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

This article is cited in 7 scientific papers (total in 7 papers)

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 НШ-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
Received: 04.02.2014
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
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\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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  crossref  elib
    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  mathnet  crossref  crossref  elib
    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  mathnet
    4. A. G. Kachurovskii, I. V. Podvigin, “Fejer sums and Fourier coefficients of periodic measures”, Dokl. Math., 98:2 (2018), 464–467  mathnet  crossref  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  scopus
    7. K. I. Knizhov, I. V. Podvigin, “O skhodimosti integrala Luzina i ego analogov”, Sib. elektron. matem. izv., 16 (2019), 85–95  mathnet  crossref
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