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Almost sure invariance principle for nonuniformly hyperbolic systems. (English) Zbl 1084.37024

Summary: We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon.
Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.

MSC:

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37A99 Ergodic theory
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
82B05 Classical equilibrium statistical mechanics (general)
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[1] Aaronson, J.: An Introduction to Infinite Ergodic Theory. Math. Surveys and Monographs 50, Providence, RI: Amer. Math. Soc., 1997 · Zbl 0882.28013
[2] Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1, 193–237 (2001) · Zbl 1039.37002 · doi:10.1142/S0219493701000114
[3] Alves, J., Luzzatto, S., Pinheiro, V.: Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré, Anal. Non Linéaire, to appear · Zbl 1134.37326
[4] Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16, Singapore: World Scientific, 2000 · Zbl 1012.37015
[5] Baladi, V.: Decay of correlations. In: Katok, A. (ed.) et al., Smooth Ergodicity Theory and its Applications Proc. Symp. Pure Math. 69, Providence, RI: Amer. Math. Soc., 2001, pp. 297–325 · Zbl 0993.37003
[6] Benedicks, M., Young, L.-S.: Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12, 13–37 (1992) · Zbl 0769.58051
[7] Benedicks, M., Young, L.-S.: Sinai-Bowen-Ruelle measures for certain Hénon maps. Invent. Math. 112, 541–576 (1993) · Zbl 0796.58025 · doi:10.1007/BF01232446
[8] Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. 470, Berlin: Springer, 1975 · Zbl 0308.28010
[9] Bruin, H., Holland, M., Nicol, M.: Livsic regularity for Markov systems. Ergod. Th. and Dyn. Syst. To appear · Zbl 1083.37029
[10] Bruin, H., Luzzatto, S., van Strien, S.: Decay of correlations in one-dimensional dynamics. Ann. Sci. École Norm. Sup. 36, 621–646 (2003) · Zbl 1039.37021
[11] Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk 46, 43–92 (1991) · Zbl 0748.58014
[12] Castro, A.: Backward inducing and exponential decay of correlations for partially hyperbolic attractors with mostly contracting direction. Ph. D. Thesis, IMPA (1998)
[13] Chernov, N.: Statistical properties of piecewise smooth hyperbolic systems in high dimensions. Discrete Contin. Dynam. Systems 5, 425–448 (1999) · Zbl 0965.37004 · doi:10.3934/dcds.1999.5.425
[14] Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999) · Zbl 1047.37503 · doi:10.1023/A:1004581304939
[15] Chernov, N., Young, L.S.: Decay of correlations for Lorentz gases and hard balls. In: Hard ball systems and the Lorentz gas. Encyclopaedia Math. Sci. 101, Berlin: Springer, 2000, pp. 89–120 · Zbl 0977.37001
[16] Conze, J.-P., Le Borgne, S.: Méthode de martingales et flow géodésique sur une surface de courbure constante négative. Ergod. Th. & Dyn. Sys. 21, 421–441 (2001) · Zbl 0983.37034
[17] Denker, M., Philipp, W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. & Dyn. Sys. 4, 541–552 (1984) · Zbl 0554.60077
[18] Dolgopyat, D.: On dynamics of mostly contracting diffeomorphisms. Commun. Math. Phys. 213, 181–201 (2000) · Zbl 0964.37020 · doi:10.1007/s002200000238
[19] Field, M.J., Melbourne, I., Török, A.: Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions. Ergod. Th. & Dyn. Sys. 23, 87–110 (2003) · Zbl 1140.37315
[20] Gordin, M.I.: The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176 (1969) · Zbl 0212.50005
[21] Gouëzel, S., Statistical properties of a skew product with a curve of neutral points. Preprint, 2004 · Zbl 1070.37003
[22] Gouëzel, S.: Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes. Ph. D. Thesis, Ecole Normale Supérieure, 2004
[23] Hennion, H.: Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118, 627–634 (1993) · Zbl 0772.60049
[24] Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982) · Zbl 0485.28016 · doi:10.1007/BF01215004
[25] Keller, G.: Un théorème de la limite centrale pour une classe de transformations monotones per morceaux. C. R. Acad. Sci. Paris 291, 155–158 (1980) · Zbl 0446.60013
[26] Liverani, C.: Central limit theorem for deterministic systems. In: International Conference on Dynamical Systems (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Research Notes in Math. 362, Harlow: Longman Group Ltd, 1996, pp. 56–75 · Zbl 0871.58055
[27] Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergod Th. and Dyn. Sys. 19, 671–685 (1999) · Zbl 0988.37035 · doi:10.1017/S0143385799133856
[28] Melbourne, I., Nicol, M.: Statistical properties of endomorphisms and compact group extensions. J. London Math. Soc. 70, 427–446 (2004) · Zbl 1160.37331 · doi:10.1112/S0024610704005587
[29] Melbourne, I., Török, A.: Central limit theorems and invariance principles for time-one maps of hyperbolic flows. Commun. Math. Phys. 229, 57–71 (2002) · Zbl 1098.37501 · doi:10.1007/s00220-002-0676-5
[30] Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 194, 191–210 (2004) · Zbl 1252.37010 · doi:10.1007/BF02916712
[31] Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérique 187–188, Montrouge: Société Mathématique de France, 1990
[32] Philipp, W., Stout, W.F.: Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables. Mem. of the Amer. Math. Soc. 161, Providence, RI: Amer. Math. Soc., 1975 · Zbl 0361.60007
[33] Pollicott, M., Sharp, R.: Invariance principles for interval maps with an indifferent fixed point. Commun. Math. Phys. 229, 337–346 (2002) · Zbl 1074.37007 · doi:10.1007/s00220-002-0685-4
[34] Ratner, M.: The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973) · Zbl 0283.58010 · doi:10.1007/BF02757869
[35] Ruelle, D.: Thermodynamic Formalism. Encyclopedia of Math. and its Applications 5, Reading, Massachusetts: Addison Wesley, 1978 · Zbl 0401.28016
[36] Sinai, Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25, 141–192 (1970) · Zbl 0252.58005
[37] Sinai, Y.G.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–70 (1972) · Zbl 0246.28008 · doi:10.1070/RM1972v027n04ABEH001383
[38] Viana, M.: Stochastic dynamics of deterministic systems. Col. Bras. de Matemática, 1997
[39] Walkden, C.P.: Invariance principles for iterated maps that contract on average. Preprint, 2003 · Zbl 1110.60024
[40] Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147, 585–650 (1998) · Zbl 0945.37009 · doi:10.2307/120960
[41] Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999) · Zbl 0983.37005 · doi:10.1007/BF02808180
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