Advertisement

Large Deviations, Fluctuations and Shrinking Intervals

  • Mark Pollicott
  • Richard SharpEmail author
Article

Abstract

This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem.

Keywords

Maximal Entropy Periodic Point Transfer Operator Time Reversal Symmetry Exponential Growth Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bowen R.: Markov partitions for Axiom A diffeomorphisms. Amer. J. Math. 92, 907–918 (1970)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Bowen R.: ω-limit sets for axiom A diffeomorphisms. J. Differ. Eq. 18, 333–339 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dolgopyat D.: Prevalence of rapid mixing in hyperbolic flows. Erg. Th. Dynam. Sys. 18, 1097–1114 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gallavotti G.: Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Electron. J. 1, 1–12 (1995)MathSciNetGoogle Scholar
  5. 5.
    Gallavotti G., Cohen E.: Dynamical ensembles of stationary states. J. Stat. Phys. 80, 931–970 (1995)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Gentile G.: Large deviation rule for Anosov flows. Forum Math. 10, 89–118 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Guivarc’h Y., Hardy J.: Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré Probab. Statist. 24, 73–98 (1988)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Jiang, D.-Q., Qian, M., Qian, M.-P.: Mathematical theory of nonequilibrium steady states. Lecture Notes in Mathematics, 1833, Berlin: Springer 2004Google Scholar
  9. 9.
    Kifer Y.: Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321, 505–524 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lalley, S.: Ruelle’s Perron-Frobenius theorem and the central limit theorem for additive functionals of one-dimensional Gibbs states. In: Adaptive statistical procedures and related topics (Upton, N.Y., 1985) IMS Lecture Notes Monogr. Ser., 8, Hayward, CA: Inst. Math. Statist., 1986, pp. 428–446Google Scholar
  11. 11.
    Maes C., Verbitskiy E.: Large deviations and a fluctuation symmetry for chaotic homeomorphisms. Commun. Math. Phys. 233, 137–151 (2003)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Melbourne I.: Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Amer. Math. Soc. 359, 2421–2441 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Orey S., Pelikan S.: Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms. Trans. Amer. Math. Soc. 315, 741–753 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Pollicott M., Sharp R.: Rates of recurrence for \({\mathbb {Z}^q}\) and \({\mathbb {R}^q}\) extensions of subshifts of finite type. J. London Math. Soc. 49, 401–418 (1994)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Pollicott M., Sharp R.: Error terms for closed orbits of hyperbolic flows. Erg. Th. Dynam. Sys. 21, 545–562 (2001)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Pollicott, M., Sharp, R.: Distribution of ergodic sums for hyperbolic maps. In: Representation theory, dynamical systems, and asymptotic combinatorics, Amer. Math. Soc. Transl. Ser. 2, 217, Providence, RI: Amer. Math. Soc., 2006, pp. 167–183Google Scholar
  17. 17.
    Rey-Bellet L., Young L.-S.: Large deviations in non-uniformly hyperbolic dynamical systems. Erg. Th. Dynam. Sys. 28, 587–612 (2008)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Ruelle D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Statist. Phys. 85, 1–23 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Ruelle D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Statist. Phys. 95, 393–468 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Wojtkowski, M.: Abstract fluctuation theorems. Ergod. Th. Dynam. Sys., to appear:doi: 10.1017/S014338570800163, 2008

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WarwickCoventryUK
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations