Nilpotent Structures in Ergodic Theory (Mathematical Surveys and Monographs) (Mathematical Surveys and Monographs, 236)

✍ Scribed by Bernard Host, Bryna Kra

• Publisher: American Mathematical Society
• Year: 2018
• Tongue: English
• Leaves: 442
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Title page
Chapter 1. Introduction
1. Characteristic factors
2. Towers of factors
3. Cubes, norms, nilfactors, and structure theorems
4. Nilsequences in ergodic theory and in combinatorics
Organization of the book
Acknowledgments
Part 1 . Basics
Chapter 2. Background material
1. Groups and commutators
2. Probability spaces
3. Polish, locally compact, and compact abelian groups
4. Averages on a locally compact group
References and further comments
Chapter 3. Dynamical Background
1. Topological dynamical systems
2. Ergodic theory
3. The Ergodic Theorems
4. Multiple recurrence and convergence
5. Joinings
6. Inverse limits of dynamical systems
References and further comments
Chapter 4. Rotations
1. Topological and measurable rotations
2. The Kronecker factor
3. Decomposition of a system via the Kronecker
References and further comments
Chapter 5. Group Extensions
1. Group extensions
2. Extensions by a compact abelian group
3. Cocycles and coboundaries
References and further comments
Part 2 . Cubes
Chapter 6. Cubes in an algebraic setting
1. Basics of algebraic cubes
2. Cubes in an abelian group
3. Cubes in nonabelian groups
4. Cubes in homogeneous spaces
References and further comments
Chapter 7. Dynamical cubes
1. Basics of dynamical cubes
2. Properties of topological dynamical cubes
References and further comments
Chapter 8. Cubes in ergodic theory
1. Initializing the construction: the measure 𝜇\type2 and the seminorm \nnorm⋅₂
2. Construction of the measures 𝜇\type𝑘
3. The seminorms \nnorm⋅{𝑘}
4. Dynamical dual functions
References and further comments
Chapter 9. The Structure factors
1. Construction of the structure factors
2. Structured systems
3. Ergodic seminorms and the centralizer
References and further comments
Part 3 . Nilmanifolds and nilsystems
Chapter 10. Nilmanifolds
1. Nilpotent Lie groups
2. Nilmanifolds
3. Subnilmanifolds
4. Bases and generators
5. Countability of nilmanifolds
References and further comments
Chapter 11. Nilsystems
1. Topological and measure theoretic nilsystems
2. Ergodic and minimal nilsystems
3. Applications and generalizations
4. Unipotent affine transformations of a nilmanifold
References and further comments
Chapter 12. Cubic structures in nilmanifolds
1. Cubes in nilmanifolds and nilsystems
2. Gowers seminorms for functions on a nilmanifold
3. Algebraic dual functions
4. The order 𝑘 Fourier algebra of a nilmanifold
5. Some properties of the Fourier algebra of order 𝑘
References and further comments
Chapter 13. Factors of nilsystems
1. Basics of factors of nilsystems
2. Quotient by a compact subgroup of the centralizer
3. Inverse limits of nilsystems and their intrinsic topology
References and further comments
Chapter 14. Polynomials in nilmanifolds and nilsystems
1. Polynomial sequences in a group
2. Polynomial orbits in a nilmanifold
3. Dynamical applications
References and further comments
Chapter 15. Arithmetic progressions in nilsystems
1. Arithmetic progressions in nilmanifolds and nilsystems
2. Ergodic decomposition
3. References and further comments
Part 4 . Structure Theorems
Chapter 16. The Ergodic Structure Theorem
1. Various forms of the Ergodic Structure Theorem
2. Nilsequences and a nonergodic Structure Theorem
3. Factors of inverse limits of nilsystems
References and further comments
Chapter 17. Other structure theorems
1. A Topological Structure Theorem
2. The Inverse Theorem for Gowers norms
References and further comments
Chapter 18. Relations between consecutive factors
1. Starting the induction and an overview of the proof
2. First properties of the extension between consecutive factors
3. Cocycles of type 𝑘
4. From cocycles of type 𝑘 to systems of order 𝑘
5. Connectedness
References and further comments
Chapter 19. The Structure Theorem in a particular case
1. Strategy and preliminaries
2. Construction of a group of transformations
3. 𝑋 is a nilsystem
References and further comments
Chapter 20. The Structure Theorem in the general case
1. Further understanding of cocycles of type 𝑘
2. Countability
3. General cocycles and the Structure Theorem
References and further comments
Part 5 . Applications
Chapter 21. The method of characteristic factors
1. The van der Corput Lemma
2. Arithmetic progressions and linear patterns
3. Convergence of polynomial averages
References and further comments
Chapter 22. Uniformity seminorms on ℓ^{∞} and pointwise convergence of cubic averages
1. Uniformity seminorms along a sequence of intervals
2. Relations with Gowers norms on \Z{𝑁}
3. Pointwise convergence of cubic averages
References and further comments
Chapter 23. Multiple correlations, good weights, and anti-uniformity
1. Decompositions for multicorrelations
2. Bounding weighted ergodic averages
3. Anti-uniformity
4. A nilsequence version of the Wiener-Wintner Theorem
References and further comments
Chapter 24. Inverse results for uniformity seminorms and applications
1. Inverse results for uniformity seminorms
2. Characterization of good weights for Multiple Ergodic Theorems
3. Correlation sequences and nilsequences
References and further comments
Chapter 25. The comparison method
1. Recurrence and convergence for the primes
2. Multiple polynomial averages along the primes
References and further comments
Bibliography
Index of Terms
Index of Symbols
Back Cover