Notes on the Recurrence of Birkhoff Sums

Given a compact metric space with Carathéodory measure, we consider ergodic transformations of the space that are measure-preserving, but not necessarily invertible. The behavior of the Birkhoff sums for integrable and almost everywhere bounded functions with zero mean value in terms of the Carathéodory measure is studied. It is shown that for almost all points of the metric space there is an infinite sequence of “time instants” along which the Birkhoff sums tend to zero and the trajectory points at the these instants approach their initial position as close as possible (as in the Poincaré recurrence theorem). As an example, we consider the transformation $$x \mapsto 2x$$ $$\bmod 1$$ of the unit interval $$0\,\leqslant \,x\,\leqslant \,1$$ closely related to Bernoulli trials.