Outer Billiards on Regular Polygons
Abstract
In 1973, J. Moser proposed that his Twist Theorem could be used to show that orbits of the outer billiards map on a sufficiently smooth closed curve were always bounded. Five years later Moser asked the same question for a convex polygon. In 1987 F. Vivaldi and A. Shaidenko showed that all orbits for a regular polygon must be bounded. R. Schwartz recently showed that a quadrilateral known as a Penrose Kite has unbounded orbits and he proposed that 'most' convex polygons support unbounded orbits. Except for a few special cases, very little is known about the dynamics of the outer billiards map on regular polygons. In this paper we present a unified approach to the analysis of regular polygons  using the canonical 'resonances' which are shared by all regular Ngons. In the case of the regular pentagon and regular octagon these resonances exist on all scales and the fractal structure is well documented, but these are the only nontrivial cases that have been analyzed. We present a partial analysis of the regular heptagon, but the limiting structure is poorly understood and this does not bode well for the remaining regular polygons. The minimal polynomial for the vertices of a regular Ngon has degree Phi(N)/2 where Phi is the Euler totient function, so N = 5, 7 and 11 are respectively quadratic, cubic and quintic. In the words of R. Schwartz, "A case such as N = 11 seems beyond the reach of current technology." Some of the graphics have embedded highresolution versions so this file is about 39Mb in size. This file and a smaller version can be downloaded at dynamicsofpolygons.org. Just click on PDFs. This paper is dedicated to the memory of Eugene Gutkin (19462013) who made fundamental contributions to both inner and outer billiards.
 Publication:

arXiv eprints
 Pub Date:
 November 2013
 arXiv:
 arXiv:1311.6763
 Bibcode:
 2013arXiv1311.6763H
 Keywords:

 Mathematics  Dynamical Systems;
 37E40;
 37J10;
 37K55;
 52A10;
 53A60;
 52C15;
 52C23
 EPrint:
 75 pages with embedded graphics. New material on semiregular polygons in Appendix G