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Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates

Nikos Frantzikinakis, Borys Kuca

TL;DR

The paper develops a comprehensive framework to analyze ergodic averages with commuting transformations and polynomial iterates that may be dependent. It proves that such averages are governed by Gowers-Host-Kra seminorms under mild ergodicity, and provides a necessary-and-sufficient criterion for joint ergodicity by combining seminorm control with eigenfunction criteria. The core technical contribution is a detailed induction (ping-pong smoothing) scheme, augmented by a flipping technique and invariance properties, which reduces complex longer families to basic-type cases. This leads to a robust proof of joint and weak joint ergodicity for broad polynomial families and yields a stronger form of a Donoso–Koutsogiannis–Sun conjecture, with extensions to Følner and multivariate settings. The results bridge polynomial Szemerédi-type averages in ergodic theory with Host-Kra structure theory and deepen understanding of when dependent polynomial iterates exhibit joint convergence to product measures.

Abstract

We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers-Host-Kra seminorms whenever the system satisfies some mild ergodicity assumptions. Combining this result with the general criteria for joint ergodicity established in our earlier work, we determine a necessary and sufficient condition under which such averages are jointly ergodic, in the sense that they converge in the mean to the product of integrals, or weakly jointly ergodic, in that they converge to the product of conditional expectations. As a corollary, we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a stronger form.

Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates

TL;DR

The paper develops a comprehensive framework to analyze ergodic averages with commuting transformations and polynomial iterates that may be dependent. It proves that such averages are governed by Gowers-Host-Kra seminorms under mild ergodicity, and provides a necessary-and-sufficient criterion for joint ergodicity by combining seminorm control with eigenfunction criteria. The core technical contribution is a detailed induction (ping-pong smoothing) scheme, augmented by a flipping technique and invariance properties, which reduces complex longer families to basic-type cases. This leads to a robust proof of joint and weak joint ergodicity for broad polynomial families and yields a stronger form of a Donoso–Koutsogiannis–Sun conjecture, with extensions to Følner and multivariate settings. The results bridge polynomial Szemerédi-type averages in ergodic theory with Host-Kra structure theory and deepen understanding of when dependent polynomial iterates exhibit joint convergence to product measures.

Abstract

We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers-Host-Kra seminorms whenever the system satisfies some mild ergodicity assumptions. Combining this result with the general criteria for joint ergodicity established in our earlier work, we determine a necessary and sufficient condition under which such averages are jointly ergodic, in the sense that they converge in the mean to the product of integrals, or weakly jointly ergodic, in that they converge to the product of conditional expectations. As a corollary, we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a stronger form.
Paper Structure (26 sections, 38 theorems, 236 equations)

This paper contains 26 sections, 38 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Extensions to other averaging schemes
  4. Outline of the article
  5. Ergodic background and definitions
  6. Basic notation
  7. Ergodic seminorms
  8. Gowers-Host-Kra seminorms
  9. Box seminorms
  10. Dual functions and sequences
  11. Eigenfunctions and criterion for weak joint ergodicity
  12. Preliminary results
  13. Two motivating examples
  14. Formalism and general strategy for longer families
  15. The formalism behind the induction scheme
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\ell\in{\mathbb N}$ and $p_1, \ldots, p_\ell\in {\mathbb Z}[n]$ be polynomials with the good ergodicity property for a system $(X,{\mathcal{X}}, \mu,T_1, \ldots, T_\ell)$. Then there exists $s\in{\mathbb N}$, depending only on $p_1, \ldots, p_\ell$, such that for all functions $f_1, \ldots, f_\ whenever $\lvert\!|\!| f_j|\!|\!\rvert_{s, T_j} = 0$ for some $j\in\{1, \ldots, \ell\}$.

Theorems & Definitions (81)

  • Definition : Good and very good ergodicity property
  • Remark
  • Theorem 1.1: Seminorm control
  • Theorem 1.2: Weak joint ergodicity
  • Remark
  • Corollary 1.3: Joint ergodicity
  • Conjecture 1
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • ...and 71 more