Journal of Modern Dynamics (JMD)

Unbounded orbits for outer billiards I

Pages: 371 - 424, Issue 3, July 2007      doi:10.3934/jmd.2007.1.371

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

Abstract: The question of B.H. Neumann, which dates back to the 1950s, asks if there exists an outer billiards system with an unbounded orbit. We prove that outer billiards for the Penrose kite, the convex quadrilateral from the Penrose tiling, has an unbounded orbit. We also analyze some finer properties of the orbit structure, and in particular produce an uncountable family of unbounded orbits. Our methods relate outer billiards on the Penrose kite to polygon exchange maps, arithmetic dynamics, and self-similar tilings.

Keywords:  outer billiards, dual billiards, unbounded orbits, Penrose kite, polygon exchange, arithmetic graph.
Mathematics Subject Classification:  Primary: 37E99; Secondary: 52C23.

Received: January 2007;      Revised: February 2007;      Available Online: April 2007.