Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton
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.
Keywords
Cite
@article{arxiv.1102.4635,
title = {Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:1102.4635},
year = {2011}
}
Comments
This is an extremely long paper -- 142 pages. It might be better as a monograph. I have tried to make the paper as modular as possible, isolating the computer-aided parts into clearly-defined units