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Finite blocking property versus pure periodicity

Published online by Cambridge University Press:  01 June 2009

THIERRY MONTEIL
THIERRY MONTEIL*
Affiliation:
CNRS – LIRMM – Univ. Montpellier 2, 161 rue Ada, 34392 Montpellier, Francehttp://www.lirmm.fr/∼monteil
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Abstract

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A translation surface 𝒮 is said to have the finite blocking property if for every pair (O,A) of points in 𝒮 there exists a finite number of ‘blocking’ points B1,…,Bn such that every geodesic from O to A meets one of the Bis. 𝒮 is said to be purely periodic if the directional flow is periodic in each direction whose directional flow contains a periodic trajectory (this implies that 𝒮 admits a cylinder decomposition in such directions). We will prove that the finite blocking property implies pure periodicity. We will also classify the surfaces that have the finite blocking property in genus two: such surfaces are exactly the torus branched coverings. Moreover, we prove that in every stratum such surfaces form a set of null measure. In Appendix A, we prove that completely periodic translation surfaces form a set of null measure in every stratum.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009, pp. 983 - 996
Copyright
Copyright © Cambridge University Press 2009

References

[1]Boca, F., Gologan, R. and Zaharescu, A.. The statistics of the trajectory of a certain billiard in a flat torus. Comm. Math. Phys. 240 (2003), 53–76.CrossRefGoogle Scholar
[2]Calta, K.. Veech surfaces and complete periodicity in genus 2. J. Amer. Math. Soc. 17(4) (2004), 871–908.Google Scholar
[3]Cheung, Y.. Hausdorff dimension of the set of nonergodic directions. Ann. of Math. (2) 158 (2003), 661–678.CrossRefGoogle Scholar
[4]Chernov, N. and Galperin, G.. Search light in billiard tables. Regul. Chaotic Dyn. 8(2) (2003), 225–241.CrossRefGoogle Scholar
[5]Fomin, D. and Kirichenko, A.. Leningrad Mathematical Olympiads 1987–1991. MathPro Press, 1994, p. 197.Google Scholar
[6]Gutkin, E.. Blocking of billiard orbits and security for polygons and flat surfaces. Geom. Funct. Anal. 15(1) (2005), 83–105.CrossRefGoogle Scholar
[7]Hiemer, P. and Snurnikov, V.. Polygonal billiards with small obstacles. J. Stat. Phys. 90(1–2) (1998), 453–466.Google Scholar
[8]Klee, V.. Is every polygonal region illuminable from some point?  Amer. Math. Monthly 76(80) (1969).CrossRefGoogle Scholar
[9]Kerckhoff, S., Masur, H. and Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. 124(2) (1986), 293–311.CrossRefGoogle Scholar
[10]Kontsevich, M. and Zorich, A.. Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153(3) (2003), 631–678.Google Scholar
[11]Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115(1) (1982), 169–200.CrossRefGoogle Scholar
[12]Masur, H.. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66(3) (1992), 387–442.CrossRefGoogle Scholar
[13]Masur, H. and Smillie, J.. Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. 134(3) (1991), 455–543.CrossRefGoogle Scholar
[14]Masur, H. and Tabachnikov, S.. Rational Billiards and Flat Structures (Handbook on Dynamical Systems, 1A). North-Holland, Amsterdam, 2002, pp. 1015–1089.Google Scholar
[15]Monteil, T.. A counter-example to the theorem of Hiemer and Snurnikov. J. Stat. Phys. 114(5–6) (2004), 1619–1623.CrossRefGoogle Scholar
[16]Monteil, T.. On the finite blocking property. Ann. Inst. Fourier (Grenoble) 55(4) (2005), 1195–1217.CrossRefGoogle Scholar
[17]Monteil, T.. A homological condition for a dynamical and illuminatory classification of torus branched coverings. Preprint http://arxiv.org/abs/math.DS/0603352.Google Scholar
[18]Tokarsky, G.. Polygonal rooms not illuminable from every point. Amer. Math. Monthly 102(10) (1995), 867–879.CrossRefGoogle Scholar
[19]Veech, W.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201–242.Google Scholar
[20]Veech, W.. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3) (1989), 553–583.Google Scholar
[21]Vorobets, Y.. Planar structures and billiards in rational polygons: the Veech alternative. Russian Math. Surveys 51(5) (1996), 779–817.CrossRefGoogle Scholar
[22]Zemljakov, A. and Katok, A.. Topological transitivity of billiards in polygons. Mat. Zametki 18(2) (1975), 291–300.Google Scholar