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Exponential instability for a class of dispersing billiards. (English) Zbl 0923.58028
A billiard in the exterior of a finite disjoint union $$K$$ of strictly convex bodies in $$\mathbb R^d$$ is studied. The existence of $$\delta>0$$ and $$C>0$$ is proved such that if two billiard trajectories have $$n$$ successive reflections at the same convex components of $$K$$, then the distance between their $$j$$th reflection points is less than $$C(\delta^j+\delta^{n-j})$$. Consequently, the billiard map is expansive. As an application, it is proved that the topological entropy $$h$$ of the billiard flow does not exceed $$\log(s-1)/a$$, where $$s$$ is the number of convex components of $$K$$ and $$a$$ is the minimal distance between them. Under the additional ‘no eclipse’ assumption (meaning that the convex hull of any two components of $$K$$ is disjoint from any other component of $$K$$), an asymptotic formula for the number of prime closed billiard orbits is proved: $$\#\{\gamma: e^{hT_{\gamma}}\leq\lambda\}\log\lambda/\lambda\to 1$$ as $$\lambda\to\infty$$, where $$T_{\gamma}$$ is the period of a closed orbit $$\gamma$$.

##### MSC:
 37A99 Ergodic theory 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 54H20 Topological dynamics (MSC2010) 54C70 Entropy in general topology
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