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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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