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On the finite blocking property. (English) Zbl 1076.37029

A polygon is said to have the finite blocking property if every billiard orbit connecting two arbitrary points \(A\) and \(O\) on the boundary must intersect a finite set of boundary points. Such set depends on \(A\) and \(O\), but does not contain them. The author proves – among other things – that the only regular polygons having such property are the triangle, the square, and the hexagon. This definition extends to translation surfaces, for which some useful characterizations are derived.

MSC:

37E35 Flows on surfaces
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
30F30 Differentials on Riemann surfaces
51M99 Real and complex geometry
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