On the convergence of multiple ergodic means

Abstract

Given sequence of measure preserving transformations $\{U_k:k=1,2,\dots ,n\}$ 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 $L{\log }^{n-1}(X)$.

15 pages