We study pointwise convergence of entangled averages of the form \[ \frac{ 1}{ N^k}\sum_{ 1\leq n_1,\ldots, n_k\leq N} T_m^{ n_{ \alpha(m)}}A_{ m-1}T^{ n_{ \alpha(m-1)}}_{ m-1}\ldots A_2T_2^{ n_{ \alpha(2)}}A_1T_1^{ n_{ \alpha(1)}} f, \] where $f\in L^2(X,\mu)$, $\alpha:\left\{ 1,\ldots,m\right\}\to\left\{ 1,\ldots,k\right\}$, 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 ($1\leq p<\infty$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.
Categories: math.DS math.FA