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Higher Order Wiener-Wintner systems: examples and applications

Idris Assani, Jacob Folks, Ryo Moore

Abstract

We will construct ``higher-dimensional" versions of the Wiener-Wintner dynamical system that was originally studied by I. Assani in 2003. We will show that on these systems we can provide very simple proofs of the a.e. convergence of the multiple recurrence averages, as well as the multiple recurrence return times averages. We will do so by obtaining a quantitative control of the multiple ergodic averages by extending the estimate for the double recurrence that was attained by J. Bourgain. We will also observe that this class of dynamical systems contains numerous examples that are not bounded by the standard classifications (e.g. entropy, mixing), such as Kolmogorov systems, classical skew products, as well as systems for which the a.e. convergence of multiple recurrence is not currently known. Along our way, we will also provide alternative characteristics of the Host-Kra-Ziegler factors from the point of view of the uniform Wiener-Wintner theorem.

Higher Order Wiener-Wintner systems: examples and applications

Abstract

We will construct ``higher-dimensional" versions of the Wiener-Wintner dynamical system that was originally studied by I. Assani in 2003. We will show that on these systems we can provide very simple proofs of the a.e. convergence of the multiple recurrence averages, as well as the multiple recurrence return times averages. We will do so by obtaining a quantitative control of the multiple ergodic averages by extending the estimate for the double recurrence that was attained by J. Bourgain. We will also observe that this class of dynamical systems contains numerous examples that are not bounded by the standard classifications (e.g. entropy, mixing), such as Kolmogorov systems, classical skew products, as well as systems for which the a.e. convergence of multiple recurrence is not currently known. Along our way, we will also provide alternative characteristics of the Host-Kra-Ziegler factors from the point of view of the uniform Wiener-Wintner theorem.
Paper Structure (25 sections, 34 theorems, 241 equations)

This paper contains 25 sections, 34 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Background and contents
  3. Goals and outline
  4. Acknowledgment
  5. Preliminary
  6. Notations and conventions
  7. Nilsystems and Host-Kra-Ziegler factors
  8. Key estimates
  9. Higher order Wiener-Wintner functions and dynamical systems
  10. Definitions
  11. Multilinearity concerns
  12. Conjugacy class of a WW system
  13. Examples
  14. Pinsker algebras and Kolmogorov automorphisms
  15. Classical skew products
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(X, \mathcal{F}, \mu, T)$ be a measure-preserving system, and let $f \in L^1(\mu)$. There exists a set $X_f \in \mathcal{F}$ such that $\mu(X_f) = 1$, and for every $x \in X_f$ and for every $t \in \mathbb{R}$, the limit exists.

Theorems & Definitions (82)

  • Theorem 1.1: Wiener-Wintner ergodic theorem
  • Theorem 1.2: Uniform Wiener-Wintner ergodic theorem
  • Theorem 1.3: Return times theorem
  • Lemma 2.1: Van der Corput's estimate
  • Lemma 2.2: Hölder's inequality on averages
  • Lemma 2.3: Maximal inequality
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • ...and 72 more