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Blocking of billiard orbits and security for polygons and flat surfaces

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Geometric & Functional Analysis GAFA Aims and scope Submit manuscript

Abstract.

We introduce and study the notion of security for polygons and flat surfaces. Let P be one. For x, y ∈ P let G(x, y) be the set of geodesics connecting x and y. We say that P is secure if for any x, y ∈ P all geodesics in G(x, y) can be blocked by a finite set B ⊂ P. We prove, in particular, that a lattice polygon is secure iff it is arithmetic.

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Authors and Affiliations

  1. Department of Mathematics, PennState University, University Park, PA, 16802, USA

    E. Gutkin

  2. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320

    E. Gutkin

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  1. E. Gutkin
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Correspondence to E. Gutkin.

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Received: September 2003 Revision: March 2004 Accepted: March 2004

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Gutkin, E. Blocking of billiard orbits and security for polygons and flat surfaces. GAFA, Geom. funct. anal. 15, 83–105 (2005). https://doi.org/10.1007/s00039-005-0502-2

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  • DOI: https://doi.org/10.1007/s00039-005-0502-2

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