Almost everywhere convergence of entangled ergodic averages

TL;DR

This paper proves almost everywhere convergence of complex entangled ergodic averages for functions in L^2 and L^p spaces, extending classical ergodic theorems to more intricate, multi-parameter settings with polynomial and flow variants.

Contribution

It introduces new joint boundedness and twisted compactness conditions ensuring pointwise convergence of entangled averages in ergodic theory.

Findings

Almost everywhere convergence for L^2 functions.

Extension to L^p spaces and polynomial powers.

Results for continuous ergodic flows.

Abstract

We study pointwise convergence of entangled averages of the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f, \] where $f\in L^2(X,\mu)$, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$, 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 ($1\leq p<\infty$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.