Abstract
Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove “regularity” properties.
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Sinai, Ya. G., Dynamical Systems with Elastic Reflections. Ergodic Properties of Dispersing Billiards, Uspehi Mat. Nauk, 1970, vol. 25, no. 2(152), pp. 141–192.
Kerckhoff, S., Masur, H., and Smillie, J., Ergodicity of Billiard Flows and Quadratic Differentials, Ann. of Math., 1986, vol. 124, pp. 293–311.
Lazutkin, V.F., On the Existence of Caustics for the Billiard Ball Problem in a Convex Domain, Izv. Akad. Nauk SSSR Ser. Mat., 1973, vol. 37, pp. 186–216.
Bunimovich, L.A., On Billiards Close to Dispersing, Mat. Sb., 1974, vol. 23, pp. 45–67.
Bunimovich, L.A., The Ergodic Properties of Certain Billiards, Funk. Anal. Prilozh., 1974, vol. 8, pp. 73–74.
Bunimovich, L. A., On Ergodic Properties of Nowhere Dispersing Billiards, Comm. Math. Phys., 1979, vol. 65, pp. 295–312.
Bunimovich, L. A., Sinai, Ya. G., and Chernov, N. I., Markov Partitions for Two-Dimensional Billiards, Uspekhi Mat. Nauk, 1990, vol. 45, no. 3(273), pp. 97–134 [Russ. Math. Surv. (Engl. Transl.) 1990, vol. 45, no. 3, pp. 105–152].
Bunimovich, L. A., Sinai, Ya. G., and Chernov, N. I., Statistical Properties of Two-Dimensional Hyperbolic Billiards, Uspekhi Mat. Nauk, 1991, vol. 46, no. 4(280), pp. 43–92 [Russ. Math. Surv. (Engl. Transl.) 1991, vol. 46, no. 4, pp. 47–106].
Chernov, N. and Zhang, H.-K., Billiards with Polynomial Mixing Rates, Nonlinearity, 2005, vol. 18, pp. 1527–1553.
Chernov, N. and Zhang, H.-K., Optimal Estimates for Correlations in Billiards, Comm. Math. Phys., (to appear).
Bálint, P. and Gouèzel, S., Limit Theorems in the Stadium Billiard, Comm. Math. Phys., 2006, vol. 263, pp. 461–512.
Wojtkowski, M., Principles for the Design of Billiards with Nonvanishing Lyapunov Exponents, Comm. Math. Phys., 1986, vol. 105, pp. 391–414.
Markarian, R., Billiards with Pesin Region of Measure One, Comm. Math. Phys., 1988, vol. 118, pp. 87–97.
Bunimovich, L. A., On Absolutely Focusing Mirros, in Ergodic Theory and related topics, III (Gùstrow, 1990), vol. 1514 of Lecture Notes in Math., Berlin: Springer-Verlag, 1992, pp. 62–82.
Donnay, V., Using Integrability to Produce Chaos: Billiards with Positive Entropy, Comm. Math. Phys., 1991, vol. 141, pp. 225–257.
Szász, D., On the K-Property of Some Planar Hyperbolic Billiards, Comm. Math. Phys., 1992, vol. 145, pp. 595–604.
Chernov, N., Decay of Correlations and Dispersing Billiards, J. Statist. Phys., 1999, vol. 94, pp. 513–556.
Chernov, N. and Sinai, Ya. G., Billiards under Small External Forces, Ann. Henri Poincaré, 2001, vol. 2, pp. 197–236.
Chernov, N. and Markarian, R., Chaotic Billiards, vol. 127 of Mathernatical Surveys and Monographs, Providence: AMS, 2006.
Friedman, N., Kaplan, A., Carasso, D., and Davidson, N., Observation of Chaotic and Regular Dynamics in Atom-Optics Billards, Phys. Rev. Lett., 2001, vol. 86, pp. 1518–1521.
Katok, A., Strelcyn, J.-M., Ledrappier, F., and Przytycki, F., Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, vol. 1222 of Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1986.
Young, L.-S., Statistical Properties of Dynamical Systems with Some Hyperbolicity, Ann. of Math., 1998, vol. 147, pp. 585–650.
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Chernov, N., Zhang, H.K. Regularity of Bunimovich’s stadia. Regul. Chaot. Dyn. 12, 335–356 (2007). https://doi.org/10.1134/S1560354707030057
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DOI: https://doi.org/10.1134/S1560354707030057