Outer billiards and the pinwheel map

Richard Evan Schwartz

doi:10.3934/jmd.2011.5.255

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2011, Volume 5, Issue 2: 255-283. Doi: 10.3934/jmd.2011.5.255

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Outer billiards and the pinwheel map

Richard Evan Schwartz^1,

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

Received: June 30, 2010

Revised: February 28, 2011

Published: June 2011

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Abstract

In this paper we establish an equivalence between an outer billiards system based on a convex polygon $P$ and an auxiliary system, which we call the pinwheel map, that is based on $P$ in a different way. The pinwheel map is akin to a first-return map of the outer billiards map. The virtue of our result is that most of the main questions about outer billiards can be formulated in terms of the pinwheel map, and the pinwheel map is simpler and seems more amenable to fruitful analysis.

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Mathematics Subject Classification: Primary: 37E15; Secondary: 37E99.

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