Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations

TL;DR

This paper proves pointwise convergence of multiple ergodic averages involving non-commuting transformations, using uniformity estimates, decomposition, and equidistribution on nilmanifolds, extending previous results in ergodic theory.

Contribution

It establishes the first pointwise convergence results for cubic and polynomial ergodic averages with non-commuting transformations, answering open questions.

Findings

Proved pointwise convergence for averages along combinatorial parallelepipeds.

Extended convergence results to averages along shifted polynomials.

Provided new techniques involving uniformity estimates and nilmanifold equidistribution.

Abstract

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I.Assani in the former case, and extending results of B.Host-B.Kra and A.Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.