Pointwise recurrence for commuting measure preserving transformations

Abstract

Let $(X,\mathcal{A},\mu )$ be a probability measure space and let $T_i,$ $1\le i\le H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages $$\frac{1}{N}\sum _{n=1}^N{f}_1(T_1^{n}x)f_2(T_2^{n}x)\cdots f_H(T_H^{n}x)$$ converge a.e. for every function $f_i\in L^{\infty }(\mu )$ .\\ As a consequence if $T_i=T^i$ for $1\le i\le H$ where $T$ is an invertible measure preserving transformation on $(X,\mathcal{A},\mu )$ then the averages $$\frac{1}{N}\sum _{n=1}^N{f}_1(T^{n}x)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ 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.

This replaces the paper arXiv:1312.5270

References in corpus (3)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• On the pointwise convergence of multiple ergodic averages
• A proof of Walsh's convergence theorem using couplings

Cited by in corpus (4)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• On the pointwise convergence of multiple ergodic averages
• Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages
• Coboundaries of nonconventional ergodic averages