Description

Book Synopsis
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.

Table of Contents
  • 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.

    Nilpotent Structures in Ergodic Theory

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      A Hardback by Bernard Host, Bryna Kra

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        Publisher: MP-AMM American Mathematical
        Publication Date: Publication Date: 2/28/2019 12:00:00 AM
        ISBN13: 9781470447809, 978-1470447809
        ISBN10: 1470447800

        Description

        Book Synopsis
        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.

        Table of Contents
        • 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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