Outer Billiards and the Pinwheel Map

TL;DR

This paper establishes a bijection between the orbits of polygonal outer billiards and a simpler system called the pinwheel map, enabling easier analysis of orbit boundedness in outer billiards.

Contribution

It introduces a bijection linking outer billiards orbits to the pinwheel map, simplifying the study of orbit behavior in polygonal systems.

Findings

Outer billiards has unbounded orbits iff the pinwheel map does

Bijection simplifies analysis of outer billiards

Potential to address main questions in polygonal outer billiards

Abstract

In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer billiards system has unbounded orbits if and only if the pinwheel map has unbounded orbits. As the pinwheel map is much easier to analyze directly, we think that this bijection will be helpful in attacking some of the main questions about polyonal outer billiards.