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

Abstract

We propose a new method to handle the n-dirnensional 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.© 2008 Society for Industrial and Applied Mathematics.