On general rotation sets of open billiards

Abstract

We investigate the rotation sets of open billiards in 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 φ is dense in it. We provide a constructive proof which illustrates that the set P φ 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. Mathematics Subject Classificationis defined for all n ∈ Z, which is called the non-wandering set. We call the points q ∈ M 0 with B k (q) = q for some k > 0 periodic points of period k of B.