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 Tr. Mosk. Mat. Obs., 2016, Volume 77, Issue 1, Pages 1–66 (Mi mmo581)

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

A. G. Kachurovskiia, I. V. Podviginb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Faculty of Physics, Novosibirsk State University, Novosibirsk, Russia

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.

Key words and phrases: convergence rates in ergodic theorems, correlation decay, large deviation decay, billiard, Anosov system.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÍØ-5998.2012.1 The research was supported by the Program for State Support of Leading Scientific Schools of the Russian Federation (grant NSh-5998.2012.1).

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English version:
Transactions of the Moscow Mathematical Society, 2016, 77, 1–53

UDC: 517.987+519.214
MSC: Primary 37A30; Secondary 37D20, 37D50, 60G10
Revised: 20.03.2014

Citation: A. G. Kachurovskii, I. V. Podvigin, “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Tr. Mosk. Mat. Obs., 77, no. 1, MCCME, M., 2016, 1–66; Trans. Moscow Math. Soc., 77 (2016), 1–53

Citation in format AMSBIB
\Bibitem{KacPod16} \by A.~G.~Kachurovskii, I.~V.~Podvigin \paper Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems \serial Tr. Mosk. Mat. Obs. \yr 2016 \vol 77 \issue 1 \pages 1--66 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo581} \elib{http://elibrary.ru/item.asp?id=28931382} \transl \jour Trans. Moscow Math. Soc. \yr 2016 \vol 77 \pages 1--53 \crossref{https://doi.org/10.1090/mosc/256} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85001930550} 

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This publication is cited in the following articles:
1. A. G. Kachurovskiǐ, I. V. Podvigin, “Large deviations of the ergodic averages: from Hölder continuity to continuity almost everywhere”, Siberian Adv. Math., 28:1 (2018), 23–38
2. I. V. Podvigin, “Estimates for correlation in dynamical systems: from Hölder continuous functions to general observables”, Siberian Adv. Math., 28:3 (2018), 187–206
3. A. G. Kachurovskii, “Integraly Feiera i ergodicheskaya teorema fon Neimana s nepreryvnym vremenem”, Veroyatnost i statistika. 27, Zap. nauchn. sem. POMI, 474, POMI, SPb., 2018, 171–182
4. A. G. Kachurovskii, I. V. Podvigin, “Fejer sums and Fourier coefficients of periodic measures”, Dokl. Math., 98:2 (2018), 464–467
5. A. G. Kachurovskii, I. V. Podvigin, “Fejer sums for periodic measures and the von Neumann ergodic theorem”, Dokl. Math., 98:1 (2018), 344–347
6. A. G. Kachurovskii, K. I. Knizhov, “Deviations of Fejer sums and rates of convergence in the von Neumann ergodic theorem”, Dokl. Math., 97:3 (2018), 211–214
7. K. I. Knizhov, I. V. Podvigin, “O skhodimosti integrala Luzina i ego analogov”, Sib. elektron. matem. izv., 16 (2019), 85–95
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