Rotation Sets of Open Billiards

TL;DR

This paper studies the properties of rotation sets in open billiards, proving convexity and density of periodic trajectory rotation vectors, and explores complex behaviors in specific billiard configurations.

Contribution

It establishes convexity and density properties of rotation sets in open billiards and provides constructive proofs and examples of complex rotation behaviors.

Findings

Rotation set is convex.

Periodic trajectory rotation vectors are dense in the rotation set.

Existence of sequences with undefined rotation vectors in certain 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.