A pointwise ergodic theorem along return times of rapidly mixing systems

Abstract We introduce a new class of sparse sequences that are ergodic and pointwise universally ‐good for ergodic averages; that is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions. These sequences are generated randomly as return or hitting times in systems exhibiting a rapid correlation decay. This can be seen as a natural variant of Bourgain's return times theorem. For example, we obtain that for any , the sequence is ergodic and pointwise universally ‐good for Lebesgue almost every . Our approach builds on techniques developed by Frantzikinakis, Lesigne and Wierdl in their study of sequences generated by independent random variables, which we adapt to the non‐independent case.