Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-04T21:48:59.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations in non-uniformly hyperbolic dynamical systems

Published online by Cambridge University Press:  01 April 2008

LUC REY-BELLET  and
LAI-SANG YOUNG
LUC REY-BELLET
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA (email: [email protected])
LAI-SANG YOUNG
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 587 - 612
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Araújo, V. and Pacifico, M. J.. Large deviations for non-uniformly expanding maps. J. Stat. Phys. 125 (2006), 415–457.Google Scholar
[2]Bahadur, R. R. and Ranga Rao, R.. On deviations from the sample mean. Ann. Math. Statist. 31 (1960), 1015–1027.CrossRefGoogle Scholar
[3]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, River Edge, NJ, 2000.Google Scholar
[4]Baxter, J. R., Jain, N. C. and Varadhan, S. R. S.. Some familiar examples for which the large deviation principle does not hold. Comm. Pure Appl. Math. 44 (1991), 911–923.CrossRefGoogle Scholar
[5]Benedicks, M. and Young, L.-S.. Markov extensions and decay of correlations for certain Hénon maps. Géométrie complexe et systèmes dynamiques (Orsay, 1995). Astérisque 261 (2000), 13–56.Google Scholar
[6]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[7]Broise, A.. Transformations dilatantes de l’intervalle et théorèmes limites. Etudes spectrales d’opérateurs de transfert et applications. Astérisque 238 (1996), 1–109.Google Scholar
[8]Bryc, W.. A remark on the connection between the large deviation principle and the central limit theorem. Statist. Probab. Lett. 18 (1993), 253–256.Google Scholar
[9]Bryc, W. and Dembo, A.. Large deviations and strong mixing. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), 549–569.Google Scholar
[10]Chaganthy, N. R. and Sethuraman, J.. Strong large deviations and local limit theorems. Ann. Probab. 21 (1993), 1671–1690.Google Scholar
[11]Chernov, N. I.. Statistical properties of piecewise smooth hyperbolic systems in high dimensions. Discrete Contin. Dyn. Syst. 5 (1999), 425–448.Google Scholar
[12]Chernov, N. I.. Sinai billiards under small external forces. Ann. Henri Poincaré 2 (2001), 197–236.CrossRefGoogle Scholar
[13]Chernov, N. I.. Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122 (2006), 1061–1094.CrossRefGoogle Scholar
[14]Chernov, N. I., Eyink, G. L., Lebowitz, J. L. and Sinai, Ya. G.. Steady-state electrical conduction in the periodic Lorentz gas. Comm. Math. Phys. 154 (1993), 569–601.Google Scholar
[15]Chernov, N. I. and Young, L.-S.. Decay of correlations for Lorentz gases and hard balls. Encycl. of Math. Sc., Math. Phys. II, vol. 101. Ed. D. Szasz. 2001, pp. 89–120.Google Scholar
[16]Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21 (2001), 401–420.Google Scholar
[17]de Acosta, A. and Chen, X.. Moderate deviations for empirical measures of Markov chains. J. Theoret. Probab. 11 (1995), 1075–1110.Google Scholar
[18]Dembo, A. and Zeitouni, O.. Large Deviations Techniques and Applications, 2nd edn(Applications of Mathematics, 38). Springer, New York, 1998.CrossRefGoogle Scholar
[19]Dolgopyat, D., Szasz, D. and Varju, T.. Recurrence properties of Lorentz gas. Duke Math. J. to appear.Google Scholar
[20]Donsker, M. D. and Varadhan, S. R. S.. Asymptotic evaluation of certain Markov process expectations for large time. I. II. Comm. Pure Appl. Math. 28 (1975), 1–47, 279–301Google Scholar
[21]Hennion, H. and Herv, L.. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness (Lecture Notes in Mathematics, 1766). Springer, Berlin, 2001.Google Scholar
[22]Kato, T.. Perturbation Theory for Linear Operators, 2nd edn(Grundlehren der mathematischen Wissenchaften, 132). Springer, Berlin, 1984.Google Scholar
[23]Keller, G.. Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Amer. Math. Soc. 314 (1989), 433–497.CrossRefGoogle Scholar
[24]Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321 (1990), 505–524.CrossRefGoogle Scholar
[25]Kontoyiannis, I. and Meyn, S. P.. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003), 304–362.Google Scholar
[26]Kontoyiannis, I. and Meyn, S. P.. Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10 (2005), 61–123.CrossRefGoogle Scholar
[27]Melbourne, I. and Nicol, M.. Almost sure invariance principle for non-uniformly hyperbolic systems. Comm. Math. Phys. 260 (2005), 131–146.CrossRefGoogle Scholar
[28]Melbourne, I. and Nicol, M.. Large deviations for non-uniformly hyperbolic systems. Preprint.Google Scholar
[29]Ney, P. and Nummelin, E.. Markov additive processes (I): Eigenvalue properties and limit theorems: (II) Large deviations. Ann. Probab. 15 (1987), 561–592, 593–609.Google Scholar
[30]Orey, S. and Pelikan, S.. Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms. Trans. Amer. Math. Soc. 315 (1989), 741–753.Google Scholar
[31]Pollicott, M., Sharp, R. and Yuri, M.. Large deviations for maps with indifferent fixed points. Nonlinearity 11 (1998), 1173–1184.CrossRefGoogle Scholar
[32]Ruelle, D.. Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (CRM Monograph Series, 4). American Mathematical Society, Providence, RI, 1994.Google Scholar
[33]Schaefer, H. H.. Banach Lattices and Positive Operators (Die Grundlagen der mathematischen Wissenschaft, 215). Springer, Berlin, 1974.Google Scholar
[34]Szasz, D. and Varju, T.. Local limit theorem for the Lorentz process and its recurrence in the plane. Ergod. Th. & Dynam. Sys. 24 (2004), 257–278.CrossRefGoogle Scholar
[35]Wang, Q. and Young, L.-S.. Strange attractors with one direction of instability. Comm. Math. Phys. 218 (2001), 1–97.Google Scholar
[36]Wang, Q. and Young, L.-S.. From invariant curves to strange attractors. Comm. Math. Phys. 225 (2002), 275–304.CrossRefGoogle Scholar
[37]Wang, Q. and Young, L.-S.. Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Comm. Math. Phys. 240 (2003), 509–529.Google Scholar
[38]Wang, Q. and Young, L.-S.. Nonuniformly expanding 1D maps. Comm. Math. Phys. 264 (2006), 255–282.Google Scholar
[39]Wang, Q. and Young, L.-S.. Toward a theory of rank one maps. Ann. Math. (2005), to appear.Google Scholar
[40]Wu, L.. Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 (2004), 255–321.CrossRefGoogle Scholar
[41]Wu, L.. Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23 (1995), 420–445.Google Scholar
[42]Young, L.-S.. Large deviations in dynamical systems. Trans. Amer. Math. Soc. 318 (1990), 525–543.Google Scholar
[43]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998), 585–650.CrossRefGoogle Scholar
[44]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar