Degenerate billiards

TL;DR

This paper investigates degenerate billiard systems where the scatterer has codimension greater than one, proving the existence of trajectories with infinitely many collisions by using an anti-integrable limit approach.

Contribution

It introduces a framework for analyzing degenerate billiards as limits of systems with elastic reflections or celestial singularities, establishing the existence of complex trajectories.

Findings

Existence of trajectories with infinite collisions in degenerate billiards.

Degenerate billiards can be approximated by systems with elastic reflections.

Application of anti-integrable limit method to billiard dynamics.

Abstract

In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Then collisions are rare. We study trajectories of degenerate billiards which have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an anti-integrable limit.