Exponential instability for a class of dispersing billiards

The billiard in the exterior of a finite disjoint union K of strictly conve x bodies in R-d with smooth boundaries is considered. The existence of glob al constants 0 < delta < 1 and C > 0 is established such that if two billia rd trajectories have n successive reflections from the same convex componen ts of K, then the distance between their jth reflection points is less than C(delta(j) + delta(n-j)) for a sequence of integers j with uniform density in (1,2,..., n). Consequently, the billiard ball map (although not continu ous in general) is expansive. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of Morita [Mor], and it is shown that the topological entropy of the billia rd flow does not exceed log(s - I)/a, where s is the number of convex compo nents of K and a is the minimal distance between different convex component s of K.