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Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems


Authors: Aleksandr G. Kachurovskii and Ivan V. Podvigin
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 77 (2016), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2016, 1-53
MSC (2010): Primary 37A30; Secondary 37D20, 37D50, 60G10
DOI: https://doi.org/10.1090/mosc/256
Published electronically: November 28, 2016
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Abstract: We present estimates (which are necessarily spectral) of the rate of convergence in the von Neumann ergodic theorem in terms of the singularity at zero of the spectral measure of the function to be averaged with respect to the corresponding dynamical system as well as in terms of the decay rate of the correlations (i.e., the Fourier coefficients of this measure). Estimates of the rate of convergence in the Birkhoff ergodic theorem are given in terms of the rate of convergence in the von Neumann ergodic theorem as well as in terms of the decay rate of the large deviation probabilities. We give estimates of the rate of convergence in both ergodic theorems for some classes of dynamical systems popular in applications, including some well-known billiards and Anosov systems.


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Additional Information

Aleksandr G. Kachurovskii
Affiliation: Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Email: agk@math.nsc.ru

Ivan V. Podvigin
Affiliation: Faculty of Physics, Novosibirsk State University, Novosibirsk, Russia
Email: ivan_podvigin@ngs.ru

DOI: https://doi.org/10.1090/mosc/256
Keywords: Convergence rates in ergodic theorems, correlation decay, large deviation decay, billiard, Anosov system.
Published electronically: November 28, 2016
Additional Notes: The research was supported by the Program for State Support of Leading Scientific Schools of the Russian Federation (grant NSh-5998.2012.1).
Article copyright: © Copyright 2016 American Mathematical Society