Abstract
The billiard in the exterior of a finite disjoint union K of strictly convex bodies in ℝ d with smooth boundaries is considered. The existence of global constants 0 < δ < 1 and C > 0 is established, such that if two billiard trajectories have reflections at the same convex components of K from time t = 0 to t = T, then the distance between the corresponding points on the two trajectories at time t is less than C( δ t + δ T−1 for t ε [0,T]. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of T. Morita, and it is shown that the topological entropy of the billiard flow does not exceed log ( s − 1 ) α , where s is the number of convex components of K and a is the minimal distance between different convex components of K.
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