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Journal of Statistical Physics

, Volume 114, Issue 5–6, pp 1619–1623 | Cite as

A Counter-Example to the Theorem of Hiemer and Snurnikov

  • Thierry Monteil
Article

Abstract

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 B1,...,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 Hiemer and Snurnikov, we construct a family of rational billiards that lack the finite blocking property.

rational polygonal billiards translation surfaces blocking property 

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REFERENCES

  1. 1.
    P. Hiemer and V. Snurnikov, Polygonal billiards with small obstacles, J. Stat. Phys. 90:453–466 (1998).Google Scholar
  2. 2.
    H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook on Dynamical Systems, Vol. 1A (North-Holland, Amsterdam, 2002), pp. 1015–1089.Google Scholar
  3. 3.
    T. Monteil, On the finite blocking property, preprint.Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Thierry Monteil
    • 1
  1. 1.CNRS UPR 9016, Case 907Institut de Mathématiques de LuminyMarseille cedex 09France

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