Seminorms for multiple averages along polynomials and applications to joint ergodicity

Abstract

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria of joint ergodicity for sequences of the form (T-1(p1j(n))... T-d(pdj(n)))(n is an element of Z), 1 <= j <= k, where T-1, ...,T-d are commuting measure preserving transformations on a probability measure space and p(ij) are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic, giving also a characterization for sequences of the form (T-i(p(n)))(n is an element of Z) 1 <= i <= d to be jointly ergodic, answering a question due to Bergelson.