Exponential instability for a class of dispersing billiards

Abstract. The billiard in the exterior of a finite disjoint union K of strictly convex bodies in IRd 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 n successive reflections from the same convex components of K, then the distance between their jth reflection points is less than C(δj + δn−j) for a sequence of integers j with uniform density in 1, 2,..., n. Consequently, the billiard ball map (though not continuous in general) is expansive. As applications, an asymp-totic of the number of prime closed billiard trajectories is proved which generalizes a result of T. Morita [Mor], and it is shown that the topological entropy of the billiard flow does not ex-ceed log(s−1)a, where s is the number of convex components of K and a is the minimal distance between different convex components of K. 1