Outer Billiards with Contraction: Regular Polygons

TL;DR

This paper investigates the dynamics of outer billiards with contraction outside regular polygons, showing convergence to classical billiards for specific polygons and discussing exceptions.

Contribution

It establishes convergence results for regular polygons with certain symmetries and explores cases where convergence may not occur, expanding understanding of billiard dynamics.

Findings

Convergence of dynamics as contraction approaches 1 for polygons with specific symmetries.

Identification of polygons where convergence to classical billiards occurs.

Discussion of potential divergence in the case of heptagons.

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 $n \geq 3$ with $[\mathbb{Q}(e^{2\pi i/n}):\mathbb{Q}] \leq 2$. Then we discuss how such convergence may fail in the case of $n=7$.