Polynomial ergodic averages for certain countable ring actions

Abstract

A recent result of Frantzikinakis establishes sufficient conditions for joint ergodicity in the setting of $\mathbb{Z}$-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action $(T_n{)}_{n\in F}$ of a countable field $F$ with characteristic zero on a probability space $(X,\mathcal{B},\mu )$ and a family $\{p_1,\dots ,p_k\}$ of independent polynomials, we have \[ \lim_{N \to \infty} \frac{1}{|Φ_N|}\sum_{n \in Φ_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod_{j=1}^k \int_X f_i \ dμ,\] where $f_i\in L^{\infty }(\mu )$, $({\Phi }_N)$ is a Fø lner sequence of $(F,+)$, and the convergence takes place in $L^2(\mu )$. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.

35 pages. One definition corrected from journal version, all claimed results from journal version preserved