Pointwise convergence of multiple ergodic averages and strictly ergodic models

TL;DR

This paper proves pointwise convergence of multiple ergodic averages in ergodic systems using strictly ergodic models, and confirms convergence for distal systems, advancing understanding of ergodic behavior in dynamical systems.

Contribution

It introduces new strictly ergodic models to establish pointwise convergence of multiple averages and confirms convergence in distal systems, addressing open questions.

Findings

Pointwise convergence of multiple ergodic averages in ergodic systems.

Positive answer to convergence question in distal systems.

Construction of strictly ergodic models for ergodic systems.

Abstract

By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},\mu, T)$, $d\in{\mathbb N}$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, the averages $$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) $$ converge $\mu$ a.e. Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge a.e. That is, we show that if $(X,\mathcal{X},\mu, T)$ is an ergodic distal system, and $f_1, \ldots, f_d \in L^{\infty}(\mu)$, then multiple ergodic averages $$\frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x) $$ converge $\mu$ a.e.