Journal of Statistical Physics

, Volume 172, Issue 6, pp 1499–1524 | Cite as

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

  • Jianyu ChenEmail author
  • Yun Yang
  • Hong-Kun Zhang


We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically Hölder observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.


ASIP Non-stationarity Hyperbolicity Singularities 

Mathematics Subject Classification

37D50 37A25 60F17 



The authors would like to thank Nicolai Haydn and Huyi Hu for helpful discussions and suggestions.


  1. 1.
    Baladi, V., Tsujii, M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier (Grenoble) 57(1), 127–154 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baladi, V., Gouëzel, S.: Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1453–1481 (2009)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Baladi, V., Gouëzel, S.: Banach spaces for piecewise cone-hyperbolic maps. J. Mod. Dyn. 4(1), 91–137 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Balint, P., Gouëzel, S.: Limit theorems in the stadium billiard. Commun. Math. Phys. 263(2), 461–512 (2006)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Balint, P., Chernov, N., Dolgopyat, D.: Limit theorems for dispersing billiards with cusps. Commun. Math. Phys. 308, 479–510 (2011)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29–54 (1979)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blank, M., Keller, G., Liverani, C.: Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6), 1905–1973 (2002)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bowen, R.: Markov partitions for Axiom A diffeomorphisms. Am. J. Math. 92, 725–747 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975)CrossRefGoogle Scholar
  10. 10.
    Bunimovich, L.A., Sinai, Ya G.: Markov partitions for dispersing billiards. Commun. Math. Phys. 73, 247–280 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russian Math. Surveys 45, 105–152 (1990)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys. 46, 47–106 (1991)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Chernov, N.I.: Limit theorems and Markov approximations for chaotic dynamical systems. Probab. Theory Relat. Fields 101(3), 321–362 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chernov, N.I.: Decay of correlations in dispersing billiards. J. Stat. Phys. 94, 513–556 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    Chernov, N.I.: Sinai billiards under small external forces. Ann. Henri Poincare 2, 197–236 (2001)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Chernov, N.I.: Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122, 1061–1094 (2006)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Chernov, N.I.: Sinai billiards under small external forces II. Ann. Henri Poincare 9, 91–107 (2008)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Chernov, N.I., Dolgopyat, D.: Brownian Brownian Motion-I, vol. 198. Memoirs of AMS, Providence (2009)zbMATHGoogle Scholar
  19. 19.
    Chernov, N.I., Kleinbock, D.: Dynamical Borel-Cantelli lemmas for Gibbs measures. Israel J. Math. 122, 1–27 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chernov, N.I., Markarian, R.: Chaotic Billiards, Mathematical Surveys Monographs, vol. 127. AMS, Providence (2006)CrossRefGoogle Scholar
  21. 21.
    Chernov, N., Markarian, R.: Dispersing billiards with cusps: slow decay of correlations. Commun. Math. Phys. 270, 727–758 (2007)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlineartity 4, 1527–1553 (2005)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Chernov, N., Zhang, H.-K.: A family of chaotic billiards with variable mixing rates. Stoch. Dyn. 5, 535–553 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chernov, N., Zhang, H.-K.: Improved estimate for correlations in billiards. Commun. Math. Phys. 277, 305–321 (2008)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Chernov, N., Zhang, H.-K.: On statistical properties of hyperbolic systems with singularities. J. Stat. Phys. 136, 615–642 (2009)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Cuny, C.: Pointwise ergodic theorems with rate with applications to limit theorems for stationary processes. Stoch. Dyn. 11(1), 135–155 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Cuny, C., Merlevède, F.: Strong invariance principles with rate for “reverse” martingale differences and applications. J. Theor. Probab. 28(1), 137–183 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Demers, M., Liverani, C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Am. Math. Soc. 360, 4777–4814 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Demers, M., Zhang, H.-K.: Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5, 665–709 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Demers, M., Zhang, H.-K.: A functional analytic approach to perturbations of the Lorentz gas. Commun. Math. Phys. 324, 767–830 (2013)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Demers, M., Zhang, H.-K.: Spectral analysis of hyperbolic systems with singularities. Nonlinearity 27, 379–433 (2014)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Denker, M.: The Central Limit Theorem for Dynamical Systems, Dynamical Systems and Ergodic Theory (Warsaw, 1986), vol. 23, pp. 33–62. Banach Center Publications, PWN, Warsaw (1989)Google Scholar
  33. 33.
    Eberlein, E.: On strong invariance principles under dependence assumptions. Ann. Probab. 14(1), 260–270 (1986)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Fayad, B.: Mixing in the absence of the shrinking target property. Bull. Lond. Math. Soc. 38(5), 829–838 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Gallavotti, G., Ornstein, D.: Billiards and Bernoulli schemes. Commun. Math. Phys. 38, 83–101 (1974)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Gouëzel, S.: Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38(4), 1639–1671 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Haydn, N., Nicol, M., Vaienti, S., Zhang, L.: Central limit theorems for the shrinking target problem. J. Stat. Phys. 153(5), 864–887 (2013)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Haydn, N., Nicol, M., Török, A., Vaienti, S.: Almost sure invariance principle for sequential and non-stationary dynamical systems. Trans. Am. Math. Soc. 369(8), 5293–5316 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Hill, R., Velani, S.: Ergodic theory of shrinking targets. Invent. Math. 119(1), 175–198 (1995)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Gröningen (1971)zbMATHGoogle Scholar
  41. 41.
    Katok, A., Strelcyn, J.M.: Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Lecture Notes in Mathematics, vol. 1222. Springer, New York (1986)CrossRefGoogle Scholar
  42. 42.
    Korepanov, A.: Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle. Commun. Math. Phys. 359(3), 1123–1138 (2018)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Marcinkiewicz, J., Zygmund, A.: Sur les fonctions indépendantes. Fund. Math. 29, 60–90 (1937)zbMATHGoogle Scholar
  44. 44.
    Melbourne, I., Nicol, M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260, 131–146 (2005)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Melbourne, I., Nicol, M.: A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37(2), 478–505 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Markarian, R.: Billiards with polynomial decay of correlations. Ergod. Theory Dyn. Syst. 24, 177–197 (2004)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Pesin, Ya.: Characteristic Lyapunov exponents, and smooth ergodic theory. Russ. Math. Surv. 32(4), 55–114 (1977)CrossRefGoogle Scholar
  48. 48.
    Pesin, Y.: Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Theory Dyn. Syst. 12, 123–152 (1992)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Philipp, W., Stout, W.: Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, vol. 161. Memoirs of the American Mathematical Society, Providence (1975)zbMATHGoogle Scholar
  50. 50.
    Ruelle, D.: Thermodynamic Formalism, Encyclopedia of Mathematics and Its Applications, vol. 5. Addison-Wesley, Boston (1978)Google Scholar
  51. 51.
    Sataev, E.: Invariant measures for hyperbolic maps with singularities. Russ. Math. Surv. 47, 191–251 (1992)CrossRefGoogle Scholar
  52. 52.
    Sinai, Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Sinai, Y.G.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Sarig, O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Shao, Q.M.: Almost sure invariance principles for mixing sequences of random variables. Stoch. Process. Appl. 48(2), 319–334 (1993)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Shao, Q.M., Lu, C.R.: Strong approximations for partial sums of weakly dependent random variables. Sci. Sinica Ser. A 30(6), 575–587 (1987)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Stenlund, M.: A vector-valued almost sure invariance principle for Sinai billiards with random scatterers. Commun. Math. Phys. 325(3), 879–916 (2014)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Stenlund, M., Young, L.S., Zhang, H.-K.: Dispersing billiards with moving scatterers. Commun. Math. Phys. 322, 909–955 (2013)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Szász, D., Varjú, T.: Local limit theorem for Lorentz process and its recurrence in the plane. Ergod. Theory Dyn. Syst. 24, 257–278 (2004)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Wu, W.B.: Strong invariance principles for dependent random variables. Ann. Probab. 35(6), 2294–2320 (2007)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Yokoyama, R.: Moment bounds for stationary mixing sequences. Z. Wahrsch. Verw. Gebiete 52(1), 45–57 (1980)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Young, L.S.: Statistical properties of systems with some hyperbolicity including certain billiards. Ann. Math. 147, 585–650 (1998)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Young, L.S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of Massachusetts AmherstAmherstUSA
  2. 2.Mathematics Department, The Graduate CenterCity University of New YorkNew YorkUSA

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