Hyperbolic billiards with nearly flat focusing boundaries. I

Details Hyperbolic billiards with nearly flat focusing boundaries. I Hyperbolic billiards with nearly flat focusing boundaries. I

2007-12-21

arxiv.org/abs/0712.3802v1

Physica D 237 (2008), no. 18, 2272-2281

Mathematics

Dynamical Systems

21 pages, 9 figures

Scientific paper

10.1016/j.physd.2008.02.006

The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstardard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms.

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