Journal of Modern Dynamics (JMD)

Outer billiards and the pinwheel map

Pages: 255 - 283, Issue 2, April 2011      doi:10.3934/jmd.2011.5.255

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

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.

Keywords:  Outer billiards, piecewise translations, pinwheel map.
Mathematics Subject Classification:  Primary: 37E15; Secondary: 37E99.

Received: July 2010;      Revised: March 2011;      Available Online: July 2011.