Notes on the recurrence of the Birkhoff sums

The measure-preserving, but not necessarily invertible, ergodic transformations of the compact metric space with the Caratheodory measure are considered. The behavior of the Birkhoff sums for integrable and almost everywhere bounded functions with zero mean value in terms of the Caratheodory measure is studied. It is shown that for almost all points of the metric space there is an infinite sequence of “moments of time”; along which the Birkhoff sums tend to zero and at the same moments the trajectory points approach their initial position as close as possible (as in the Poincare return theorem). As an example, we consider the transformation $x\mapsto 2x$ mod 1; of the single segment 0 $\le x \le 1$ closely related to Bernoulli tests.