Skip to main content
SpringerLink
Log in

Statistical limit theorems for suspension flows

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. In this paper, we give conditions under which statistical laws, such as the central limit theorem and almost sure invariance principle, for the underlying discrete time system are inherited by the suspension flow. As a consequence, we give a simpler proof of the results of Ratner (1973) and recover the results of Denker and Philipp (1984) for Axiom A flows. Morcover, we obtain several new results for nonuniformly and partially hyperbolic flows, including frame flows on negatively curved manifolds satisfying a pinching condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Aaronson,An Introduction to Infinite Ergodic Theory, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  2. P. Ashwin, I. Melbourne and M. Nicol,Hypermeander of spirals; local bifurcations and statistical properties, Physica D156 (2001), 364–382.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Bowen,Symbolic dynamics for hyperbolic flows, American Journal of Mathematics95 (1973), 429–460.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Brin and M. Gromov,On the ergodicity of frame flows, Inventiones Mathematicae60 (1980), 1–7.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Brin and H. Karcher,Frame flows on manifolds with pinched negative curvature, Compositio Mathematica52 (1984), 275–297.

    MathSciNet  MATH  Google Scholar 

  6. K. Burns and M. Pollicott,Stable ergodicity and frame flows, Geometriae Dedicata98 (2003), 189–210.

    Article  MathSciNet  MATH  Google Scholar 

  7. J.-P. Conze and S. Le Borgne,Méthode de martingales et flow géodésique sur une surface de courbure constante négative, Ergodic Theory and Dynamical Systems21 (2001), 421–441.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. P. Cornfeld, S. V. Fomin and Y. G. Sinai,Ergodic Theory, Grundlehren der mathematischen Wissenschaften245, Springer, New York, 1982.

    MATH  Google Scholar 

  9. M. Denker and W. Philipp,Approximation by Brownian motion for Gibbs measures and flows under a function, Ergodic Theory and Dynamical Systems4 (1984), 541–552.

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Dolgopyat,On mixing properties of compact group extensions of hyperbolic systems, Israel Journal of Mathematics130 (2002), 157–205.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Dolgopyat,Limit theorems for partially hyperbolic systems, Transactions of the American Mathematical Society356 (2004), 1637–1689.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. K. Eagleson,Some simple conditions for limit theorems to be mixing, Teoriya Veroyatnosteii i Primeneniya21 (1976), 653–660. (English translation: Theory of Probability and its Applications21 (1976), 637–642, 1977)

    MathSciNet  Google Scholar 

  13. W. Feller,An Introduction to Probability Theory and its Applications, II, Wiley, New York, 1966.

    Google Scholar 

  14. M. J. Field, I. Melbourne and A. Török,Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions, Ergodic Theory and Dynamical Systems23 (2003), 87–110.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. J. Field, I. Melbourne and A. Török,Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets, Ergodic Theory and Dynamical Systems, to appear.

  16. M. I. Gordin,The central limit theorem for stationary processes, Soviet Mathematics Doklady10 (1969), 1174–1176.

    MATH  Google Scholar 

  17. S. Gouëzel,Central limit theorem and stable laws for intermittent maps. Probability Theory and Related Fields128 (2004), 82–122.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Gouëzel,Statistical properties of a skew product with a curve of neutral points, Preprint, 2004.

  19. F. Hofbauer and G. Keller,Ergodic properties of invariant measures for piecewise monotonic transformations, Mathematische Zeitschrift180 (1982), 119–140.

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Liverani,Central limit theorem for deterministic systems, inInternational Conference on Dynamical Systems (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Research Notes in Mathematics362, Longman Group Ltd, Harlow, 1996, pp. 56–75.

    Google Scholar 

  21. I. Melbourne and M. Nicol,Statistical properties of endomorphisms and compact group extensions, Journal of the London Mathematical Society, to appear.

  22. I. Melbourne and A. Török,Central limit theorems and invariance principles for time-one maps of hyperbolic flows, Communications in Mathematical Physics229 (2002), 57–71.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Nicol, I. Melbourne and P. Ashwin,Euclidean extensions of dynamical systems, Nonlinearity14 (2001), 275–300.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Philipp and W. F. Stout,Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, Memoirs of the American Mathematical Society161, American Mathematical Society, Providence, RI, 1975.

    Google Scholar 

  25. M. Ratner,The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel Journal of Mathematics16 (1973), 181–197.

    Article  MathSciNet  Google Scholar 

  26. A. Rényi,On mixing sequences of sets, Acta Mathematica Academiae Scientiarum Ilungaricae9 (1958), 215–228.

    Article  MATH  Google Scholar 

  27. D. Ruelle,Thermodynamic Formalism, inEncyclopedia of Mathematics and its Applications 5, Addison Wesley, Massachusetts, 1978.

    Google Scholar 

  28. D. Ruelle,The thermodynamic formalism for expanding maps, Communications in Mathematical Physics125 (1989), 239–262.

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Wong,Law of the iterated logarithm for transitive C 2 Anosov flows and semiflows over maps of the interval, Monatshefte für Mathematik94 (1982), 163–173.

    Article  MATH  Google Scholar 

  30. L.-S. Young,Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics147 (1998), 585–650.

    Article  MathSciNet  MATH  Google Scholar 

  31. L.-S. Young,Recurrence times and rates of mixing, Israel Journal of Mathematics110 (1999), 153–188.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Maths and Stats, University of Surrey, GU2 7XH, Guildford, UK

    Ian Melbourne

  2. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA

    Andrei Török

  3. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania

    Andrei Török

Authors
  1. Ian Melbourne
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrei Török
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ian Melbourne.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Melbourne, I., Török, A. Statistical limit theorems for suspension flows. Isr. J. Math. 144, 191–209 (2004). https://doi.org/10.1007/BF02916712

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02916712

Keywords

Search

Navigation