Ergodic Theorems, Almost Invariant Sets, Value Distributions of Time Averagings

Abstract

It is shown how some deviations of ergodic averages can be structured. The deviations tend to zero almost everywhere. They are asymptotically almost invariant with respect to the action due to averaging. In this situation, the question of the distribution of the values of such deviations is meaningful. It turns out that for any free ergodic $Z^d$-action these distributions can be weakly close to any given distribution if we change the scale on the value line. This article shortly provides related proofs of the ergodic theorems of von Neumann, Birkhoff, Wiener, and Rokhlin's lemma for $Z^d$-actions with an invariant measure.