Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables

摘要

Two area-preserving twist maps are associated to a smooth closed convex table: the (classical) billiard map and the dual billiard map. When the table is circular, these maps are integrable and their phase spaces are foliated by invariant curves. The invariant curves with rational rotation numbers are resonant and do not persist under generic perturbations of the circle. We present a sufficient condition for the break-up of these curves. This condition is expressed directly in terms of the Fourier coefficients of the perturbation. It follows from a standard Melinkov argument. (c) 2005 Elsevier B.V. All rights reserved.