Tabachnikov, Serge Birkhoff billiards are insecure. (English) Zbl 1160.37025 Discrete Contin. Dyn. Syst. 23, No. 3, 1035-1040 (2009). 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. Reviewer: Mircea Crâşmăreanu (Iaşi) Cited in 1 ReviewCited in 3 Documents 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 Keywords:Birkhoff billiard; security; finite blocking; rational approximation PDFBibTeX XMLCite \textit{S. Tabachnikov}, Discrete Contin. Dyn. Syst. 23, No. 3, 1035--1040 (2009; Zbl 1160.37025) Full Text: DOI arXiv