Criterion of absolute focusing for focusing component of billiards

摘要

We show that a focusing component I" 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 I" is monotonic. That is, if I" is absolutely focusing this implies monotonicity of curvatures of the wave fronts in the series of reflections off I" and therefore explains why and how the absolutely focusing components may generate hyperbolicity of billiards.