Pointwise recurrence for commuting measure preserving transformations

TL;DR

This paper proves the almost everywhere convergence of certain multiple ergodic averages involving commuting measure-preserving transformations, resolving a long-standing open problem and providing a new proof for a special case of Bourgain's theorem.

Contribution

It establishes pointwise convergence of nonconventional ergodic averages for commuting transformations, solving a major open question in ergodic theory.

Findings

Almost everywhere convergence of averages for commuting transformations

Convergence of polynomial iterates for a single transformation

New proof of Bourgain's double recurrence theorem

Abstract

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages $$\frac{1}{N}\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\cdots f_H(T_H^nx)$$ converge a.e. for every function $f_i \in L^{\infty}(\mu)$ .\\ As a consequence if $T_i = T^i$ for $1\leq i \leq H$ where $T$ is an invertible measure preserving transformation on $(X, \mathcal{A}, \mu)$ then the averages $$\frac{1}{N}\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ converge a.e. This solves a long open question on the pointwise convergence of nonconventional ergodic averages. For $H=2$ it provides another proof of J. Bourgain's a.e. double recurrence theorem.