Abstract
A range of issues related to the recurrence of integrals of conditionally periodic functions with zero mean value is discussed. In the case of smooth functions on the torus, the recurrence of integrals obviously holds for all initial phases. A new observation is that, for almost all initial phases, the recurrence 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 extended to the general case of ergodic transformations of compact metric spaces with Carathéodory measure.
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ACKNOWLEDGMENTS
The author is grateful to Academician V.V. Kozlov for helpful discussions of the issues considered in this paper.
Funding
This work was supported by the Russian Science Foundation, project no. 21-71-30011.
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Translated by I. Ruzanova
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Denisova, N.V. Recurrence of Integrals of Conditionally Periodic Functions. Dokl. Math. 108, 316–319 (2023). https://doi.org/10.1134/S1064562423700849
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DOI: https://doi.org/10.1134/S1064562423700849