Document Type

Article

Publication Date

1997

Keywords

Billiard ball map, Integrable, KAM, Caustics, Ellipse

Abstract

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.

Publication Title

New York Journal of Mathematics

Volume

3

First Page

32

Last Page

47

Required Publisher's Statement

The New York Journal of Mathematics uses a copyright agreement that allows authors to retain copyright.

http://nyjm.albany.edu/

http://www.emis.de/journals/NYJM/j/1997/3-2.pdf

Language

English

Format

application/pdf

Included in

Mathematics Commons

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