Pointwise entangled ergodic theorems for Dunford–Schwartz operators

Abstract

We investigate pointwise convergence of entangled ergodic averages of Dunford–Schwartz operators $T_0,T_1,\dots ,T_m$ on a Borel probability space. These averages take the form

$$\frac{1}{N^k}{\sum }_{1\le n_1,\dots ,n_k\le N}{T}_m^{n_{\alpha \left(m\right)}}{A}_{m-1}{T}_{m-1}^{n_{\alpha (m-1)}}\cdots A_2{T}_2^{n_{\alpha \left(2\right)}}{A}_1{T}_1^{n_{\alpha \left(1\right)}}f,$$ where $f\in L^p(X,\mu )$ for some $1\le p<\infty$, and $\alpha :\{1,\dots ,m\}\to \{1,\dots ,k\}$ encodes the entanglement. We prove that, under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, convergence holds almost everywhere for all $f\in L^p$. We also present an extension to polynomial powers in the case $p=2$, in addition to a continuous version concerning Dunford–Schwartz $C_0$-semigroups.