Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton

TL;DR

This paper thoroughly analyzes outer billiards on the Penrose kite, revealing a 3D compactification with renormalization, and provides insights into orbit distribution, geometry, and symmetries using computer-aided proofs.

Contribution

It introduces a 3D compactification and renormalization scheme for outer billiards on the Penrose kite, advancing understanding of its geometric and algebraic properties.

Findings

Unbounded orbits have Hausdorff dimension 1

Established a compactification with a polyhedron exchange map

Proved renormalization scheme using computer-aided exact calculations

Abstract

In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us to make some sharp statements concerning the distribution, large-scale geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits. For instance, one of our results is that the union of the unbounded orbits has Hausdorff dimension 1. We give a computer-aided proof of the results concerning the compactification and the renormalization. This proof involves finitely many calculations done with exact integer arithmetic.