Lyapunov Exponents for Open Billiards in the Exterior of Balls

In this paper we consider the billiard flow in the exterior of several (at least three) balls in ${\mathbb{R}}^3$ with centres lying on a plane. We assume that the balls satisfy the no eclipse condition (H) and their radii are small compared to the distances between their centres. We prove that with respect to any Gibbs measure on the non-wandering set of the billiard flow the two positive Lyapunov exponents are different: ${\lambda }_1>{\lambda }_2>0$.