Santalo's formula and stability of trapping sets of positive measure

TL;DR

This paper generalizes Santalo's formula for billiard trajectories around obstacles on Riemannian manifolds and shows the stability of the measure of trapped points under boundary perturbations, with implications for inverse scattering problems.

Contribution

It extends Santalo's formula to Riemannian manifolds and demonstrates the continuous dependence of trapped set measures on obstacle boundary perturbations.

Findings

The measure of trapped points remains positive under small boundary perturbations.

The measure of trapped points depends continuously on obstacle boundary changes.

Obstacle volume can be uniquely determined by average scattering times in Euclidean space.

Abstract

Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle $K$ with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then for all sufficiently small smooth perturbations of the boundary of $K$ the set of trapped points for the new obstacle obtained in this way also has positive measure. More generally the measure of the set of trapped points depends continuously on perturbations of the obstacle $K$. Some consequences of the generalised Santalo's formula are derived in the case of scattering by an obstacle $K$ in an Euclidean space. For example, it is shown that, for a large class of obstacles $K$, the volume of $K$ is uniquely determined by the average travelling times of scattering rays in the exterior of $K$.