Almost everywhere convergence of entangled ergodic averages

Abstract

We study pointwise convergence of entangled averages of the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{α(m)}}A_{m-1}T^{n_{α(m-1)}}_{m-1}\ldots A_2T_2^{n_{α(2)}}A_1T_1^{n_{α(1)}} f, \] where $f\in L^2(X,\mu )$, $\alpha :\left\{1,\dots ,m\right\}\to \left\{1,\dots ,k\right\}$, 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.

16 pages, to appear in Integral Equations and Operator Theory