We show that a focusing component $\Gamma$ 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 $\Gamma$ is monotonic. That is, if $\Gamma$ is absolutely focusing this implies monotonicity of curvatures of the wave fronts in the series of reflections off $\Gamma$ and therefore explains why and how the absolutely focusing components may generate hyperbolicity of billiards.
Keywords:
billiards, continued fractions, dispersing, focusing, defocusing, absolute focusing
Citation:
Bunimovich L. A., Criterion of Absolute Focusing for Focusing Component of Billiards, Regular and Chaotic Dynamics,
2009, Volume 14, Number 1,
pp. 42-48