Pointwise entangled ergodic theorems for Dunford-Schwartz operators

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 $$\frac{1}{N^k}\sum _{1\le n_1,\dots ,n_k\le N}{T}_m^{n_{\alpha (m)}}{A}_{m-1}{T}_{m-1}^{n_{\alpha (m-1)}}\dots A_2{T}_2^{n_{\alpha (2)}}{A}_1{T}_1^{n_{\alpha (1)}}f,$$ where $f\in L^p(X,\mu )$ for some $1\le p<\mathrm{\infty }$, 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 $(A_i,T_i)$, almost everywhere convergence holds 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.