Horns in Billiards

TL;DR

This paper demonstrates that horns in billiard tables cause all trajectories to eventually escape after finitely many collisions and introduces an adiabatic invariant related to these dynamics.

Contribution

It establishes the finite-time escape property for trajectories in billiards with horns and derives an adiabatic invariant, expanding understanding of billiard dynamics with singularities.

Findings

Trajectories in billiards with horns escape after finitely many collisions.

An adiabatic invariant is constructed for billiards with horns.

Horns behave similarly to cusps in expelling trajectories.

Abstract

We show that, like cusps, horns in billiards expel every trajectory after finitely many collisions. We further produce an adiabatic invariant.