CF-Nil systems and convergence of two-dimensional ergodic averages

TL;DR

The paper develops the CF-Nil framework to align measurable and topological pro-nilfactors and constructs strictly ergodic CF-Nil($\infty$) models for ergodic systems. This enables pointwise convergence of two-dimensional ergodic averages weighted by nilsequences and yields $L^2$ convergence results for two-variable polynomial averages without requiring zero-entropy in general non-commuting settings. It also provides a detailed analysis via Furstenberg systems and Pinsker theory, linking higher-order structure to randomness in multi-parameter averages. Finally, it demonstrates limitations by constructing Floyd–Auslander-type CF-Nil systems where universal multi-parameter convergence fails, highlighting the nuanced boundary of higher-order ergodic phenomena.

Abstract

A topological dynamical system $(X,T)$ is called CF-Nil($k$) if it is strictly ergodic and the maximal measurable and maximal topological $k$-step pro-nilfactors coincide as measure preserving systems. Through constructing specific ``CF-Nil'' models, we prove that for any ergodic system $(X,\mathcal{X},μ,T)$, any nilsequence $\{ψ(m,n)\}_{m,n\in\mathbb{Z}}$ and any $f_1,\dots,f_d\in L^{\infty}(μ)$, the averages \begin{equation*} \dfrac{1}{N^{2}} \sum_{m,n=0}^{N-1} ψ(m,n)\prod_{j=1}^{d}f_{j}(T^{m+jn}x) \end{equation*} converge pointwise as $N$ goes to infinity. Moreover, we show the $L^2$-convergence of a certain two-dimensional averages for non-commuting transformations without zero entropy condition.

Theorems & Definitions (79)

• Theorem \parA
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• Theorem \parC
• Theorem \parD
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6