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On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards

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Abstract

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables—asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than 40 years ago.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Leonid Bunimovich

  2. Department of Mathematics and Statistics, UMass Amherst, Amherst, MA, 01003, USA

    Hong-Kun Zhang

  3. Department of Mathematics, University of Houston, Houston, TX, 77004, USA

    Pengfei Zhang

Authors
  1. Leonid Bunimovich
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  2. Hong-Kun Zhang
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  3. Pengfei Zhang
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Correspondence to Pengfei Zhang.

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Communicated by K. Khanin

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Bunimovich, L., Zhang, HK. & Zhang, P. On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards. Commun. Math. Phys. 341, 781–803 (2016). https://doi.org/10.1007/s00220-015-2539-x

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  • DOI: https://doi.org/10.1007/s00220-015-2539-x

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