# American Institute of Mathematical Sciences

July  2011, 5(3): 473-581. doi: 10.3934/jmd.2011.5.473

## Outer billiards on the Penrose kite: Compactification and renormalization

 1 Department of Mathematics, Brown University, Providence, RI 02912, United States

Received  February 2011 Revised  August 2011 Published  November 2011

We give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this $2$-dimensional dynamical system has a $3$-dimensional compactification, a certain polyhedron exchange map defined on the $3$-torus, and that this $3$-dimensional system admits a renormalization scheme. The two features allow us to make sharp statements concerning the distribution, large- and fine-scale geometry, and hidden algebraic symmetry, of the orbits. One concrete result is that the union of the unbounded orbits has Hausdorff dimension $1$. We establish many of the results with computer-aided proofs that involve only integer arithmetic.
Citation: Richard Evan Schwartz. Outer billiards on the Penrose kite: Compactification and renormalization. Journal of Modern Dynamics, 2011, 5 (3) : 473-581. doi: 10.3934/jmd.2011.5.473
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##### References:
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