**Mathematical Surveys and Monographs**

Volume: 236;
2018;
427 pp;
Hardcover

MSC: Primary 37; 11; 28; 47;

**Print ISBN: 978-1-4704-4780-9
Product Code: SURV/236**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-5061-8
Product Code: SURV/236.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# Nilpotent Structures in Ergodic Theory

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*Bernard Host; Bryna Kra*

Nilsystems play a key role in the structure
theory of measure preserving systems, arising as the natural objects
that describe the behavior of multiple ergodic averages. This book is
a comprehensive treatment of their role in ergodic theory, covering
development of the abstract theory leading to the structural
statements, applications of these results, and connections to other
fields.

Starting with a summary of the relevant dynamical background, the
book methodically develops the theory of cubic structures that give
rise to nilpotent groups and reviews results on nilsystems and their
properties that are scattered throughout the literature. These basic
ingredients lay the groundwork for the ergodic structure theorems, and
the book includes numerous formulations of these deep results, along
with detailed proofs. The structure theorems have many applications,
both in ergodic theory and in related fields; the book develops the
connections to topological dynamics, combinatorics, and number theory,
including an overview of the role of nilsystems in each of these
areas. The final section is devoted to applications of the structure
theory, covering numerous convergence and recurrence results.

The book is aimed at graduate students and researchers in ergodic
theory, along with those who work in the related areas of arithmetic
combinatorics, harmonic analysis, and number theory.

#### Readership

Graduate students and researchers interested in ergodic theory and its connections to combinatorics and number theory.

#### Table of Contents

# Table of Contents

## Nilpotent Structures in Ergodic Theory

- Cover Cover11
- Title page iii4
- Chapter 1. Introduction 112
- Part 1 . Basics 920
- Part 2 . Cubes 8192
- Part 3 . Nilmanifolds and nilsystems 151162
- Part 4 . Structure Theorems 265276
- Part 5 . Applications 327338
- Chapter 21. The method of characteristic factors 329340
- Chapter 22. Uniformity seminorms on ℓ^{∞} and pointwise convergence of cubic averages 349360
- Chapter 23. Multiple correlations, good weights, and anti-uniformity 365376
- Chapter 24. Inverse results for uniformity seminorms and applications 385396
- Chapter 25. The comparison method 399410
- Bibliography 409420
- Index of Terms 419430
- Index of Symbols 425436

- Back Cover Back Cover1442