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
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.
PDF file (555 kB)
Transactions of the Moscow Mathematical Society, 2016, 77, 1–53
MSC: Primary 37A30; Secondary 37D20, 37D50, 60G10
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
\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.
\jour Trans. Moscow Math. Soc.
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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
I. V. Podvigin, “Estimates for correlation in dynamical systems: from Hölder continuous functions to general observables”, Siberian Adv. Math., 28:3 (2018), 187–206
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
A. G. Kachurovskii, I. V. Podvigin, “Fejer sums and Fourier coefficients of periodic measures”, Dokl. Math., 98:2 (2018), 464–467
A. G. Kachurovskii, I. V. Podvigin, “Fejer sums for periodic measures and the von Neumann ergodic theorem”, Dokl. Math., 98:1 (2018), 344–347
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
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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