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Hyperbolic polygonal billiards with finitely many ergodic SRB measures

  • GIANLUIGI DEL MAGNO (a1), JOÃO LOPES DIAS (a2), PEDRO DUARTE (a3) and JOSÉ PEDRO GAIVÃO (a4)

Abstract

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai–Ruelle–Bowen measures whose basins cover a set of full Lebesgue measure.

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[1] Arroyo, A., Markarian, R. and Sanders, D. P.. Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries. Nonlinearity 22(7) (2009), 14991522.
[2] Arroyo, A., Markarian, R. and Sanders, D. P.. Structure and evolution of strange attractors in non-elastic triangular billiards. Chaos 22 (2012), 026107.
[3] Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157193.
[4] Chernov, N.. Decay of correlations and dispersing billiards. J. Stat. Phys. 94(3–4) (1999), 513556.
[5] Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127) . American Mathematical Society, Providence, RI, 2006.
[6] Chernov, N. and Zhang, H.-K.. Billiards with polynomial mixing rates. Nonlinearity 18(4) (2005), 15271553.
[7] Chernov, N. and Zhang, H.-K.. On statistical properties of hyperbolic systems with singularities. J. Stat. Phys. 136(4) (2009), 615642.
[8] Del Magno, G., Lopes Dias, J., Duarte, P., Gaivão, J. P. and Pinheiro, D.. Chaos in the square billiard with a modified reflection law. Chaos 22 (2012), 026106.
[9] Del Magno, G., Lopes Dias, J., Duarte, P., Gaivão, J. P. and Pinheiro, D.. SRB measures for polygonal billiards with contracting reflection laws. Comm. Math. Phys. 329 (2014), 687723.
[10] Del Magno, G., Lopes Dias, J., Duarte, P. and Gaivão, J. P.. Ergodicity of polygonal slap maps. Nonlinearity 27(8) (2014), 19691983.
[11] Del Magno, G., Lopes Dias, J., Duarte, P. and Gaivão, J. P.. Ergodic properties of polygonal billiards with strongly contracting reflection laws, Preprint, 2015, arXiv:1501.03697.
[12] Erickson, L. H. and La Valle, S. M.. Toward the design and analysis of blind, bouncing robots. IEEE Int. Conf. on Robotics and Automation (Karlsruhe, Germany, 6–10 May 2013). IEEE, Piscataway, NJ, 2013.
[13] Katok, A. et al. . Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222) . Springer, Berlin, 1986.
[14] Markarian, R., Pujals, E. J. and Sambarino, M.. Pinball billiards with dominated splitting. Ergod. Th. & Dynam. Sys. 30(6) (2010), 17571786.
[15] Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 339351.
[16] Pesin, Y. B.. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. & Dynam. Sys. 12(1) (1992), 123151.
[17] Sataev, E. A.. Invariant measures for hyperbolic mappings with singularities. Uspekhi Mat. Nauk 47(1(283)) (1992), 147202, 240; Engl. trans. Russian Math. Surveys 47(1) (1992), 191–251.
[18] Sheriff, R. E. and Geldart, L. P.. Exploration Seismology. Cambridge University Press, Cambridge, 1995.

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