Unbounded Orbits for Outer Billiards

TL;DR

This paper proves the existence of unbounded orbits in outer billiards systems, specifically relative to the Penrose kite, and analyzes the complex structure of these orbits using advanced mathematical tools.

Contribution

It demonstrates for the first time that outer billiards on the Penrose kite have unbounded orbits and explores their intricate structure using self-similar tilings and arithmetic dynamics.

Findings

Unbounded orbits exist for outer billiards on the Penrose kite.

There is an uncountable set of such unbounded orbits.

The orbit structure relates to self-similar tilings and polygon exchange maps.

Abstract

Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has been: is there an outer billiards system with an unbounded orbit. We answer this question by proving that outer billiards defined relative to the Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that appears in the famous Penrose tiling. We also analyze some of the finer orbit structure of outer billiards on the penrose kite. This analysis shows that there is an uncountable set of unbounded orbits. Our method of proof relates the problem to self-similar tilings, polygon exchange maps, and arithmetic dynamics.