Chaotic n-dimensional euclidean and hyperbolic open billiards and chaotic spinning planar billiards

Abstract

We propose a new method to handle the n-dimensional billiard problem in the exterior of a finite mutually disjoint union of convex ( but not necessarily strictly convex) smooth obstacles without eclipse in the Euclidean or hyperbolic n-space, and we prove that there exist trajectories visiting the obstacles in any given doubly infinite prescribed order ( with the obvious restriction of no consecutive repetition). As an interesting variant of planar billiards, we consider spinning obstacles and particles and prove that any forward sequence of obstacles has a trajectory that follows it.