Abstract
We study the separation from zero of a sequence \( \phi \) to obtain the estimates of the form \( {\phi(n)/n} \) for the rate of pointwise convergence of ergodic averages. Each of these \( \phi \) is shown to be separated from zero for mixings which is not always so for weak mixings. Moreover, for the characteristic function of a nontrivial set, it is shown that there exists a measure preserving transformation with arbitrarily slow decay of ergodic averages.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rokhlin V. A., On the Fundamental Ideas of Measure Theory, Amer. Math. Soc., Providence (1952).
Kornfeld I. P., Sinai Ya. G., and Fomin S. V., Ergodic Theory, Springer, New York (1981).
Kachurovskii A. G., “The rate of convergence in ergodic theorems,” Russian Math. Surveys, vol. 51, no. 4, 653–703 (1996).
Derriennic Y., “Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the central limit theorem,” Discrete Cont. Dyn. Syst., vol. 15, no. 1, 143–158 (2006).
Kachurovskii A. G. and Podvigin I. V., “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems,” Trans. Moscow Math. Soc., vol. 77, no. 1, 1–53 (2016).
Kachurovskii A. G. and Podvigin I. V., “Measuring the rate of convergence in the Birkhoff ergodic theorem,” Math. Notes, vol. 106, no. 1, 52–62 (2019).
Podvigin I. V., “Lower bound of the supremum of ergodic averages for \( {^{d}} \) and \( {^{d}} \)-actions,” Sib. Electr. Math. Reports, vol. 17, 626–636 (2020).
Blum J. R. and Hanson D. L., “On the mean ergodic theorem for subsequences,” Bull. Amer. Math. Soc., vol. 66, no. 6, 308–311 (1960).
Bergelson V., del Junco A., Lemanczyk M., and Rosenblatt J., “Rigidity and non-recurrence along sequences,” Ergodic Theory Dynam. Systems, vol. 34, no. 5, 1464–1502 (2014).
Bellow A. and Losert V., “The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences,” Trans. Amer. Math. Soc., vol. 288, no. 1, 307–345 (1985).
Atkinson G., “Recurrence of co-cycles and random walks,” J. London Math. Soc. (2), vol. 13, no. 3, 486–488 (1976).
Schmidt K., “On recurrence,” Z. Wahrscheinlichkeitstheor. Verw. Geb., vol. 68, no. 1, 75–95 (1984).
Shneiberg I. Ya., “Zeros of integrals along trajectories of ergodic systems,” Funct. Anal. Appl., vol. 19, no. 2, 160–161 (1985).
Stepin A. M., “Spectral properties of generic dynamical systems,” Math. USSR-Izv., vol. 29, no. 1, 159–192 (1987).
Tikhonov S. V., “A complete metric in the set of mixing transformations,” Sb. Math., vol. 198, no. 4, 575–596 (2007).
Friedman N., Gabriel P., and King J., “An invariant for rigid rank-1 transformations,” Ergodic Theory Dinam. Systems, vol. 8, no. 1, 53–72 (1988).
Ferenczi S., “Systems of finite rank,” Colloq. Math., vol. 73, no. 1, 35–65 (1997).
James J., Koberda T., Lindsey K., Silva C. E., and Speh P., “On ergodic transformations that are both weakly mixing and uniformly rigid,” New York J. Math., vol. 15, 393–403 (2009).
Yancey K. B., “On weakly mixing homeomorphisms of the two-torus that are uniformly rigid,” J. Math. Anal. Appl., vol. 399, no. 2, 524–541 (2013).
Kunde P., “Uniform rigidity sequences for weak mixing diffeomorphisms on \( 𝔻^{2},𝔸 \) and \( 𝕋^{2} \),” J. Math. Anal. Appl., vol. 429, no. 1, 111–130 (2015).
Eisner T. and Grivaux S., “Hilbertian Jamison sequences and rigid dynamical systems,” J. Funct. Anal., vol. 261, no. 7, 2013–2052 (2011).
Adams T., “Tower multiplexing and slow weak mixing,” Colloq. Math., vol. 138, no. 1, 47–71 (2015).
Kachurovskii A. G., Podvigin I. V., and Svishchev A. A., “Zero-one law for the rates of convergence in the Birkhoff ergodic theorem with continuous time,” Mat. Tr., vol. 24, no. 2, 65–80 (2021).
Halasz G., “Remarks on the remainder in Birkhoff’s ergodic theorem,” Acta Math. Acad. Scient. Hung., vol. 28, no. 3, 389–395 (1976).
Krengel U., “On the speed of convergence in the ergodic theorem,” Mon. Math., vol. 86, 3–6 (1978).
Kakutani S. and Petersen K., “The speed of convergence in the ergodic theorem,” Monatsh. Math., vol. 91, 11–18 (1981).
Kachurovskii A. G., Podvigin I. V., and Svishchev A. A., “The maximum pointwise rate of convergence in Birkhoff’s ergodic theorem,” Zap. Nauchn. Sem. POMI, vol. 498, 18–25 (2020).
Del Junco A. and Rosenblatt J. M., “Counterexamples in ergodic theory and number theory,” Math. Ann., vol. 245, 185–197 (1979).
Volný D., “On limit theorems and category for dynamical systems,” Yokohama Math. J., vol. 38, 29–35 (1990).
Volny D. and Weiss B., “Coboundaries in \( {L^{\infty}_{0}} \),” Ann. Inst. H. Poincaré Probab. Stat., vol. 40, no. 6, 771–778 (2004).
Kwapien S., “Linear functionals invariant under measure preserving transformations,” Math. Nachr., vol. 119, no. 1, 175–179 (1984).
Halmos P. R., Lectures on Ergodic Theory, Chelsea, New York (1960).
Adams T. and Rosenblatt J., “Joint coboundaries,” in: Dynamical Systems, Ergodic Theory, and Probability: In Memory of Kolya Chernov, vol. 698, Amer. Math. Soc., Providence (2017), 5–33.
Ber A. F., Borst M. J., and Sukochev F. A., “Full proof of Kwapien’s theorem on representing bounded mean zero functions on \( {[0,1]} \),” Studia Math., vol. 259, no. 3, 241–270 (2021).
Funding
The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project 0314–2019–0005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 2, pp. 379–391. https://doi.org/10.33048/smzh.2022.63.209
Rights and permissions
About this article
Cite this article
Podvigin, I.V. On Possible Estimates of the Rate of Pointwise Convergence in the Birkhoff Ergodic Theorem. Sib Math J 63, 316–325 (2022). https://doi.org/10.1134/S0037446622020094
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622020094