Nilpotent Structures in Ergodic Theory

Bernard Host, Bryna Kra (Autoren)

Buch | Softcover

427 Seiten

2018
American Mathematical Society (Verlag)
978-1-4704-8470-5 (ISBN)

Infos

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.

Autor

Bernard Host, Universite Paris-Est Marne-la-Vallee, Champs-sur-Marne, France Bryna Kra, Northwestern University, Evanston, IL

Inhalt

Introduction
Part 1. Basics
Background material
Dynamical background
Rotations
Group extensions
Part 2. Cubes
Cubes in an algebraic setting
Dynamical cubes
Cubes in ergodic theory
The structure factors
Part 3. Nilmanifolds and nilsystems
Nilmanifolds
Nilsystems
Cubic structures in nilmanifolds
Factors of nilsystems
Polynomials in nilmanifolds and nilsystems
Arithmetic progressions in nilsystems
Part 4. Structure theorems
The ergodic structure theorem
Other structure theorems
Relations between consecutive factors
The structure theorem in a particular case
The structure theorem in the general case
Part 5. Applications
The method of characteristic factors
Uniformity seminorms on $/ell ^/infty $ and pointwise convergence of cubic averages
Multiple correlations, good weights, and anti-uniformity
Inverse results for uniformity seminorms and applications
The comparison method

Produkt-Details

• Erscheinungsdatum: 25.10.2025
• Reihe/Serie: Mathematical Surveys and Monographs
• Verlagsort: Providence
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• ISBN-10: 1-4704-8470-6 / 1470484706
• ISBN-13: 978-1-4704-8470-5 / 9781470484705
• Zustand: Neuware