Uniform Estimates on the Number of Collisions in Semi-Dispersing Billiards

Abstract

Estimating the number of collisions in a neighborhood of a given is a central problem in the theory of billiard systems. Moreover, the existence of a uniform estimate on the number of collisions is related to various important properties of a billiard system. For example, the Sinai-Chernov formulas for the metric entropy of billiards are proved under the assumption that such an estimate exists ([3], [9]). It is quite obvious that if the boundary of the billiard has any concave parts then a trajectory can have arbitrarily many collisions in a neighborhood of a point of concavity, and sometimes even infinitely many collisions (see, for example [11]). Therefore, the class of billiards for which uniform estimates are possible are the semi-dispersing billiards, which have been studied extensively in the numerous works (see the review in [7]). The most important examples of semi-dispersing billiards are the billiards that correspond to hard balls gas models. Vaserstein (1979, [11]) and Gal'perin (1981, [4]) proved that in semidispersing billiards any trajectory has only finitely many collisions in any finite period of time. Sinai (1978, [8]) proved the existence of a uniform estimate for billiards inside polyhedral angles, and pointed out that his results should also hold for semi-dispersing billiards in a neighborhood of a x with linearly independent normals to the walls of the billiard at x (see the remark at the end of [8]). In this paper we deal with semi-dispersing billiards on arbitrary manifolds. All our results apply to the usual billiards in Rk and Wk. First we prove that for semi-dispersing billiards on arbitrary manifolds any trajectory has only a finite number of collisions in a finite period of time.