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arXiv · 1102.4635

Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton

Richard Evan Schwartz
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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.

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Richard Evan Schwartz. 2011-02-22. Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton. https://arxiv.org/abs/1102.4635

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