Journal of Statistical Physics

, Volume 125, Issue 2, pp 411–453

Large Deviations for Non-Uniformly Expanding Maps

Article

Abstract

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states.

Keywords

non-uniform expansion physical measures hyperbolic times large deviations 

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References

  1. 1.
    J. Alves. Statistical analysis of non-uniformly expanding dynamical systems. Publicaçes Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, 2003. 24° Colóquio Brasileiro de Matematica. [24th Brazilian Mathematics Colloquium].Google Scholar
  2. 2.
    J. F. Alves. SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. école Norm. Sup. 33:1–32 (2000).MATHMathSciNetGoogle Scholar
  3. 3.
    J. F. Alves and V. Araujo. Random perturbations of nonuniformly expanding maps. Astérisque 286:25–62 (2003).MATHMathSciNetGoogle Scholar
  4. 4.
    J. F. Alves, V. Araiijo, and B. Saussol. On the uniform hyperbolicity of nonuniformly hyperbolic systems. Proc. Am. Math. Soc. 131(4):1303–1309 (2003).MATHCrossRefGoogle Scholar
  5. 5.
    J. F. Alves and V. Araújo. Hyperbolic times: frequency versus integrability. Ergodic Theory Dyn. Syst. 24:1–18 (2004).CrossRefGoogle Scholar
  6. 6.
    J. F. Alves, C. Bonatti, and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2):351–398 (2000).MATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    J. F. Alves, S. Luzzatto, and V. Pinheiro. Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6):817–839 (2005).MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    V. Araujo and M. J. Pacifico. Physical measures for infinite-modal maps. Preprint IMPA Serie A, 328 (2004).Google Scholar
  9. 9.
    V. Araújo and A. Tahzibi. Stochastic stability at the boundary of expanding maps. Nonlinearity 18:939–959 (2005).MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    A. Arbieto and C. Matheus. Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials. http://arxiv.org/abs/math.DS/0603629 (2006).
  11. 11.
    A. Arbieto, C. Matheus, S. Senti, and M. Viana. Maximal entropy measures for viana maps. Discrete and Continuous Dynamical Systems, to appear (2006).Google Scholar
  12. 12.
    M. Benedicks and L. Carleson. On iterations of 1 − ax2 on (−1, 1). Annals Math. 122:1–25 (1985).MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Benedicks and L. Carleson. The dynamics of the Hénon map. Annals Math. 133:73–169 (1991).MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Benedicks and L.-S. Young. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12:13–37 (1992).MATHMathSciNetGoogle Scholar
  15. 15.
    M. Benedicks and L.-S. Young. SBR-measures for certain Hénon maps. Invent. Math. 112:541–576 (1993).MATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    M. Benedicks and L.-S. Young. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261:13–56 (2000).MATHMathSciNetGoogle Scholar
  17. 17.
    C. Bonatti, L. J. Díaz, and M. Viana. Dynamics beyond uniform hyperbolicity, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective, Mathematical Physics, III.Google Scholar
  18. 18.
    C. Bonatti and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115:157–193 (2000).MATHMathSciNetGoogle Scholar
  19. 19.
    R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lect. Notes in Math. Springer Verlag (1975).Google Scholar
  20. 20.
    R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Invent. Math. 29:181–202 (1975).MATHMathSciNetCrossRefADSGoogle Scholar
  21. 21.
    L. Breiman. The individual ergodic theorem of information theory. Ann. Math. Statist. 28:809–811 (1957).MATHMathSciNetGoogle Scholar
  22. 22.
    M. Brin and A. Katok. On local entropy. In Geometric dynamics (Rio de. Janeiro, 1981), volume 1007 of Lecture Notes in Math., pp. 30–38. Springer, Berlin (1983).Google Scholar
  23. 23.
    J. Buzzi and O. Sari. Uniqueness of equilibrium measures for countable markov shifts and multidimensional piecewise expanding maps. Ergodic Theory Dynam. Systems 23(5):1383–1400 (2003).MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Buzzi, O. Sester, and M. Tsujii. Weakly expanding skew-products of quadratic maps. Ergodic Theory Dynam. Systems 23(5):1401–1414 (2003).MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    W. de Melo and S. van Strien. One-dimensional dynamics. Springer Verlag (1993).Google Scholar
  26. 26.
    M. Denker. The central limit theorem for dynamical systems. In Dynamical Systems and Ergodic Theory, volume 23. Banach Center Publ., Warsaw (1989).Google Scholar
  27. 27.
    M. Denker and M. Kesseböhmer. Thermodynamic formalism, large deviation, and multifractals. In Stochastic climate models (Chorin, 1999), volume 49 of Progr. Probab., pp. 159–169. Birkhäuser, Base (2001).Google Scholar
  28. 28.
    R. S. Ellis. Entropy, large deviations, and statistical mechanics. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag Berlin (2006).Google Scholar
  29. 29.
    S. Gouezel. Decay of correlations for nonuniforly expanding systems. Bulletin de la Soc. Math. France 134: 1–31 (2006).Google Scholar
  30. 30.
    V. Guilleinin and A. Pollack. Differential Topology. Prentice Hall, New Jersey (1974).Google Scholar
  31. 31.
    M. Hirsch. Differential Topology. Springer-Verlag, New-York (1976).MATHGoogle Scholar
  32. 32.
    M. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81:39–88 (1981).MATHMathSciNetCrossRefADSGoogle Scholar
  33. 33.
    A. Katok and J. M. Strelcyn. Invariant manifolds, entropy and billiards. Smooth maps with singularities, volume 1222 of Lect. Notes in Math. Springer Verlag (1986).Google Scholar
  34. 34.
    Y. Kifer. Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321(2):505–524 (1990).MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    A. Lasota and J. Yorke. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186:481–488 (1973).MathSciNetCrossRefGoogle Scholar
  36. 36.
    P.-D. Liu. Pesin's Entropy Formula for endomorphisms. Nagoya Math. J. 150:197–209 (1998).MATHMathSciNetGoogle Scholar
  37. 37.
    S. Luzzatto and M. Viana. Positive Lyapunov exponents for Lorenz-like families with criticalities. Asterisque 261:201–237, 2000. Geometrie complexe et systemes dynamiques (Orsay, 1995).MATHMathSciNetGoogle Scholar
  38. 38.
    R. Mañé. Ergodic theory and differentiable dynamics. Springer Verlag, New York (1987).MATHGoogle Scholar
  39. 39.
    B. McMillan. The basic theorems of information theory. Ann. Math. Stat. 24:196–219 (1953).MATHMathSciNetGoogle Scholar
  40. 40.
    R.J. Metzger. Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows. Ann. Inst. H. Poincaré Anal. Non Lineaire 17(2):247–276 (2000).MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    L. Mora and M. Viana. Abundance of strange attractors. Acta Math. 171(1):1–71 (1993).MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    J. Munkres. Topology, 2nd edition, Prentice-Hall, New Deli (2004).Google Scholar
  43. 43.
    K. Oliveira and M. Viana. Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms. Discrete Continuous Dynam. Syst. 15(1): 225–236 (2006).Google Scholar
  44. 44.
    V. I. Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19:197–231 (1968).MATHGoogle Scholar
  45. 45.
    Y. Pesin and Y. Sinai. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2:417–438 (1982).MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    V. Pinheiro. SRB measures for weakly expanding maps. Nonlinearity 19(5):1185–1200 (2006).Google Scholar
  47. 47.
    V. Pliss. On a conjecture due to Smale. Diff. Uravnenija 8:262–268 (1972).Google Scholar
  48. 48.
    A. Rovella. The dynamics of perturbations of the contracting Lorenz attractor. Bull. Braz. Math. Soc. 24(2):233–259 (1993).MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    D. Ruelle. A measure associated with Axiom A attractors. Am. J. Math. 98:619–654 (1976).MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    D. Ruelle. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9:83–87 (1978).MATHMathSciNetGoogle Scholar
  51. 51.
    D. Ruelle. The thermodynamical formalism for expanding maps. Comm. Math. Phys. 125:239–262 (1989).MATHMathSciNetCrossRefADSGoogle Scholar
  52. 52.
    D. Ruelle. Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics. Cambridge Mathematical Library, 2nd edition, Cambridge University Press, Cambridge (2004).Google Scholar
  53. 53.
    M. Rychlik. Bounded variation and invariant measures. Studia Math. 76:69–80 (1983).MATHMathSciNetGoogle Scholar
  54. 54.
    C. E. Shannon. A mathematical theory of communication. Bell System Tech. J. 27:379–423, 623–656 (1948).MathSciNetMATHGoogle Scholar
  55. 55.
    Y. Sinai. Gibbs measures in ergodic theory. Russian Math. Surveys 27:21–69 (1972).MathSciNetCrossRefADSMATHGoogle Scholar
  56. 56.
    R. Sturman and J. Stark. Semi-uniform ergodic theorems and applications to forced systems. Nonlinearity 13:113–143 (2000).MATHMathSciNetCrossRefADSGoogle Scholar
  57. 57.
    M. Viana. Multidimensional nonhyperbolic attractors. Inst. Hautes éatudes Sci. Publ. Math. 85:63–96 (1997).MATHMathSciNetGoogle Scholar
  58. 58.
    M. Viana. Stochastic dynamics of deterministic systems. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997. 21° Colóquio Brasileiro de Matematica. [21th Brazilian Mathematics Colloquium].Google Scholar
  59. 59.
    P. Walters. Antroduction to ergodic theory. Springer Verlag (1982).Google Scholar
  60. 60.
    L.-S. Young. Some large deviation results for dynamical systems. Trans. Am. Math. Soc. 318(2):525–543 (1990).MATHCrossRefGoogle Scholar
  61. 61.
    L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. Annals Math. 147:585–650 (1998).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Centro de Matemática da Universidade do PortoPortoPortugal

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