Luchezar Stoyanov
Exponential Instability for a Class of Dispersing Billiards
(79K, AmsLaTex)
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
$0 < \delta < 1$ 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 $C(\delta^j + \delta^{n-j})$ for a sequence of integers
$j$ with uniform density in $1,2,\ldots,n$. Consequently, the
billiard ball map (though not continuous in general) is expansive. As applications,
an asymptotic of the number of prime closed billiard trajectories is proved which generalizes
a result of T. Morita \cite{kn:Mor}, and it is shown that the topological entropy of the billiard
flow does not exceed $\frac{\log (s-1)}{a}$, where $s$ is the number of convex components of $K$
and $a$ is the minimal distance between different convex components of $K$.