95-540 Luchezar Stoyanov

Exponential Instability for a Class of Dispersing Billiards (79K, AmsLaTex) Dec 21, 95

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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 $K$.

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