Abstract:
We show that a focusing component Γ of the boundary of a billiard table is absolutely focusing iff a sequence of convergents of a continued fraction corresponding to any series of consecutive reflections off Γ is monotonic. That is, if Γ is absolutely focusing this implies monotonicity of curvatures of the wave fronts in the series of reflections off Γ and therefore explains why and how the absolutely focusing components may generate hyperbolicity of billiards.
Keywords:
billiards, continued fractions, dispersing, focusing, defocusing, absolute focusing.
\Bibitem{Bun09}
\by L. A. Bunimovich
\paper Criterion of Absolute Focusing for Focusing Component of Billiards
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 1
\pages 42--48
\mathnet{http://mi.mathnet.ru/rcd539}
\crossref{https://doi.org/10.1134/S1560354709010055}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2480951}
\zmath{https://zbmath.org/?q=an:1229.37030}
Linking options:
https://www.mathnet.ru/eng/rcd539
https://www.mathnet.ru/eng/rcd/v14/i1/p42
This publication is cited in the following 1 articles:
V. Lopac, A. Šimić, “Chaotic properties of the truncated elliptical billiards”, Communications in Nonlinear Science and Numerical Simulation, 16:1 (2011), 309
Citation in format AMSBIB
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