A sharp asymptotic for the lengths of certain scattering rays in the exterior of two convex domains

Abstract

Billiard trajectories in the exterior of two strictly convex domains in the plane are considered incoming from a given source S and arriving at a given target P. An asymptotic with an exponentially small error term for the sequence of travelling times of these trajectories is obtained involving the distance between the two domains and the curvatures at the ends of the shortest segment connecting the domains.