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 the criteria of joint ergodicity for sequences of the form (Tp1,j(n)1⋅…⋅Tpd,j(n)d)n∈ℤ, 1≤j≤k, where T1,…,Td are commuting measure preserving transformations on a probability measure space and pi,j are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic. We also give a characterization for sequences of the form (Tp(n)i)n∈ℤ,1≤i≤d to be jointly ergodic, answering a question due to Bergelson.