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.
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