Pointwise convergence of some multiple ergodic averages

Abstract

We show that for every ergodic system $(X,\mu ,T_1,\dots ,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \prod_{j=1}^d T_j^{n_j}x)\cdots f_d(T_d^n \prod_{j=1}^d T_j^{n_j}x). \] converges for $\mu$-a.e. $x\in X$ as $N\to \infty$. If $X$ is distal, we prove that the average \[\frac{1}{N}\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\cdots f_d(T_d^nx) \] converges for $\mu$-a.e. $x\in X$ as $N\to \infty$. We also establish the pointwise convergence of averages along cubical configurations arising from a system commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.

References in corpus (5)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Multiple recurrence and the structure of probability-preserving systems
• Dynamical cubes and a criteria for systems having products extensions
• Pointwise multiple averages for systems with two commuting transformations
• A pointwise cubic average for two commuting transformations

Cited by in corpus (3)

• Topological correspondence of multiple ergodic averages of nilpotent group actions
• Almost sure convergence of the multiple ergodic average for certain weakly mixing systems
• Pointwise multiple averages for sublinear functions