Outer Billiards on Kites

TL;DR

This paper investigates the dynamics of outer billiards on kites, proving the existence of unbounded orbits for irrational kites and describing their structure and dimensions, revealing deep connections to various mathematical areas.

Contribution

It proves unbounded orbits exist for all irrational kites and characterizes their structure and Hausdorff dimension, linking outer billiards to modular groups and self-similar tilings.

Findings

Unbounded orbits exist for any irrational kite.

Detailed description of the set of unbounded orbits.

Connections established with modular groups and Diophantine approximation.

Abstract

Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The first half of this monograph proves that outer billiards has an unbounded orbit defined relative to any irrational kite. The second half of the monograph gives a very sharp description of the set of unbounded orbits, both in terms of the dynamics and the Hausdorff dimension. The analysis in both halves reveals a close connection between outer billiards on kites and the modular group, as well as connections to self-similar tilings, polytope exchange maps, Diophantine approximation, and odometers.