Annales Henri Poincaré

, Volume 19, Issue 4, pp 979–1042 | Cite as

Equidistribution for Standard Pairs in Planar Dispersing Billiard Flows

  • Péter Bálint
  • Péter Nándori
  • Domokos Szász
  • Imre Péter Tóth


We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi et al. (Invent Math, 211:39–117, 2018. obtained for Hölder continuous observables in these billiards. The model dependence of the bounds is also discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baladi, V., Demers, M.F., Liverani, C.: Exponential decay of correlations for finite horizon Sinai billiard flows. Invent. Math. 211, 39–177 (2018).
  2. 2.
    Baladi, V., Kuna, T., Lucarini, V.: Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables. Nonlinearity 30(3), 1204 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Baladi, V., Kuna, T., Lucarini, V.: Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables. Nonlinearity 30(8), C4 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bálint, P., Gilbert, T., Nándori, P., Szász, D., Tóth, I.P.: On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas. J. Stat. Phys. 166, 903–925 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bálint, P., Nándori, P., Szász, D., Tóth, I. P.: Stochastic dynamics from a Newtonian one. Work in progressGoogle Scholar
  6. 6.
    Bálint, P., Chernov, N., Dolgopyat, D.: Limit theorems for dispersing billiards with cusps. Commun. Math. Phys. 308, 479–510 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bálint, P., Chernov, N., Dolgopyat, D.: Convergence of moments for dispersing billiards with cusps. Contemp. Math. 698, 35–69 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chernov, N.: Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122, 1061–1094 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chernov, N.: A stretched exponential bound on time correlations for billiard flows. J. Stat. Phys. 127, 21–50 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chernov, N., Dolgopyat, D.: Particle’s drift in self-similar billiards. Ergod. Theory Dyn. Syst. 28, 389–403 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chernov, N., Dolgopyat, D.: Brownian Brownian motion. I. Mem. Am. Math. Soc. 927, 198 (2009)MathSciNetMATHGoogle Scholar
  13. 13.
    Chernov, N., Dolgopyat, D.: Galton board: limit theorems and recurrence. Journal AMS 22, 821–858 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Chernov, N., Dolgopyat, D.: Anomalous current in periodic Lorentz gases with infinite horizon. Rus. Math. Surv. 64, 651–699 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chernov, N., Markarian, R.: Chaotic billiards. In: Mathematical Surveys and Monographs, vol. 127. American Mathematical Society (2006)Google Scholar
  16. 16.
    Climenhaga, V., Dolgopyat, D., Pesin, Y.: Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps. Commun. Math. Phys. 346, 553–602 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Climenhaga, V., Pesin Y., Zelerowicz, A.: A geometric approach to equilibrium measures via Charathéodory construction. In preparation, (2017), pp. 36Google Scholar
  18. 18.
    Demers, M., Zhang, H.-K.: A functional analytic approach to perturbations of the Lorentz gas. Commun. Math. Phys. 324, 767–863 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dolgopyat, D.: Limit theorems for partially hyperbolic systems. Trans. Am. Math. Soc. 356, 1637–1689 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Dolgopyat, D.: Averaging and Invariant measures. Mosc. Math. J. 5, 537–576 (2005)MathSciNetMATHGoogle Scholar
  21. 21.
    Dolgopyat, D., Liverani, C.: Energy transfer in a fast-slow Hamiltonian system. Commun. Math. Phys. 328, 201–225 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Dolgopyat, D., Nándori, P.: Non equilibrium density profiles in Lorentz tubes with thermostated boundaries. CPAM 69, 649–692 (2016)MATHGoogle Scholar
  23. 23.
    Dolgopyat, D., Nándori, P.: The first encounter of two billiard particles of small radius.
  24. 24.
    Dolgopyat, D., de Simoi, J.: Dynamics of some piecewise smooth Fermi-Ulam Models. Chaos 22, paper 026124 (2012)Google Scholar
  25. 25.
    Dolgopyat, D., Szász, D., Varjú, T.: Recurrence properties of Lorentz gas. Duke Math. J. 142, 241–281 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Dolgopyat, D., Szász, D., Varjú, T.: Limit theorems for locally perturbed Lorentz processes. Duke Math. J. 148, 459–499 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Eslami, P.: Stretched-exponential mixing for \(C^{1+\alpha }\) skew products with discontinuities. Ergod. Theory Dyn. Syst. 369(2), 783–803 (2017)Google Scholar
  28. 28.
    Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergod. Theory Dyn. Syst. 26, 189–217 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrscheinlichkeitstheorie verw. Geb. 69, 461–478 (1985)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Krámli, A., Simányi, N., and Szász, D.: A ,transversal’ fundamental theorem for semi-dispersing billiards. Commun. Math. Phys. 129, 535–560 (1990). Erratum: ibidem 129, 207–208 (1991)Google Scholar
  31. 31.
    Melbourne, I.: Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Am. Math. Soc. 359, 2421–2441 (2007)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Porte, M.: Linear response for Dirac observables of Anosov diffeomorphisms.
  33. 33.
    Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Isr. J. Math. 116, 223–248 (2000)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Shah, K., Turaev, D., Gelreich, V., and Rom-Kedar, V.: Equilibration of energy in slow-fast systems. In: Proceedings of the national academy of sciences.
  35. 35.
    Tóth, IP.: Generalized Hölder continuity and oscillation functions.
  36. 36.
    Siani, Y.G.: Dynamical systems with elastic reflections. Rus. Mat. Surv. 25, 137–189 (1970)CrossRefGoogle Scholar
  37. 37.
    Young, L.S.: Statistical properties of systems with some hyperbolicity including certain billiards. Ann. Math. 147, 585–650 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Péter Bálint
    • 1
    • 2
  • Péter Nándori
    • 3
  • Domokos Szász
    • 1
  • Imre Péter Tóth
    • 1
    • 2
  1. 1.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.MTA-BME Stochastics Research GroupBudapestHungary
  3. 3.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations