Seminorms for multiple averages along polynomials and applications to joint ergodicity

Abstract

Exploiting the recent work of Tao and Ziegler on a 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 $(T_1^{p_{1,j}(n)}\cdot \dots \cdot T_d^{p_{d,j}(n)}{)}_{n\in \mathbb{Z}},$ $1\le j\le k$, where $T_1,\dots ,T_d$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ 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 $(T_i^{p(n)}{)}_{n\in \mathbb{Z}},1\le i\le d$ to be jointly ergodic, answering a question due to Bergelson.

Some changes have been made correcting minor issues

References in corpus (6)

• Universal Characteristic Factors and Furstenberg Averages
• Ergodic averages of commuting transformations with distinct degree polynomial iterates
• Nilfactors of R^m-actions and configurations in sets of positive upper density in R^m
• Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors
• Integer part polynomial correlation sequences
• Weak ergodic averages over dilated measures