Stable Laws for Chaotic Billiards with Cusps at Flat Points


We consider billiards with a single cusp where the walls meeting at the vertex of the cusp have zero one-sided curvature, thus forming a flat point at the vertex. For Hölder continuous observables, we show that properly normalized Birkhoff sums, with respect to the billiard map, converge in law to a totally skewed α-stable law.

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  1. 1.

    Adler, R.J.: Weak convergence results for extremal processes generated by dependent random variables. Ann. Probab. 6(4), 660–667 (1978)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bálint, P., Chernov, N., Dolgopyat, D.: Limit theorems for dispersing billiards with cusps. Commun. Math. Phys. 308(2), 479–510 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Bernstein, S.: Sur l’extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Ann. 97(1), 1–59 (1927)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1999)

    Google Scholar 

  5. 5.

    Chernov, N., Dolgopyat, D.: Brownian Brownian Motion-I. American Mathematical Soc., Providence (2009)

    Google Scholar 

  6. 6.

    Chernov, N., Markarian, R.: Chaotic Billiards. American Mathematical Society, Providence (2006)

    Google Scholar 

  7. 7.

    Chernov, N., Markarian, R.: Dispersing billiards with cusps: slow decay of correlations. Commun. Math. Phys. 270(3), 727–758 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlinearity 18(4), 1527 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Chernov, N., Zhang, H.-K.: On statistical properties of hyperbolic systems with singularities. J. Stat. Phys. 136(4), 615–642 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Durrett, R.: Probability: Theory and Examples. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  11. 11.

    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters, Noordhoff Pub, Groningen (1971)

    Google Scholar 

  12. 12.

    Kallenberg, O.: Characterization and convergence of random measures and point processes. Probab. Theory Relat. Fields 27(1), 9–21 (1973)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)

    Google Scholar 

  14. 14.

    Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics. Springer, Berlin (1983)

    Google Scholar 

  15. 15.

    Markarian, R.: Billiards with polynomial decay of correlations. Ergod. Theory Dyn. Syst. 24(01), 177–197 (2004)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall/CRC, Boca Raton (1994)

    Google Scholar 

  17. 17.

    Zhang, H.-K.: Decay of correlations for billiards with flat points II: cusps effect. Contemp. Math. (2017)

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The research of H. Zhang was supported in part by NSF Grant DMS-1151762 and also in part by a grant from the Simons Foundation (337646, HZ). The research of P. Jung was supported in part by NSA Grant H98230-14-1-0144 and NRF Grant N01170220. We would like to thank Dmitry Dolgopyat for posing the questions and also suggesting the main results discussed in this paper, i.e., the emergence of stable laws in billiard systems exhibiting slow decay of correlations. H. Zhang also thanks him for many invaluable discussions and suggestions.

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Correspondence to Paul Jung.

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Communicated by Dmitry Dolgopyat.

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Jung, P., Zhang, HK. Stable Laws for Chaotic Billiards with Cusps at Flat Points. Ann. Henri Poincaré 19, 3815–3853 (2018).

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