Rotation Sets of Open Billiards

Abstract

We investigate the rotation sets of open billiards in $\mathbb{R}^N$ for the natural observable related to a starting point of a given billiard trajectory. We prove that the general rotation set is convex and the set of all convex combinations of rotation vectors of periodic trajectory $P_\phi$ is dense in it. We provide a constructive proof which illustrates that the set $P_\phi$ is dense in the pointwise rotation set, and the closure of the pointwise rotation set is convex. We also consider a class of billiards consisting of three obstacles and construct a sequence in the symbol space such that its rotation vector is not defined.