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

Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton