Journal of Modern Dynamics (JMD)

Outer billiards on the Penrose kite: Compactification and renormalization

Pages: 473 - 581, Issue 3, July 2011      doi:10.3934/jmd.2011.5.473

       Abstract        References        Full Text (1247.0K)       Related Articles       

Richard Evan Schwartz - Department of Mathematics, Brown University, Providence, RI 02912, United States (email)

Abstract: 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.

Keywords:  Dynamics, outer billiards, Penrose kite, compactification, renormalization, polytope exchange, piecewise translation.
Mathematics Subject Classification:  Primary: 37E15; Secondary: 37E99.

Received: February 2011;      Revised: August 2011;      Available Online: November 2011.