Outer Billiards with Contraction: Regular Polygons

Abstract

We study outer billiards with contraction outside regular polygons. For regular $n$-gons with $n=3,4,5,6,8$, and $12$, we show that as the contraction rate approaches $1$, dynamics of the system converges, in a certain sense, to that of the usual outer billiards map. These are precisely the values of $n\ge 3$ with $[\mathbb{Q}(e^{2\pi i/n}):\mathbb{Q}]\le 2$. Then we discuss how such convergence may fail in the case of $n=7$.

17 pages

References in corpus (2)

• Outer Billiards, Arithmetic Graphs, and the Octagon
• Outer Billiards with Contraction: Attracting Cantor Sets