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Nilpotent Structures in Ergodic Theory

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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.
Bernard Host, Universite Paris-Est Marne-la-Vallee, Champs-sur-Marne, France. Bryna Kra, Northwestern University, Evanston, IL.
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 Bibliography Index of terms Index of symbols.
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