ergodic theory, dynamical systems and/or probability.

#### Reviews

This book is a research monograph and contains an impressive amount of material. The presentation is careful, well organized, and reliable. This monograph is definitely a valuable complement to the ergodic theory literature. It will be useful to graduate students and researchers in ergodic theory and related fields.

-- Bulletin of the London Mathematical Society

Accessible to readers with a firm background in measure-theoretic probability … carefully organized and well written … invaluable both as an introduction and as a reference work on its subject, and this definitely is not just because it is the only one at the moment.

-- Zentralblatt MATH

This book is devoted mainly to the ergodic theory of transformations preserving an infinite measure, and as such it is a welcome addition to the literature. [O]verall this book fills important gaps in the literature and is recommended to researchers and advanced students.

-- Mathematical Reviews