Rotation Sets of Billiards with N Obstacles on a Torus

TL;DR

This paper investigates the properties of admissible trajectories in billiards with N obstacles on a torus, proving convexity of the admissible rotation set, density of periodic trajectories, and its relation to the general rotation set.

Contribution

It establishes the convexity of the admissible rotation set and shows that periodic admissible trajectories are dense within it, highlighting new geometric properties.

Findings

Admissible rotation set is convex.

Periodic admissible trajectories are dense in the rotation set.

Admissible rotation set is a proper subset of the general rotation set.

Abstract

For billiards with $N$ obstacles on a torus, we study the behavior of specific kind of its trajectories, \emph{the so called admissible trajectories}. Using the methods developed in \cite{1}, we prove that the \emph{admissible rotation set} is convex, and the periodic trajectories of admissible type are dense in the admissible rotation set. In addition, we show that the admissible rotation set is a proper subset of the general rotation set.