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Integrable Smooth Planar Billiards And Evolutes, Edoh Y. Amiran
Integrable Smooth Planar Billiards And Evolutes, Edoh Y. Amiran
Mathematics Faculty Publications
Any elliptic region is an example of an integrable domain: the set of tangents to a confocal ellipse or hyperbola remains invariant under reflection across the normal to the boundary. The main result states that when Ω is a strictly convex bounded planar domain with a smooth boundary and is integrable near the boundary, its boundary is necessarily an ellipse. The proof is based on the fact that ellipses satisfy a certain “transitivity property”, and that this characterizes ellipses among smooth strictly convex closed planar curves. To establish the transitivity property, KAM theory is used with a perturbation of the …