Illinois Journal of Mathematics

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

Michael C. R. Johnson

Full-text: Open access

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 $\mathbb{Z}^d$.

Article information

Source
Illinois J. Math., Volume 53, Number 3 (2009), 865-882.

Dates
First available in Project Euclid: 4 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1286212920

Digital Object Identifier
doi:10.1215/ijm/1286212920

Mathematical Reviews number (MathSciNet)
MR2727359

Zentralblatt MATH identifier
1210.28017

Subjects
Primary: 28D05: Measure-preserving transformations 37A15: General groups of measure-preserving transformations [See mainly 22Fxx]

Citation

Johnson, Michael C. R. Convergence of polynomial ergodic averages of several variables for some commuting transformations. Illinois J. Math. 53 (2009), no. 3, 865--882. doi:10.1215/ijm/1286212920. https://projecteuclid.org/euclid.ijm/1286212920


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