Convergence of Polynomial Ergodic Averages of Several Variables for some Commuting Transformations

Abstract

Let $(X,\mathcal{B},\mu )$ be a probability space and let $T_1,...,T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1}...T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\ne (0,...,0)$, then the averages $\frac{1}{|{\Phi }_N|}{\sum }_{u\in {\Phi }_N}{\prod }_{i=1}^r{T}_1^{p_{i1}(u)}...T_l^{p_{il}(u)}{f}_i$ converge in $L^2(\mu )$ for all polynomials $p_{ij}:{\mathbb{Z}}^d\to \mathbb{Z}$, all $f_i\in L^{\infty }(\mu )$, and all Følner sequences $\{{\Phi }_N{\}}_{N=1}^{\infty }$ in ${\mathbb{Z}}^d$.