Recurrence of integrals of conditionally periodic functions

A range of issues related to the returnability of integrals of conditionally periodic functions with zero mean value is discussed. In the case of smooth functions on the torus, the returnability of integrals obviously holds for all initial phases. A new observation is that for almost all initial phases, the returnability property simultaneously holds not only for integrals, but also for phase points on the torus. Moreover, this result is also valid in the case where the corresponding functions on the torus are only continuous. These observations are transferred to the general case ergodic transformations of compact metric spaces with Carateodori measure.