Outer Billiards on Kites

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.