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

TL;DR

This paper introduces CF-Nil systems and proves the pointwise convergence of two-dimensional ergodic averages involving nilsequences, extending results to non-commuting transformations without zero entropy.

Contribution

It constructs CF-Nil models and establishes convergence of two-dimensional averages for ergodic systems with nilsequences, advancing understanding of multiple ergodic averages.

Findings

Pointwise convergence of two-dimensional averages with nilsequences.

L^2 convergence of averages for non-commuting transformations.

CF-Nil models unify topological and measure-theoretic perspectives.

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 $(X,\mathcal{X},\mu,T)$, any nilsequence $\{\psi(m,n)\}_{m,n\in\mathbb{Z}}$ and any $f_1,\dots,f_d\in L^{\infty}(\mu)$, the averages \begin{equation*} \dfrac{1}{N^{2}} \sum_{m,n=0}^{N-1} \psi(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.