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

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. 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 Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behaviour of multiple ergodic averages. This book provides a comprehensive treatment of their role in ergodic theory.