On the pointwise entangled ergodic theorem

TL;DR

This paper establishes conditions for the almost everywhere convergence of entangled ergodic averages involving Dunford-Schwartz operators, extending classical ergodic theorems to more complex operator compositions.

Contribution

It introduces twisted compactness conditions ensuring convergence of entangled ergodic averages and provides a continuous version of the theorem.

Findings

Established convergence conditions for entangled ergodic averages

Provided examples illustrating the conditions

Extended results to a continuous setting

Abstract

We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\ldots, T_a$ on a Borel probability space of the form $$ \sum_{n=1}^N T_a^n A_{a-1}T_{a-1}^nA_{a-1}\cdot \ldots \cdot A_0 T_0^nf$$ for $f\in L^p(X,\mu)$, $p\geq 1$. We also discuss examples and present a continuous version of the result.