Polynomial ergodic averages for certain countable ring actions

<p style='text-indent:20px;'>A recent result of Frantzikinakis in [<xref ref-type="bibr" rid="b17">17</xref>] establishes sufficient conditions for joint ergodicity the setting <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions. We generalize this actions second-countable locally compact abelian groups. obtain two applications result. First, we show that, given an ergodic action id="M2">\begin{document}$ (T_n)_{n \in F} $\end{document}</tex-math></inline-formula> a countable field id="M3">\begin{document}$ F with characteristic zero on probability space id="M4">\begin{document}$ (X,\mathcal{B},\mu) and family id="M5">\begin{document}$ \{p_1,\dots,p_k\} independent polynomials, have</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \lim\limits_{N \to \infty} \frac{1}{|\Phi_N|}\sum\limits_{n \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod\limits_{j 1}^k \int_X f_i d\mu, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where id="M6">\begin{document}$ L^{\infty}(\mu) $\end{document}</tex-math></inline-formula>, id="M7">\begin{document}$ (\Phi_N) is Følner sequence id="M8">\begin{document}$ (F,+) convergence takes place id="M9">\begin{document}$ L^2(\mu) $\end{document}</tex-math></inline-formula>. This yields corollaries combinatorics topological dynamics. Second, prove that similar holds totally suitable rings.</p>