On the finite blocking property

Thierry Monteil[1]

  • [1] Institut de Mathématiques de Luminy, CNRS UMR 6206, Case 907, 163 Avenue de Luminy, 13288 Marseille cedex 09 (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 4, page 1195-1217
  • ISSN: 0373-0956

Abstract

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A planar polygonal billiard 𝒫 is said to have the finite blocking property if for every pair ( O , A ) of points in 𝒫 there exists a finite number of “blocking” points B 1 , ⋯ , B n such that every billiard trajectory from O to A meets one of the B i ’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces. We prove that the only Veech surfaces with the finite blocking property are the torus branched coverings. We also provide a local sufficient condition for a translation surface to fail the finite blocking property. This enables us to give a complete classification for the L-shaped surfaces as well as to obtain a density result in the space of translation surfaces in every genus g ≥ 2 .

How to cite

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Monteil, Thierry. "On the finite blocking property." Annales de l’institut Fourier 55.4 (2005): 1195-1217. <http://eudml.org/doc/116217>.

@article{Monteil2005,
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 $B_1, \dots , B_n$ such that every billiard trajectory from $O$ to $A$ meets one of the $B_i$’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces. We prove that the only Veech surfaces with the finite blocking property are the torus branched coverings. We also provide a local sufficient condition for a translation surface to fail the finite blocking property. This enables us to give a complete classification for the L-shaped surfaces as well as to obtain a density result in the space of translation surfaces in every genus $g\ge 2$.},
affiliation = {Institut de Mathématiques de Luminy, CNRS UMR 6206, Case 907, 163 Avenue de Luminy, 13288 Marseille cedex 09 (France)},
author = {Monteil, Thierry},
journal = {Annales de l’institut Fourier},
keywords = {Blocking property; polygonal billiards; regular polygons; translation surfaces; Veech surfaces; torus branched covering; illumination; quadratic differentials; blocking property},
language = {eng},
number = {4},
pages = {1195-1217},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the finite blocking property},
url = {http://eudml.org/doc/116217},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Monteil, Thierry
TI - On the finite blocking property
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 4
SP - 1195
EP - 1217
AB - 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. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces. We prove that the only Veech surfaces with the finite blocking property are the torus branched coverings. We also provide a local sufficient condition for a translation surface to fail the finite blocking property. This enables us to give a complete classification for the L-shaped surfaces as well as to obtain a density result in the space of translation surfaces in every genus $g\ge 2$.
LA - eng
KW - Blocking property; polygonal billiards; regular polygons; translation surfaces; Veech surfaces; torus branched covering; illumination; quadratic differentials; blocking property
UR - http://eudml.org/doc/116217
ER -

References

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  1. M. Boshernitzan, Correspondence with Howard Masur, (1986) 
  2. N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6, (1981), Masson Zbl0483.22001MR647314
  3. K. Calta, Veech surfaces and complete periodicity in genus 2  Zbl1073.37032MR2083470
  4. D. Cartwright, A brief introduction to buildings, Harmonic functions on trees and buildings 206 (1997), 45-77, Amer. Math. Soc. Zbl0943.51011
  5. A. Eskin, A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001), 59-103 Zbl1019.32014MR1839286
  6. D. Fomin, (1990) 
  7. P. Hiemer, V. Snurnikov, Polygonal billiards with small obstacles, J. Statist. Phys. 90 (1998), 453-466 Zbl0995.37019MR1611100
  8. V. Klee, Is every polygonal region illuminable from some point?, Amer. Math. Monthly 76 (1969) MR1535289
  9. V. Klee, S. Wagon, Old and new unsolved problems in plane geometry and number theory, (1991), Math. Assoc. America Zbl0784.51002MR1133201
  10. S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. Math. 124 (1986), 293-311 Zbl0637.58010MR855297
  11. M. Kontsevich, Lyapunov exponents and Hodge theory, The mathematical beauty of physics (Saclay, 1996) 24 (1997), 318-332, World Sci. Publishing Zbl1058.37508
  12. M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631-678 Zbl1087.32010MR2000471
  13. H. Masur, S. Tabachnikov, Rational billiards and flat structures, 1A (2002), 1015-1089, North-Holland Zbl1057.37034
  14. C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), 857-885 Zbl1030.32012MR1992827
  15. T. Monteil, A counter-example to the theorem of Hiemer and Snurnikov, J. Statist. Phys. 114 (2004), 1619-1623 Zbl1058.37040MR2039490
  16. T. Monteil, Finite blocking property versus pure periodicity 
  17. J. Schmeling, S. Troubetzkoy, Inhomogeneous Diophantine Approximation and Angular Recurrence for Polygonal Billiards, Mat. Sb. 194 (2003), 129-144 Zbl1043.37028MR1992153
  18. G. Tokarsky, Polygonal rooms not illuminable from every point, Amer. Math. Monthly 102 (1995), 867-879 Zbl0849.52007MR1366048
  19. W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 553-583 Zbl0676.32006MR1005006
  20. Y. Vorobets, Planar structures and billiards in rational polygons: the Veech alternative, Russian Math. Surveys 51 (1996), 779-817 Zbl0897.58029MR1436653
  21. A. Zemljakov, A. Katok, Topological transitivity of billiards in polygons, Mat. Zametki 18 (1975), 291-300 Zbl0323.58012MR399423
  22. A. Zorich, Private communication, (2003) 

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