Perturbations of elliptic billiards

Abstract

The dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied. In the case of an elliptic billiard table, the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points. Using a discrete analog to the Melnikov method, we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table. From the formulas that we get, we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table. In addition, we also include a characterization of the period two periodic points for any billiard table.