Almost everywhere convergence of entangled ergodic averages

We study pointwise convergence of entangled averages of 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 $f\in L^2(X,\mu )$, $\alpha :\{1,\dots ,m\}\to \{1,\dots ,k\}$, 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 ($1\le p<\mathrm{\infty }$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.