Convergence of polynomial ergodic averages of several variables for some commuting transformations

Abstract

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,\ldots , T_l$ be $l$ commuting invertible measure preserving transformations of $X$. We show that if $T_1^{c_1} \ldots T_l^{c_l}$ is ergodic for each $(c_1,\ldots ,c_l)\neq(0,\ldots,0)$, then the averages $\frac{1}{|\Phi_N|}\sum_{u\in\Phi_N}\prod _{i=1}^r T_1^{p_{i1}(u)}\ldots T_l^{p_{il}(u)}f_i$ 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 $\mathbb{Z}^d$.