Outer billiards

TL;DR

This paper investigates the dynamics of outer billiards outside regular polygons, proving the existence of aperiodic points for certain shapes and characterizing periodic points for the octagon.

Contribution

It provides new proofs of aperiodic points existence and characterizes periodic points for outer billiards outside regular polygons, especially the octagon.

Findings

Existence of aperiodic points outside regular octagon and dodecagon.

Set of periodic points has full measure for regular octagon.

Complete list of possible periods for the regular octagon.

Abstract

In this paper outer, or dual, billiards outside regular polygons are studied; in particular, periodic points for cases of strictly convex "tables" and for regular n-gons with n = 3,4,6,8,12 are discussed. The main results of the paper are: 1) proof of an existence of aperiodic points for outer billiards outside regular octagon and dodecagon; 2) proof of the fact that for regular octagon, set of periodic points is of full measure; 3) list of all possible periods for case of regular octagon.