Polynomial ergodic averages for certain countable ring actions

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 $(X,\mathcal{B},μ)$ 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 $f_i \in L^{\infty}(μ)$, $(Φ_N)$ is a Fø lner sequence of $(F,+)$, and the convergence takes place in $L^2(μ)$. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.

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