Unbounded orbits for outer billiards I

Richard Evan Schwartz

doi:10.3934/jmd.2007.1.371

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2007, Volume 1, Issue 3: 371-424. Doi: 10.3934/jmd.2007.1.371

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Unbounded orbits for outer billiards I

Richard Evan Schwartz^1,

1.
Department of Mathematics, Brown University, Providence, RI 02912

Received: December 31, 2006

Revised: January 31, 2007

Published: March 2007

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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.

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Mathematics Subject Classification: Primary: 37E99; Secondary: 52C23.

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