New York Journal of Mathematics
Volume 3 (1997) 32-47


Edoh Y. Amiran

Integrable Smooth Planar Billiards and Evolutes

Published: April 21, 1997
Keywords: Billiard ball map, integrable, KAM, caustics, ellipse
Subject: primary 58F17, secondary 53C22

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.

Author information

Mathematics Department Western Washington University Bellingham, WA 98225-9063