Document Type

Article

Publication Date

1997

Keywords

Billiard ball map, Integrable, KAM, Caustics

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

Subjects - Topical (LCSH)

Kolmogorov-Arnold-Moser theory; Perturbation (Mathematics); Calculus, Integral; Ellipse

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

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Mathematics Commons

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