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

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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

## Bibliographic 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
- 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).
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Moscow Math. Soc.
**2016**, 1-53 - MSC (2010): Primary 37A30; Secondary 37D20, 37D50, 60G10
- DOI: https://doi.org/10.1090/mosc/256
- MathSciNet review: 3643963