Almost Everywhere Convergence of Entangled Ergodic Averages

Abstract

We study pointwise convergence of entangled averages of the form

$$\begin{aligned} \frac{1}{N^k}\sum _{1\le n_1,\ldots , n_k\le N} T_m^{n_{\alpha (m)}}A_{m-1}T^{n_{\alpha (m-1)}}_{m-1}\cdots A_2T_2^{n_{\alpha (2)}}A_1T_1^{n_{\alpha (1)}} f, \end{aligned}$$

where $f\in L^2(X,\mu )$, $\alpha :\left\{ 1,\ldots ,m\right\} \rightarrow \left\{ 1,\ldots ,k\right\}$, and the $T_i$ are ergodic measure preserving transformations on the standard probability space $(X,\mu )$. We show that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for all $f\in L^2$. We also present results for the general $L^p$ case ($1\le p<\infty$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.