Criterion of Absolute Focusing for Focusing Component of Billiards

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.