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Horns in Billiards

David de Frutos Ostrander, Boris Hasselblatt, Mark Levi

Abstract

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

Horns in Billiards

Abstract

We show that, like cusps, horns in billiards expel every trajectory after finitely many collisions. We further produce an adiabatic invariant.
Paper Structure (3 sections, 3 theorems, 24 equations, 4 figures)

This paper contains 3 sections, 3 theorems, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Adiabatic invariant

Key Result

Theorem A

As in FIGHorn, consider two $C^3$ arcs in the plane with a common end-point $O$ at which they share a tangent line and have different curvatures of the same sign. Then there exists a neighborhood of $O$ such that any billiard trajectory starting in it will leave this neighborhood after finitely many

Figures (4)

  • Figure 1: A horn
  • Figure 2: A horn with centers of osculating circles and the "middle" circle.
  • Figure 3: Proof that the angular momentum relative to $\overline O$ is repelled from the tip of the cusp.
  • Figure 4: The focusing boundary and the middle circle (radius $R= R_+-d$).

Theorems & Definitions (7)

  • Theorem A
  • Remark 1
  • Lemma 1
  • proof : Proof of \ref{['lem:key']}
  • Remark 2
  • proof : Proof of Theorem \ref{['thm:main']}
  • Theorem B