Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. von Neumann, Physical applications of the ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), no. 3, 263-266.
  • 2. A. G. Kachurovskii, Rates of convergence in ergodic theorems, Uspekhi Mat. Nauk 51 (1996), no. 4(310), 73-124; English transl., Russian Math. Surveys 51 (1996), no. 4, 653-703. MR 1422228
  • 3. I. P. Kornfeld, Ya. G. Sinai, and S. V. Fomin, Ergodic theory, Nauka, Moscow, 1980; English transl., Springer, Berlin-Heidelberg-New York, 1982. MR 610981
  • 4. A. G. Kachurovskii 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; English transl., Sb. Math. 201 (2010), no. 3-4, 493-500. MR 2675340
  • 5. 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)
  • 6. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585-650. MR 1637655
  • 7. L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188. MR 1750438
  • 8. L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 587-612. MR 2408394
  • 9. I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6661-6676. MR 2434305
  • 10. I. Melbourne, Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1735-1741. MR 2470832
  • 11. V. P. Leonov, On the dispersion of time means of a stationary stochastic process, Teor. Veroyatnost. Primenen. 6 (1961), no. 1, 93-101; English transl., Theory Probab. Appl. 6 (1961), no. 1, 87-93. MR 0133173
  • 12. 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.
  • 13. A. G. Kachurovskii 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; English transl., Math. Notes 87 (2010), no. 5-6, 720-727. MR 2766588
  • 14. A. G. Kachurovskii 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; English transl., Sb. Math. 202 (2011), no. 7-8, 1105-1125. MR 2866197
  • 15. N. A. Dzhulai and A. G. Kachurovskii, 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; English transl., Sib. Math. J. 52 (2011), no. 5, 824-835. MR 2908125
  • 16. A. G. Kachurovskii and V. V. Sedalishchev, On the constants in the estimates of the rate of convergence in the Birkhoff ergodic theorem, Mat. Zametki 91 (2012), no. 4, 624-628; English transl., Math. Notes 91 (2012), no. 3-4, 582-587. MR 3201463
  • 17. 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; English transl., Sib. Math. J. 53 (2012), no. 5, 882-888. MR 3057930
  • 18. 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; English transl., Sib. Math. J. 55 (2014), no. 2, 336-348. MR 3237344
  • 19. A. G. Kachurovskii and I. V. Podvigin, Large deviations and the rate of convergence in the Birkhoff ergodic theorem, Mat. Zametki 94 (2013), no. 4, 569-577; English transl., Math. Notes 94 (2013), no. 3-4, 524-531. MR 3423283
  • 20. A. G. Kachurovskii and I. V. Podvigin, Convergence rates in ergodic theorems for some billiards and Anosov diffeomorphisms, Dokl. Akad. Nauk 451 (2013), no. 1, 11-13; English transl., Dokl. Math. 88 (2013), no. 1, 385-387. MR 3155153
  • 21. A. G. Kachurovskii 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; English transl., Dokl. Math. 89 (2014), no. 2, 139-142. MR 3237613
  • 22. A. G. Kachurovskii, On the convergence of averages in the ergodic theorem for the groups $ \mathbb{Z}^d$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 256 (1999), 121-128; English transl., J. Math. Sci. (New York) 107 (2001), no. 5, 4231-4236. MR 1708562
  • 23. A. M. Vershik and A. G. Kachurovskii, Rates of convergence in ergodic theorems for locally finite groups, and reversed martingales, Differ. Uravn. Protsessy Upr. (1999), no. 1, 19-26. (Russian) MR 1890198
  • 24. A. G. Kachurovskii, On uniform convergence in the ergodic theorem, J. Math. Sci. (New York) 95 (1999), no. 5, 2546-2551. MR 1712742
  • 25. A. N. Shiryaev, Probability, Nauka, Moscow, 1989; English transl., Springer, New York, 1995. MR 1024077
  • 26. C. Cuny and M. 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. MR 2548500
  • 27. A. Gomilko, M. Haase, and Yu. Tomilov, On rates in mean ergodic theorems, Math. Res. Lett. 18 (2011), no. 2, 201-213. MR 2784667
  • 28. A. Gomilko, M. Haase, and Yu. Tomilov, Bernstein functions and rates in mean ergodic theorems for operator semigroups, J. Anal. Math. 118 (2012), no. 2, 545-576. MR 3000691
  • 29. M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), no. 4, 739-741; English transl., Sov. Math. Dokl. 10 (1969), 1174-1176. MR 0251785
  • 30. M. Stenlund, A strong pair correlation bound implies the CLT for Sinai billiards, J. Stat. Phys. 140 (2010), no. 1, 154-169. MR 2651443
  • 31. U. Krengel, Ergodic theorems, de Gruyter Stud. in Math., vol. 6, Walter de Gruyter, Berlin, 1985.
  • 32. N. K. Bari, A treatise on trigonometric series, Fizmatgiz, Moscow, 1961; English transl., vol. I, II, Pergamon Press, Oxford-London-New York-Paris-Frankfurt, 1964. MR 0171116
  • 33. J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), no. 1, 70-82.
  • 34. I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Nauka, Moscow, 1965; English transl., Wolters-Noordhoff Publishing Company, Groningen, 1971. MR 0322926
  • 35. V. F. Gaposhkin, On the rate of decrease of the probabilities of $ \varepsilon $-deviations for means of stationary processes, Mat. Zametki 64 (1998), no. 3, 366-372; English transl., Math. Notes 64 (1998), no. 3-4, 316-321 (1999). MR 1680150
  • 36. V. F. Gaposhkin, Estimates of means for almost all realizations of stationary processes, Sibirsk. Mat. Zh. 20 (1979), no. 5, 978-989; English transl., Sib. Math. J. 20 (1980), no. 5, 691-699. MR 559060
  • 37. V. F. Gaposhkin, Convergence of series connected with stationary sequences, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 6, 1366-1392; English transl., Math. USSR-Izv. 9 (1975), no. 6, 1297-1321. MR 0402880
  • 38. F. Browder, On the iterations of transformations in noncompact minimal dynamical systems, Proc. Amer. Math. Soc. 9 (1958), no. 5, 773-780. MR 0096975
  • 39. A. Zygmund, Trigonometric series, vol. 1, At the University Press, Cambridge, 1959. MR 0107776
  • 40. I. Assani and M. Lin, On the one-sided Hilbert transform, Ergodic Theory and Related Fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, 21-39. MR 2331323
  • 41. R. E. Edwards, Fourier series. A modern introduction, vol. 1, 2, Springer-Verlag, New York-Heidelberg-Berlin, 1979, 1982. MR 545506
  • 42. A. Ya. Helemskii, Lectures and exercises on functional analysis, MCCME, Moscow, 2004; English transl., Transl. of Math. Monographs, vol. 233, Amer. Math. Soc., Providence, RI, 2006. MR 2248303
  • 43. V. F. Gaposhkin, Some examples concerning the problem of $ \varepsilon $-deviations for stationary sequences, Teor. Veroyatnost. Primenen. 46 (2001), no. 2, 370-375; English transl., Theory Probab. Appl. 46 (2003), no. 2, 341-346. MR 1968692
  • 44. V. I. Bogachev, Measure theory, vol. 1, Regul. i Khaotich. Dinamika, Moscow-Izhevsk, 2003; English transl., Springer-Verlag, Berlin-Heidelberg, 2007. MR 2267655
  • 45. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern geometry. Methods and applications, Nauka, Moscow, 1986; English transl., parts 1-3, Springer-Verlag, New York, 1992, 1985,
    1990. MR 864355
  • 46. M. L. Blank, Stability and localization in chaotic dynamics, MCCME, Moscow, 2001. (Russian)
  • 47. E. Lesigne and D. Volný, Large deviations for generic stationary processes, Colloq. Math. 84/85 (2000), part 1, 75-82. MR 1778841
  • 48. O. Sarig, Decay of correlations, Handbook of Dynamical Systems 1 (2006), part B, 244-263.
  • 49. L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys. 108 (2002), no. 5-6, 733-754. MR 1933431
  • 50. C. Liverani, B. Saussol, and S. Vaienty, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671-685. MR 1695915
  • 51. H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 495-524. MR 2054191
  • 52. M. Pollicott, R. Sharp, and M. Yuri, Large deviations for maps with indifferent fixed points, Nonlinearity 11 (1998), no. 4, 1173-1184. MR 1632614
  • 53. M. Pollicott and R. Sharp, Large deviations for intermittent maps, Nonlinearity 22 (2009), no. 9, 2079-2092. MR 2534293
  • 54. O. Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), no. 3, 629-653. MR 1946554
  • 55. S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math. 139 (2004), 29-65. MR 2041223
  • 56. R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 177-197. MR 2041267
  • 57. N. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity 18 (2005), no. 4, 1527-1553. MR 2150341
  • 58. N. Chernov and R. Markarian, Dispersing billiards with cusps: slow decay of correlations, Comm. Math. Phys. 270 (2007), no. 3, 727-758. MR 2276463
  • 59. N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards, Comm. Math. Phys. 277 (2008), no. 2, 305-321. MR 2358286
  • 60. N. Chernov and R. Markarian, Chaotic billiards, Math. Surveys and Monographs, vol. 127, Amer. Math. Soc., Providence, RI, 2006. MR 2229799
  • 61. S. Troubetzkoy, Approximation and billiards, Dynamical systems and Diophantine approximation, Sémin. congr., vol. 19, Soc. Math. France, Paris, 2009, 173-185. MR 2808408
  • 62. P. Balint and I. Melbourne, Decay of correlations and invariance principle for dispersing billiards with cusps, and related planar billiard flows, J. Stat. Phys. 133 (2008), no. 3, 435-447. MR 2448631
  • 63. V. M. Anikin and A. F. Golubentsev, Analytical models of deterministic chaos, Fizmatlit, Moscow, 2007. (Russian)
  • 64. 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.
  • 65. W. Parry, On the $ \beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), no. 3-4, 401-416. MR 0142719
  • 66. D. Mayer and G. Roepstorff, On the relaxation time of Gauss's continued-fraction map I. The Hilbert space approach (Koopmanism), J. Stat. Phys. 47 (1987), no. 1-2, 149-171. MR 892927
  • 67. D. Mayer and G. Roepstorff, On the relaxation time of Gauss's continued-fraction map II. The Banach space approach (Transfer operator method), J. Stat. Phys. 50 (1988), no. 1-2, 331-344. MR 939491
  • 68. I. Antoniou and S. Shkarin, Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map, J. Stat. Phys. 111 (2003), no. 1-2, 355-369. MR 1964274
  • 69. M. Iosifescu and C. Kraaikamp, Metrical theory and continued fractions, Mathematics and Its Applications, vol. 547, Kluwer Acad. Publ., Dordrecht, 2002. MR 1960327
  • 70. S. Waddington, Large deviation asymptotics for Anosov flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 4, 445-484. MR 1404318
  • 71. X. Bressaud and C. Liverani, Anosov diffeomorphisms and coupling, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 129-152. MR 1889567
  • 72. R. Bowen, Methods of symbolic dynamics, Mathematics, vol. 13, Mir, Moscow, 1979. (Russian)
  • 73. L. Orey and S. Pelikan, Deviation 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
  • 74. Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990), no. 2, 505-524. MR 1025756
  • 75. M. Pollicott and R. Sharp, Large deviations, fluctuations and shrinking intervals, Comm. Math. Phys. 290 (2009), no. 1, 321-334. MR 2520516
  • 76. L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525-543. MR 975689
  • 77. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014
  • 78. 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)
  • 79. N. I. Chernov, Markov approximations and decay of correlations for Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 269-324. MR 1626741
  • 80. D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357-390. MR 1626749
  • 81. D. Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097-1114. MR 1653299
  • 82. D. Dolgopyat, Prevalence of rapid mixing. II. Topological prevalence, Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1045-1059. MR 1779392
  • 83. C. Liverani, On contact Anosov flows, Ann. of Math. (2) 159 (2004), no. 3, 1275-1312. MR 2113022
  • 84. K. 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
  • 85. V. Lynch, Decay of correlations for non-Hölder observables, Discrete Contin. Dyn. Syst. 16 (2006), no. 1, 19-46. MR 2221734
  • 86. H.-K. Zhang, Decay of correlations on non-Hölder observables, Int. J. Nonlinear Sci. 10 (2010), no. 3, 359-369. MR 2755057
  • 87. L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys. 146 (1992), no. 1, 123-138. MR 1163671
  • 88. G. Keller and T. 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
  • 89. M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque 261 (2000), 13-56. MR 1755436
  • 90. N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys. 94 (1999), no. 3-4, 513-556. MR 1675363
  • 91. N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls, Encyclopaedia Math. Sci., vol. 101, Springer, Berlin, 2000, 89-120. MR 1805327
  • 92. N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys. 122 (2006), no. 6, 1061-1094. MR 2219528
  • 93. A. Avila, S. Gouëzel, and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143-211. MR 2264836
  • 94. J. S. Athreya, Quantitative recurrence and large deviations for Teichmüller geodesic flow, Geom. Dedicata 119 (2006), no. 1, 121-140. MR 2247652
  • 95. V. Araujo and A. 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

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 37A30, 37D20, 37D50, 60G10

Retrieve articles in all journals with MSC (2010): 37A30, 37D20, 37D50, 60G10


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

American Mathematical Society