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

Kachurovskii, Aleksandr; Ivan V. Podvigin

doi:10.1090/mosc/256

Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems
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by Aleksandr G. Kachurovskii and Ivan V. Podvigin
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2016, 1-53

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.

References

J. von Neumann, Physical applications of the ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), no. 3, 263–266.
A. G. Kachurovskiĭ, Rates of convergence in ergodic theorems, Uspekhi Mat. Nauk 51 (1996), no. 4(310), 73–124 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 4, 653–703. MR 1422228, DOI 10.1070/RM1996v051n04ABEH002964
I. P. Kornfel′d, Ya. G. Sinaĭ, and S. V. Fomin, Ergodicheskaya teoriya, “Nauka”, Moscow, 1980 (Russian). MR 610981
A. G. Kachurovskiĭ and A. V. Reshetenko, On the rate of convergence in von Neumann’s ergodic theorem with continuous time, Mat. Sb. 201 (2010), no. 4, 25–32 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 3-4, 493–500. MR 2675340, DOI 10.1070/SM2010v201n04ABEH004080
Ya. G. Sinai, Ergodic theory of smooth dynamical systems, Chap. 6. Stochasticity of smooth dynamical systems. Elements of KAM theory, Current problems in mathematics, Fundamental directions, vol. 2, Dynamical systems–2, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, 115–122. (Russian)
Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180
Luc Rey-Bellet and Lai-Sang Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 587–612. MR 2408394, DOI 10.1017/S0143385707000478
Ian Melbourne and Matthew Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6661–6676. MR 2434305, DOI 10.1090/S0002-9947-08-04520-0
Ian Melbourne, Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1735–1741. MR 2470832, DOI 10.1090/S0002-9939-08-09751-7
V. P. Leonov, On the dispersion of time means of a stationary stochastic process. , Teor. Verojatnost. i Primenen. 6 (1961), 93–101 (Russian, with English summary). MR 0133173
Yu. K. Belyaev, An example of a process with mixing, Teor. Veroyatnost. Primenen. 6 (1961), no. 1, 101–102; English transl., Theory Probab. Appl. 6 (1961), no. 1, 93–94.
A. G. Kachurovskiĭ and V. V. Sedalishchev, On the constants in the estimates for the rate of convergence in von Neumann’s ergodic theorem, Mat. Zametki 87 (2010), no. 5, 756–763 (Russian, with Russian summary); English transl., Math. Notes 87 (2010), no. 5-6, 720–727. MR 2766588, DOI 10.1134/S000143461005010X
A. G. Kachurovskiĭ and V. V. Sedalishchev, Constants of estimates for the rate of convergence in the von Neumann and Birkhoff ergodic theorems, Mat. Sb. 202 (2011), no. 8, 21–40 (Russian, with Russian summary); English transl., Sb. Math. 202 (2011), no. 7-8, 1105–1125. MR 2866197, DOI 10.1070/SM2011v202n08ABEH004180
N. A. Dzhulaĭ and A. G. Kachurovskiĭ, Constants in estimates for the rate of convergence in von Neumann’s ergodic theorem with continuous time, Sibirsk. Mat. Zh. 52 (2011), no. 5, 1039–1052 (Russian, with Russian summary); English transl., Sib. Math. J. 52 (2011), no. 5, 824–835. MR 2908125, DOI 10.1134/S0037446611050077
A. G. Kachurovskii and V. V. Sedalishchev, On the constants in the estimates of the rate of convergence in the Birkhoff ergodic theorem, Math. Notes 91 (2012), no. 3-4, 582–587. Translation of Mat. Zametki 91 (2012), no. 4, 624–628. MR 3201463, DOI 10.1134/S0001434612030340
V. V. Sedalishchev, Constants in estimates for the rate of convergence in Birkhoff’s ergodic theorem with continuous time, Sibirsk. Mat. Zh. 53 (2012), no. 5, 1102–1110 (Russian, with Russian summary); English transl., Sib. Math. J. 53 (2012), no. 5, 882–888. MR 3057930, DOI 10.1134/S0037446612050138
V. V. Sedalishchev, The relation between rates of convergence in the von Neumann and Birkhoff ergodic theorems in $L_p$, Sibirsk. Mat. Zh. 55 (2014), no. 2, 412–426 (Russian, with Russian summary); English transl., Sib. Math. J. 55 (2014), no. 2, 336–348. MR 3237344, DOI 10.1134/s0037446614020165
A. G. Kachurovskii and I. V. Podvigin, Large deviations and the rate of convergence in the Birkhoff ergodic theorem, Math. Notes 94 (2013), no. 3-4, 524–531. Russian version appears in Mat. Zametki 94 (2013), no. 4, 569–577. MR 3423283, DOI 10.1134/S0001434613090228
A. G. Kachurovskiĭ and I. V. Podvigin, Convergence rates in ergodic theorems for some billiards and Anosov diffeomorphisms, Dokl. Akad. Nauk 451 (2013), no. 1, 11–13 (Russian); English transl., Dokl. Math. 88 (2013), no. 1, 385–387. MR 3155153, DOI 10.1134/s1064562413040029
A. G. Kachurovskiĭ and I. V. Podvigin, Rates of convergence in ergodic theorems for the periodic Lorentz gas on the plane, Dokl. Akad. Nauk 455 (2014), no. 1, 11–14 (Russian); English transl., Dokl. Math. 89 (2014), no. 2, 139–142. MR 3237613, DOI 10.1134/s106456241402001x
A. G. Kachurovskiĭ, On the convergence of averages in the ergodic theorem for the groups $\mathbf Z^d$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 256 (1999), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3, 121–128, 266 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 107 (2001), no. 5, 4231–4236. MR 1708562, DOI 10.1023/A:1012425724804
A. M. Vershik and A. G. Kachurovskiĭ, Rates of convergence in ergodic theorems for locally finite groups, and reversed martingales, Differ. Uravn. Protsessy Upr. 1 (1999), 19–26 (Russian, with English and Russian summaries). MR 1890198
A. G. Kachurovskii, On uniform convergence in the ergodic theorem, J. Math. Sci. (New York) 95 (1999), no. 5, 2546–2551. Dynamical systems. 7. MR 1712742, DOI 10.1007/BF02169054
A. N. Shiryaev, Veroyatnost′, 2nd ed., “Nauka”, Moscow, 1989 (Russian). MR 1024077
Christophe Cuny and Michael Lin, Pointwise ergodic theorems with rate and application to the CLT for Markov chains, Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 710–733 (English, with English and French summaries). MR 2548500, DOI 10.1214/08-AIHP180
Alexander Gomilko, Markus Haase, and Yuri Tomilov, On rates in mean ergodic theorems, Math. Res. Lett. 18 (2011), no. 2, 201–213. MR 2784667, DOI 10.4310/MRL.2011.v18.n2.a2
Alexander Gomilko, Markus Haase, and Yuri Tomilov, Bernstein functions and rates in mean ergodic theorems for operator semigroups, J. Anal. Math. 118 (2012), no. 2, 545–576. MR 3000691, DOI 10.1007/s11854-012-0044-0
M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
Mikko Stenlund, A strong pair correlation bound implies the CLT for Sinai billiards, J. Stat. Phys. 140 (2010), no. 1, 154–169. MR 2651443, DOI 10.1007/s10955-010-9987-7
U. Krengel, Ergodic theorems, de Gruyter Stud. in Math., vol. 6, Walter de Gruyter, Berlin, 1985.
N. K. Bary, A treatise on trigonometric series. Vols. I, II, A Pergamon Press Book, The Macmillan Company, New York, 1964. Authorized translation by Margaret F. Mullins. MR 0171116
J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), no. 1, 70–82.
I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
V. F. Gaposhkin, On the rate of decrease of the probabilities of $\epsilon$-deviations for means of stationary processes, Mat. Zametki 64 (1998), no. 3, 366–372 (Russian, with Russian summary); English transl., Math. Notes 64 (1998), no. 3-4, 316–321 (1999). MR 1680150, DOI 10.1007/BF02314839
V. F. Gapoškin, Estimates of means for almost all realizations of stationary processes, Sibirsk. Mat. Zh. 20 (1979), no. 5, 978–989, 1165 (Russian). MR 559060
V. F. Gapoškin, Convergence of series that are connected with stationary sequences, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 6, 1366–1392, 1438 (Russian). MR 0402880
Felix E. Browder, On the iteration of transformations in noncompact minimal dynamical systems, Proc. Amer. Math. Soc. 9 (1958), 773–780. MR 96975, DOI 10.1090/S0002-9939-1958-0096975-9
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Idris Assani and Michael Lin, On the one-sided ergodic Hilbert transform, Ergodic theory and related fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 21–39. MR 2331323, DOI 10.1090/conm/430/08249
R. E. Edwards, Fourier series. A modern introduction. Vol. 1, 2nd ed., Graduate Texts in Mathematics, vol. 64, Springer-Verlag, New York-Berlin, 1979. MR 545506
A. Ya. Helemskii, Lectures and exercises on functional analysis, Translations of Mathematical Monographs, vol. 233, American Mathematical Society, Providence, RI, 2006. Translated from the 2004 Russian original by S. Akbarov. MR 2248303, DOI 10.1090/mmono/233
V. F. Gaposhkin, Some examples concerning the problem of $\epsilon$-deviations for stationary sequences, Teor. Veroyatnost. i Primenen. 46 (2001), no. 2, 370–375 (Russian, with Russian summary); English transl., Theory Probab. Appl. 46 (2003), no. 2, 341–346. MR 1968692, DOI 10.1137/S0040585X9797897X
V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Sovremennaya geometriya, 2nd ed., “Nauka”, Moscow, 1986 (Russian). Metody i prilozheniya. [Methods and applications]. MR 864355
M. L. Blank, Stability and localization in chaotic dynamics, MCCME, Moscow, 2001. (Russian)
Emmanuel Lesigne and Dalibor Volný, Large deviations for generic stationary processes. part 1, Colloq. Math. 84/85 (2000), no. part 1, 75–82. Dedicated to the memory of Anzelm Iwanik. MR 1778841, DOI 10.4064/cm-84/85-1-75-82
O. Sarig, Decay of correlations, Handbook of Dynamical Systems 1 (2006), part B, 244–263.
Lai-Sang Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys. 108 (2002), no. 5-6, 733–754. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933431, DOI 10.1023/A:1019762724717
Carlangelo Liverani, Benoît Saussol, and Sandro Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671–685. MR 1695915, DOI 10.1017/S0143385799133856
Huyi Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 495–524. MR 2054191, DOI 10.1017/S0143385703000671
Mark Pollicott, Richard Sharp, and Michiko Yuri, Large deviations for maps with indifferent fixed points, Nonlinearity 11 (1998), no. 4, 1173–1184. MR 1632614, DOI 10.1088/0951-7715/11/4/023
Mark Pollicott and Richard Sharp, Large deviations for intermittent maps, Nonlinearity 22 (2009), no. 9, 2079–2092. MR 2534293, DOI 10.1088/0951-7715/22/9/001
Omri Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), no. 3, 629–653. MR 1946554, DOI 10.1007/s00222-002-0248-5
Sébastien Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math. 139 (2004), 29–65. MR 2041223, DOI 10.1007/BF02787541
Roberto Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 177–197. MR 2041267, DOI 10.1017/S0143385703000270
N. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity 18 (2005), no. 4, 1527–1553. MR 2150341, DOI 10.1088/0951-7715/18/4/006
N. Chernov and R. Markarian, Dispersing billiards with cusps: slow decay of correlations, Comm. Math. Phys. 270 (2007), no. 3, 727–758. MR 2276463, DOI 10.1007/s00220-006-0169-z
N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards, Comm. Math. Phys. 277 (2008), no. 2, 305–321. MR 2358286, DOI 10.1007/s00220-007-0360-x
Nikolai Chernov and Roberto Markarian, Chaotic billiards, Mathematical Surveys and Monographs, vol. 127, American Mathematical Society, Providence, RI, 2006. MR 2229799, DOI 10.1090/surv/127
Serge Troubetzkoy, Approximation and billiards, Dynamical systems and Diophantine approximation, Sémin. Congr., vol. 19, Soc. Math. France, Paris, 2009, pp. 173–185 (English, with English and French summaries). MR 2808408
Péter Bálint and Ian Melbourne, Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows, J. Stat. Phys. 133 (2008), no. 3, 435–447. MR 2448631, DOI 10.1007/s10955-008-9623-y
V. M. Anikin and A. F. Golubentsev, Analytical models of deterministic chaos, Fizmatlit, Moscow, 2007. (Russian)
V. M. Anikin, S. S. Arkadakskii, and A. S. Remizov, An analytical solution to the spectral problem for the Perron-Frobenius operator of piecewise linear chaotic maps, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelinein. Din. 14 (2006), no. 2, 16–34.
W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
D. Mayer and G. Roepstorff, On the relaxation time of Gauss’s continued-fraction map. I. The Hilbert space approach (Koopmanism), J. Statist. Phys. 47 (1987), no. 1-2, 149–171. MR 892927, DOI 10.1007/BF01009039
D. Mayer and G. Roepstorff, On the relaxation time of Gauss’ continued-fraction map. II. The Banach space approach (transfer operator method), J. Statist. Phys. 50 (1988), no. 1-2, 331–344. MR 939491, DOI 10.1007/BF01022997
I. Antoniou and S. A. Shkarin, Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map, J. Statist. Phys. 111 (2003), no. 1-2, 355–369. MR 1964274, DOI 10.1023/A:1022217410549
Marius Iosifescu and Cor Kraaikamp, Metrical theory of continued fractions, Mathematics and its Applications, vol. 547, Kluwer Academic Publishers, Dordrecht, 2002. MR 1960327, DOI 10.1007/978-94-015-9940-5
Simon Waddington, Large deviation asymptotics for Anosov flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 4, 445–484. MR 1404318, DOI 10.1016/S0294-1449(16)30110-X
Xavier Bressaud and Carlangelo Liverani, Anosov diffeomorphisms and coupling, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 129–152. MR 1889567, DOI 10.1017/S0143385702000056
R. Bowen, Methods of symbolic dynamics, Mathematics, vol. 13, Mir, Moscow, 1979. (Russian)
Steven Orey and Stephan Pelikan, Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms, Trans. Amer. Math. Soc. 315 (1989), no. 2, 741–753. MR 935534, DOI 10.1090/S0002-9947-1989-0935534-4
Yuri Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990), no. 2, 505–524. MR 1025756, DOI 10.1090/S0002-9947-1990-1025756-7
Mark Pollicott and Richard Sharp, Large deviations, fluctuations and shrinking intervals, Comm. Math. Phys. 290 (2009), no. 1, 321–334. MR 2520516, DOI 10.1007/s00220-008-0725-9
Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525–543. MR 975689, DOI 10.1090/S0002-9947-1990-0975689-7
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
A. G. Kachurovskii and I. V. Podvigin, Estimates of the rate of convergence in the Birkhoff and Bowen theorems for Anosov flows, Vestnik Kemerov. Univ. 47 (2011), no. 3/1, 255–258. (Russian)
N. I. Chernov, Markov approximations and decay of correlations for Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 269–324. MR 1626741, DOI 10.2307/121010
Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
Dmitry Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097–1114. MR 1653299, DOI 10.1017/S0143385798117431
Dmitry Dolgopyat, Prevalence of rapid mixing. II. Topological prevalence, Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1045–1059. MR 1779392, DOI 10.1017/S0143385700000572
Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2) 159 (2004), no. 3, 1275–1312. MR 2113022, DOI 10.4007/annals.2004.159.1275
Karla Díaz-Ordaz, Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps, Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 159–176. MR 2191390, DOI 10.3934/dcds.2006.15.159
Vincent Lynch, Decay of correlations for non-Hölder observables, Discrete Contin. Dyn. Syst. 16 (2006), no. 1, 19–46. MR 2221734, DOI 10.3934/dcds.2006.16.19
Hong-Kun Zhang, Decay of correlations on non-Hölder observables, Int. J. Nonlinear Sci. 10 (2010), no. 3, 359–369. MR 2755057
L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys. 146 (1992), no. 1, 123–138. MR 1163671
Gerhard Keller and Tomasz Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), no. 1, 31–69. MR 1182410
Michael Benedicks and Lai-Sang Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque 261 (2000), xi, 13–56 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755436
N. Chernov, Decay of correlations and dispersing billiards, J. Statist. Phys. 94 (1999), no. 3-4, 513–556. MR 1675363, DOI 10.1023/A:1004581304939
N. Chernov and L. S. Young, Decay of correlations for Lorentz gases and hard balls, Hard ball systems and the Lorentz gas, Encyclopaedia Math. Sci., vol. 101, Springer, Berlin, 2000, pp. 89–120. MR 1805327, DOI 10.1007/978-3-662-04062-1_{5}
N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys. 122 (2006), no. 6, 1061–1094. MR 2219528, DOI 10.1007/s10955-006-9036-8
Artur Avila, Sébastien Gouëzel, and Jean-Christophe Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143–211. MR 2264836, DOI 10.1007/s10240-006-0001-5
Jayadev S. Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata 119 (2006), 121–140. MR 2247652, DOI 10.1007/s10711-006-9058-z
Vítor Araújo and Alexander I. Bufetov, A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials, Ergodic Theory Dynam. Systems 31 (2011), no. 4, 1043–1071. MR 2818685, DOI 10.1017/S0143385710000349