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arXiv 1609.02529math.DS
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某些多重遍历平均的点态收敛性

Pointwise convergence of some multiple ergodic averages

Sebastián Donoso, Wenbo Sun

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摘要：

我们证明了对每一个具有交换变换的遍历系统$(X,μ,T_1,\ldots,T_d)$，平均\[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \prod_{j=1}^d T_j^{n_j}x)\cdots f_d(T_d^n \prod_{j=1}^d T_j^{n_j}x). \]对$μ$ -a.e. $x\in X$作为$N\to\infty$。如果$X$是远端的，我们证明了平均\[\frac{1}{N}\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\cdots f_d(T_d^nx) \]收敛$μ$ -a.e. $x\in X$作为$N\to\infty$。我们还建立了点态收敛的平均沿立方配置所产生的系统交换变换。我们的方法结合了存在性的饱和和魔术扩张介绍奥斯汀和主机分别与思想的拓扑模型黄，邵和叶。

英文摘要：

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.

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