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Birkhoff billiards are insecure. (English) Zbl 1160.37025

Two points \(A, B\) of a Riemannian manifold \(M\) are called secure if there exists a finite set of points \(S\subset M\setminus \{A, B\}\) such that every geodesic connecting \(A\) and \(B\) passes through a point of \(S\). The manifold is called secure if any pair of its points is secure; for example a flat torus of any dimension is secure. In this note it is proved that a compact plane billiard domain, bounded by a smooth curve, is insecure. An appendix, written by R. Schwartz, contains an useful result on rational approximation.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37E99 Low-dimensional dynamical systems
53C22 Geodesics in global differential geometry
78A05 Geometric optics
11K60 Diophantine approximation in probabilistic number theory
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