Mathematical Research Letters

Volume 13 (2006)

Number 4

Sub-Riemannian geometry and periodic orbits in classical billiards

Pages: 587 – 598

DOI: http://dx.doi.org/10.4310/MRL.2006.v13.n4.a8

Authors

Yuliy Baryshnikov (Bell Labs, Lucent Technologies)

Vadim Zharnitsky (University of Illinois)

Abstract

Classical (Birkhoff) billiards with full 1-parameter families of periodic orbits are considered. It is shown that construction of a convex billiard with a “rational” caustic ({\em i.e.} carrying only periodic orbits ) can be reformulated as the problem of finding a closed curve tangent to a non-integrable distribution on a manifold. The properties of this distribution are described as well as the consequences for the billiards with rational caustics. A particular implication of this construction is that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist.

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