POLYNOMIAL ERGODIC AVERAGES FOR CERTAIN COUNTABLE RING ACTIONS

A recent result of Frantzikinakis in [17] establishes sufficient conditions for joint ergodicity in the setting of 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 subset of F of a countable field F with characteristic zero on a probability space (X, B, mu) and a family {p(1), ..., p(k)} of independent polynomials, we have lim(N ->infinity) 1/vertical bar Phi(N)vertical bar Sigma(n is an element of Phi N) T-p1(n) f(1) ... T-pk(n) f(k) = Pi(j=1) integral(X) f(i) d mu, where f(i) is an element of L-infinity(mu), (Phi(N)) is a Flner 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.