On the pointwise entangled ergodic theorem

We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\dots ,T_a$ on a Borel probability space of the form $$\sum _{n=1}^N{T}_a^n{A}_{a-1}{T}_{a-1}^n{A}_{a-1}\cdot \dots \cdot A_0{T}_0^{n}f$$ for $f\in L^p(X,\mu )$, $p\ge 1$. We also discuss examples and present a continuous version of the result.