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A counter-example to the theorem of Hiemer and Snurnikov. (English) Zbl 1058.37040

Summary: A planar polygonal billiard \({\mathcal P}\) is said to have the finite blocking property if for every pair \((O,A)\) of points in \({\mathcal P}\), there exists a finite number of “blocking” points \(B_1,\dots,B_n\), such that every billiard trajectory from \(O\) to \(A\) meets one of the \(B_i\)’s. As a counter-example to a theorem of P. Hiemer and V. Snurnikov [J. Stat. Phys. 90, 453–466 (1998; Zbl 0995.37019)], we construct a family of rational billiards that lack the finite blocking property.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
70H05 Hamilton’s equations

Citations:

Zbl 0995.37019
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