Billiard ball map, Integrable, KAM, Caustics, Ellipse
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 integrable billiard map.
New York Journal of Mathematics
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Amiran, Edoh Y., "Integrable Smooth Planar Billiards and Evolutes" (1997). Mathematics. 100.