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Non-persistence of resonant caustics in perturbed elliptic billiards

Published online by Cambridge University Press:  21 August 2012

SÔNIA PINTO-DE-CARVALHO  and
RAFAEL RAMÍREZ-ROS
SÔNIA PINTO-DE-CARVALHO
Affiliation:
Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123–970, Belo Horizonte, Brazil (email: [email protected])
RAFAEL RAMÍREZ-ROS
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain (email: [email protected])
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Abstract

A caustic of a billiard table is a curve such that any billiard trajectory, once tangent to the curve, stays tangent after every reflection at the boundary. When the billiard table is an ellipse, any non-singular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics—those whose tangent trajectories are closed polygons—are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1876 - 1890
Copyright
Copyright © 2012 Cambridge University Press 

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