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

TL;DR

This paper proves the convergence in $L^2$ of polynomial ergodic averages involving multiple commuting transformations under certain ergodicity conditions, extending classical results to polynomial and multi-variable settings.

Contribution

It establishes the convergence of polynomial ergodic averages for several commuting transformations with ergodic products, generalizing previous linear cases.

Findings

Averages converge in $L^2()$ for all polynomials and Ffner sequences.

Convergence holds for transformations with ergodic products.

Extends classical linear ergodic theorems to polynomial and multi-variable contexts.

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 $(c_1,...,c_l)\neq (0,...,0)$, 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 $p_{ij}\colon \mathbb{Z}^d\to\mathbb{Z}$, all $f_i\in L^{\infty}(\mu)$, and all F{\o}lner sequences $\{\Phi_N\}_{N=1}^{\infty}$ in $\mathbb{Z}^d$.