Ergodic Theorems for PSPACE functions and their converses

Abstract

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence to the ergodic averages for integrable functions can in general, be arbitrarily slow [11]. However, we show that for PSPACE L 1 functions and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXPSPACE random. Further, we show that the class of EXPSPACE randoms is a strict subset of the class of PSPACE randoms studied by Huang and Stull [9].