Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages

TL;DR

This paper establishes that ergodic systems can be modeled with strict ergodicity under group actions, leading to new results on the almost everywhere convergence of non-conventional ergodic averages, including averages along cubes.

Contribution

It proves the existence of strictly ergodic models under group actions and connects these models to the convergence of non-conventional ergodic averages.

Findings

Almost everywhere convergence of non-conventional averages for ergodic systems.

Extension of convergence results to averages along cubes.

Method applicable to various types of ergodic averages.

Abstract

The well-known Jewett-Krieger's Theorem states that each ergodic system has a strictly ergodic model. Strengthening the model by requiring that it is strictly ergodic under some group actions, and building the connection of the new model with the convergence of pointwise non-conventional ergodic averages we prove that for an ergodic system $(X,\X,\mu, T)$, $d\in\N$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, %and any tempered F{\rm ${\o}$}lner sequence $F_n$ of $\Z^2$, the averages \begin{equation*} \frac{1}{N^2} \sum_{(n,m)\in F_N} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) \end{equation*} converge $\mu$ a.e. We remark that the same method can be used to show the pointwise convergence of ergodic averages along cubes which was firstly proved by Assani and then extended to a general case by Chu and Franzikinakis.