On the convergence of multiple ergodic means

TL;DR

This paper proves almost everywhere convergence of multiple ergodic means for measure-preserving transformations on a measurable space, extending classical results to broader function classes and multiple transformations.

Contribution

It generalizes the convergence theorem of Dunford and Zygmund to multiple transformations and larger function classes.

Findings

Almost everywhere convergence of ergodic means for multiple transformations.

Extension of classical convergence results to functions in L log^{d-1}(X).

Applicable to transformations with rank up to n.

Abstract

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots s_{n}}\sum_{j_1=0}^{s_1-1}\cdots\sum_{j_n=0}^{s_n-1}f\left(U_1^{j_1}\cdots U_n^{j_n} x \right) \end{equation} as $\min_j s_j\to\infty $, for any function $f\in L\log^{d-1}(X)$, where $d\le n$ is the rank of the transformations. The result gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the convergence of the means in a narrower class of functions $L\log^{n-1}(X)$.