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Hyperbolicity and abundance of elliptical islands in annular billiards

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  25 November 2022

REGINALDO BRAZ BATISTA Open the ORCID record for REGINALDO BRAZ BATISTA [Opens in a new window] ,
MÁRIO JORGE DIAS CARNEIRO Open the ORCID record for MÁRIO JORGE DIAS CARNEIRO [Opens in a new window]  and
SYLVIE OLIFFSON KAMPHORST Open the ORCID record for SYLVIE OLIFFSON KAMPHORST [Opens in a new window]
REGINALDO BRAZ BATISTA
Affiliation:
Departamento de Matemática, ICE UFJF, Via Local 880, 36036-900 Juiz de Fora, Brazil (e-mail: reginaldo.braz@ufjf.br)
MÁRIO JORGE DIAS CARNEIRO
Affiliation:
Departamento de Matemática, ICEx UFMG, CP 702, 31270-901 Belo Horizonte, Brazil (e-mail: carneiro@mat.ufmg.br)
SYLVIE OLIFFSON KAMPHORST*
Affiliation:
Departamento de Matemática, ICEx UFMG, CP 702, 31270-901 Belo Horizonte, Brazil (e-mail: carneiro@mat.ufmg.br)
*
e-mail: syok@ufmg.br
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Abstract

We study the billiard dynamics in annular tables between two eccentric circles. As the center and the radius of the inner circle changes, a two-parameters map is defined by the first return of trajectories to the obstacle. We obtain an increasing family of hyperbolic sets, in the sense of the Hausdorff distance, as the radius goes to zero and the center of the obstacle approximates the outer boundary. The dynamics on each of these sets is conjugate to a shift with an increasing number of symbols. We also show that for many parameters, the system presents quadratic homoclinic tangencies whose bifurcation gives rise to elliptical islands (conservative Newhouse phenomenon). Thus, for many parameters, we obtain the coexistence of a ‘large’ hyperbolic set with many elliptical islands.

Keywords

billiardshyperbolic setshomoclinic tangencies

MSC classification

Secondary: 37D05: Hyperbolic orbits and sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3545 - 3577
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Bálint, P., Halász, M., Hernández-Tahuilán, J. A. and Sanders, D. P.. Chaos and stability in a two-parameter family of convex billiard tables. Nonlinearity 24(5) (2011), 1499.CrossRefGoogle Scholar
Batista, R. B.. Estudo do bilhar no anel de círculos excêntricos. Unpublished report, 2005.Google Scholar
Batista, R. B.. Estudo do bilhar no anel de círculos excêntricos. Master’s Thesis, UFMG, Brazil, 2008.Google Scholar
Batista, R. B.. Bilhares com obstáculos. PhD Thesis, UFMG, Brazil, 2015.Google Scholar
Birkhoff, G. D.. Dynamical Systems. Colloquium Publications. American Mathematical Society, Providence, RI, 1966. Original edition 1927.Google Scholar
Bolotin, S. V.. Degenerate billiards in celestial mechanics. Regul. Chaotic Dyn. 22(1) (2017), 27–53.10.1134/S1560354717010038CrossRefGoogle Scholar
Bunimovich, L. A.. On absolutely focusing mirrors. Ergodic Theory and Related Topics III. Eds. V. Warstat, K. Richter and U. Krengel. Springer, Berlin, 1992, pp. 62–82.10.1007/BFb0097528CrossRefGoogle Scholar
Bunimovich, L. A.. Mushrooms and other billiards with divided phase space. Chaos 11(4) (2001), 802–808.CrossRefGoogle ScholarPubMed
Bunimovich, L. A., Zhang, H.-K. and Zhang, P.. On another edge of defocusing: hyperbolicity of asymmetric lemon billiards. Comm. Math. Phys. 341(3) (2016), 781–803.CrossRefGoogle Scholar
Canale, E., Markarian, R., Oliffson Kamphorst, S. and Pinto-de Carvalho, S.. A lower bound for chaos on the elliptical stadium. Phys. D 115(3–4) (1998), 189–202.CrossRefGoogle Scholar
Chen, Y.-C.. On topological entropy of billiard tables with small inner scatterers. Adv. Math. 224(2) (2010), 432–460.10.1016/j.aim.2009.11.012CrossRefGoogle Scholar
Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127). American Mathematical Society, Providence, RI, 2006.10.1090/surv/127CrossRefGoogle Scholar
Correia, M. F. and Zhang, H.-K.. Stability and ergodicity of moon billiards. Chaos 25(8) (2015), 083110.CrossRefGoogle ScholarPubMed
de Moraes Costa, M. L.. Bilhar no anel de excêntrico. Master’s Thesis, Departamento de Física, UFMG, 2001.Google Scholar
Dettmann, C. P. and Fain, V.. Linear and nonlinear stability of periodic orbits in annular billiards. Chaos 27(4) (2017), 043106.10.1063/1.4979795CrossRefGoogle ScholarPubMed
Donnay, V. J.. Using integrability to produce chaos: billiards with positive entropy. Comm. Math. Phys. 141 (1991), 225–257.10.1007/BF02101504CrossRefGoogle Scholar
Duarte, P.. Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 359–409.CrossRefGoogle Scholar
Duarte, P.. Elliptic isles in families of area-preserving maps. Ergod. Th. & Dynam. Sys. 28(6) (2008), 1781–1813.CrossRefGoogle Scholar
Foltin, C.. Billiards with positive topological entropy. Nonlinearity 15(6) (2002), 2053.CrossRefGoogle Scholar
Gorodetski, A.. On stochastic sea of the standard map. Comm. Math. Phys. 309(1) (2012), 155–192.CrossRefGoogle Scholar
Gouesbet, G., Meunier-Guttin-Cluzel, S. and Gréhan, G.. Periodic orbits in hamiltonian chaos of the annular billiard. Phys. Rev. E 65(1) (2001), 016212.CrossRefGoogle ScholarPubMed
Kaloshin, V. and Sorrentino, A.. On the local Birkhoff conjecture for convex billiards. Ann. of Math. (2) 188(1) (2018), 315–380.10.4007/annals.2018.188.1.6CrossRefGoogle Scholar
Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics (Cambridge Studies in Advanced Mathematics, 35). Cambridge University Press, Cambridge, 1995.Google Scholar
Saitô, N., Hirooka, H., Ford, J., Vivaldi, F. and Walker, G.. Numerical study of billiard motion in an annulus bounded by non-concentric circles. Phys. D 5(2) (1982), 273–286.CrossRefGoogle Scholar
Sinai, Y. G.. Dynamical systems with elastic reflections. Russian Math. Surveys 25 (1970), 137–189.CrossRefGoogle Scholar
Wiggins, S.. Introduction to Applied Nonlinear Dynamical Systems and Chaos (Texts in Applied Mathematics, 2). Springer, New York, 1990.10.1007/978-1-4757-4067-7CrossRefGoogle Scholar
Wojtkowski, M.. Invariant families of cones and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5(1) (1985), 145–161.CrossRefGoogle Scholar
Wojtkowski, M.. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys. 105(3) (1986), 391–414.CrossRefGoogle Scholar