Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates

TL;DR

This paper studies the convergence of multiple ergodic averages involving commuting transformations with polynomial iterates, establishing seminorm control and conditions for joint ergodicity under mild assumptions.

Contribution

It introduces seminorm control for such averages with pairwise dependent polynomials and characterizes joint ergodicity conditions, extending previous criteria.

Findings

Averages are controlled by Gowers-Host-Kra seminorms under mild ergodicity.

Necessary and sufficient conditions for joint ergodicity are established.

A special case of a conjecture by Donoso, Koutsogiannis, and Sun is proved.

Abstract

We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers-Host-Kra seminorms whenever the system satisfies some mild ergodicity assumptions. Combining this result with the general criteria for joint ergodicity established in our earlier work, we determine a necessary and sufficient condition under which such averages are jointly ergodic, in the sense that they converge in the mean to the product of integrals, or weakly jointly ergodic, in that they converge to the product of conditional expectations. As a corollary, we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a stronger form.