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Uniform vector-valued pointwise ergodic theorems for operators

Micky Barthmann, Sohail Farhangi

TL;DR

This work develops a uniform vector-valued Wiener-Wintner framework for a broad class of operators, including compositions of ergodic Koopman operators with contractive multipliers. By introducing and leveraging the strong spaCb property, the authors establish uniform pointwise convergence results for weakly mixing and mildly mixing vector-valued functions, and extend these to structured coefficient sequences. The approach uses an ultraproduct-Shift analysis on constructed sequence spaces to translate pointwise behavior into a duality setting, enabling uniform control over unit-circle twists. The results generalize classical Wiener-Wintner theorems to vector-valued settings beyond positive contractions and yield new uniform ergodic insights for ergodic, weakly mixing, and mildly mixing Koopman dynamics. Polynomial-weight versions are shown to require additional structure beyond spaCb, highlighting the nuanced landscape of uniform weighted ergodic phenomena.

Abstract

We prove a uniform vector-valued Wiener-Wintner Theorem for a class of operators that includes compositions of ergodic Koopman operators with contractive multiplication operators. Our results are new even in the case of complex-valued functions, as they also apply to some non-positive non-contractive operators, and they give new uniform pointwise theorems for ergodic, weakly mixing, and mildly mixing Koopman operators.

Uniform vector-valued pointwise ergodic theorems for operators

TL;DR

This work develops a uniform vector-valued Wiener-Wintner framework for a broad class of operators, including compositions of ergodic Koopman operators with contractive multipliers. By introducing and leveraging the strong spaCb property, the authors establish uniform pointwise convergence results for weakly mixing and mildly mixing vector-valued functions, and extend these to structured coefficient sequences. The approach uses an ultraproduct-Shift analysis on constructed sequence spaces to translate pointwise behavior into a duality setting, enabling uniform control over unit-circle twists. The results generalize classical Wiener-Wintner theorems to vector-valued settings beyond positive contractions and yield new uniform ergodic insights for ergodic, weakly mixing, and mildly mixing Koopman dynamics. Polynomial-weight versions are shown to require additional structure beyond spaCb, highlighting the nuanced landscape of uniform weighted ergodic phenomena.

Abstract

We prove a uniform vector-valued Wiener-Wintner Theorem for a class of operators that includes compositions of ergodic Koopman operators with contractive multiplication operators. Our results are new even in the case of complex-valued functions, as they also apply to some non-positive non-contractive operators, and they give new uniform pointwise theorems for ergodic, weakly mixing, and mildly mixing Koopman operators.
Paper Structure (16 sections, 21 theorems, 86 equations)

This paper contains 16 sections, 21 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. History and statement of results
  4. Preliminaries
  5. Bochner spaces
  6. Filters and Ultrafilters
  7. Mixing properties of operators
  8. Strongly Pointwise absolutely Cesàro bounded (spaCb) operators
  9. Sequence spaces and Ultraproducts
  10. Sequence spaces
  11. Ultraproducts
  12. The shift operator
  13. Mixing sequences and pointwise orbits
  14. Polynomial pointwise theorems
  15. Main results
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(X,\mathscr{B},\mu,\varphi)$ be a probability measure preserving system, and let $f \in L^1(X,\mu)$. For a.e. $x \in X$, we have where $f^*(x) \in L^1(X,\mu)$ is such that $f^*(\varphi x) = f^*(x)$ for a.e. $x \in X$ and $\int_Af^*d\mu = \int_Afd\mu$ for every $A \in \mathscr{B}$ satisfying $A = \varphi^{-1}A$. In particular, if $\varphi$ is ergodic, then for a.e. $x \in X$ we have

Theorems & Definitions (42)

  • Theorem 1.1: Birkhoff, Birkhoff'sErgodicTheorem
  • Theorem 1.2: ChaconErgodicTheorem
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: UKErgodicThms
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.5
  • ...and 32 more