Pointwise convergence of some multiple ergodic averages

We show that for every ergodic system $(X,μ,T_1,\ldots,T_d)$ 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 $μ$-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 $μ$-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.

No reviews yet.

No comments yet.