Random ergodic theorems and real cocycles (Q696294)

Let \(T\) be an ergodic measure preserving transformation on the standard probability space \((\Omega,{\mathcal T},\mu)\), and let \(F\) be a real measurable function on \(\Omega.\) For any measure preserving invertible system \((X,{\mathcal A},\nu,\tau)\), the authors study the mean convergence of ergodic averages \[ \frac{1}{N}\sum_{n=0}^{N-1}f\circ \tau^{k_n(\omega)}, \] where \(k_n(\omega)=\sum_{i=0}^{N-1}F(T^i\omega)\) and \(f\in L^p(\nu)\), \(1\leq p<\infty.\) They prove that the sequence \(\{k_n(\omega)\}\) is almost surely universally good, i.e., for a.e. \(\omega\), the sequence of averages converges in \(L^p(\nu)\) if and only if the cocycle generated by \(F\) is either a coboundary, or the set of all \(\theta\) such that \(\theta F\) is a coboundary modulo 1 is countable. When no assumption is made on \(F\), the authors prove that \(\{k_n(\omega)\}\) is almost surely universally good for the class of mildly mixing transformations. On the other hand, this is not the case for weakly mixing transformations. They show that given any \((\Omega,{\mathcal T},\mu,T)\) that is aperiodic and ergodic, there exists an integrable function \(F\) for which \(\{k_n(\omega)\}\) is not almost surely universally good for the class of weakly mixing transformations.