Multiple recurrence and convergence without commutativity

TL;DR

This paper proves multiple recurrence and convergence for pairs of zero entropy measure preserving transformations without assuming commutativity, focusing on polynomial iterates with degree at least 2, and explores structural properties of Furstenberg systems.

Contribution

It establishes recurrence and convergence results for non-commuting transformations with polynomial iterates of degree ≥ 2, using nilpotent structure and disjointness arguments.

Findings

Recurrence and convergence hold for pairs of transformations with polynomial iterates of degree ≥ 2.

Furstenberg systems of sequences with polynomial iterates have special structural properties.

Characteristic factors with nilpotent structure are key to the proofs.

Abstract

We establish multiple recurrence and convergence results for pairs of zero entropy measure preserving transformations that do not satisfy any commutativity assumptions. Our results cover the case where the iterates of the two transformations are $n$ and $n^k$ respectively, where $k\geq 2$, and the case $k=1$ remains an open problem. Our starting point is based on the observation that Furstenberg systems of sequences of the form $(f(T^{n^k}x))$ have very special structural properties when $k\geq 2$. We use these properties and some disjointness arguments in order to get characteristic factors with nilpotent structure for the corresponding ergodic averages, and then finish the proof using some equidistribution results on nilmanifolds.