Table des matières de ce fascicule | Article précédent | Article suivant
Thierry Monteil
On the finite blocking property
(Sur la propriété de blocage fini)
Annales de l'institut Fourier, 55 no. 4 (2005), p. 1195-1217
Article: sur abonnement | Analyses MR 2157167 | Zbl 1076.37029
Class. Math.: 37E35, 37D50, 37D40, 37A10, 5199, 30F30
Mots clés: propriété de blocage, billards polygonaux, polygones réguliers, surfaces de translation, surfaces de Veech, revêtement ramifié du tore, illumination, différentielles quadratiques
On dit qu'un billard polygonal ${\cal P}$ a la propriété de blocage fini si pour tout
couple de points $(O,A)$ de ${\cal P}$ il existe un nombre fini de points ``bloquants''
$B_1, \dots , B_n$ tels que toute trajectoire de billard de $O$ à $A$ rencontre l'un des
$B_i$. En généralisant notre construction d'un contre exemple à un théorème de Hiemer et
Snurnikov, nous montrons que les seuls polygones réguliers qui ont la propriété de
blocage fini sont les carrés, le triangle équilateral et l'hexagone. Puis nous étendons
ce résultat aux surfaces de translation. Nous prouvons que les seules surfaces de Veech
jouissant de la propritété de blocage fini sont les revêtements ramifiés du tore. Nous
donnons aussi une condition suffisante locale pour qu'une surface de translation ne
jouisse pas de la propriété de blocage fini. Cela nous permet de donner une
classification complète pour les surfaces en forme de L ainsi que d'obtenir un résultat
de densité dans l'espace des surfaces de translation en tout genre $g\geq 2$.
[Bos] M. Boshernitzan, “Correspondence with Howard Masur”, (unpublished), 1986
[Bou] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6, Masson, 1981 MR 647314 | Zbl 0483.22001
[Cal] K. Calta, “Veech surfaces and complete periodicity in genus $2$”, J. Amer. Math. Soc., to appear
arXiv | MR 2083470 | Zbl 02106878
[Car] D. Cartwright, A brief introduction to buildings, Contemp. Math., Amer. Math. Soc., 1997, p. 45-77 Zbl 0943.51011
[EO] 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), p. 59-103 MR 1839286 | Zbl 1019.32014
[Fo] D. Fomin, “”, Zadaqi Leningradskih Matematitcheskih Olimpiad, Leningrad, 1990
[HS] P. Hiemer & V. Snurnikov, “Polygonal billiards with small obstacles”, J. Statist. Phys. 90 (1998), p. 453-466 MR 1611100 | Zbl 0995.37019
[Kl] V. Klee, “Is every polygonal region illuminable from some point?”, Amer. Math. Monthly 76 (1969) MR 1535289
[KW] V. Klee & S. Wagon, Old and new unsolved problems in plane geometry and number theory, Math. Assoc. America, 1991 MR 1133201 | Zbl 0784.51002
[KMS] S. Kerckhoff, H. Masur & J. Smillie, “Ergodicity of billiard flows and quadratic differentials”, Ann. Math. 124 (1986), p. 293-311 MR 855297 | Zbl 0637.58010
[Ko] M. Kontsevich, Lyapunov exponents and Hodge theory, Adv. Ser. Math. Phys., World Sci. Publishing, 1997, p. 318-332 Zbl 1058.37508
[KZ] M. Kontsevich & A. Zorich, “Connected components of the moduli spaces of Abelian differentials with prescribed singularities”, Invent. Math. 153 (2003), p. 631-678 MR 2000471 | Zbl 02001031
[MT] H. Masur & S. Tabachnikov, Rational billiards and flat structures, Handbook on dynamical systems, North-Holland, 2002, p. 1015-1089 Zbl 1057.37034
[Mc] C. McMullen, “Billiards and Teichmüller curves on Hilbert modular surfaces”, J. Amer. Math. Soc. 16 (2003), p. 857-885 MR 1992827 | Zbl 1030.32012
[Mo] T. Monteil, “A counter-example to the theorem of Hiemer and Snurnikov”, J. Statist. Phys. 114 (2004), p. 1619-1623 MR 2039490 | Zbl 1058.37040
[Mo2] T. Monteil, “Finite blocking property versus pure periodicity”, Preprint
arXiv
[ST] J. Schmeling & S. Troubetzkoy, “Inhomogeneous Diophantine Approximation and Angular Recurrence for Polygonal Billiards”, Mat. Sb. 194 (2003), p. 129-144 MR 1992153 | Zbl 1043.37028
[To] G. Tokarsky, “Polygonal rooms not illuminable from every point”, Amer. Math. Monthly 102 (1995), p. 867-879 MR 1366048 | Zbl 0849.52007
[Ve] W. Veech, “Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards”, Invent. Math. 97 (1989), p. 553-583
Article | MR 1005006 | Zbl 0676.32006
[Vo] Y. Vorobets, “Planar structures and billiards in rational polygons: the Veech alternative”, Russian Math. Surveys 51 (1996), p. 779-817 MR 1436653 | Zbl 0897.58029
[ZK] A. Zemljakov & A. Katok, “Topological transitivity of billiards in polygons”, Mat. Zametki 18 (1975), p. 291-300 MR 399423 | Zbl 0323.58012
[Zo] A. Zorich, “Private communication”, , 2003
Thierry Monteil
On the finite blocking property
(Sur la propriété de blocage fini)
Annales de l'institut Fourier, 55 no. 4 (2005), p. 1195-1217
Article: sur abonnement | Analyses MR 2157167 | Zbl 1076.37029
Class. Math.: 37E35, 37D50, 37D40, 37A10, 5199, 30F30
Mots clés: propriété de blocage, billards polygonaux, polygones réguliers, surfaces de translation, surfaces de Veech, revêtement ramifié du tore, illumination, différentielles quadratiques
Résumé - Abstract
Bibliographie
[Bou] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6, Masson, 1981 MR 647314 | Zbl 0483.22001
[Cal] K. Calta, “Veech surfaces and complete periodicity in genus $2$”, J. Amer. Math. Soc., to appear
arXiv | MR 2083470 | Zbl 02106878
[Car] D. Cartwright, A brief introduction to buildings, Contemp. Math., Amer. Math. Soc., 1997, p. 45-77 Zbl 0943.51011
[EO] 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), p. 59-103 MR 1839286 | Zbl 1019.32014
[Fo] D. Fomin, “”, Zadaqi Leningradskih Matematitcheskih Olimpiad, Leningrad, 1990
[HS] P. Hiemer & V. Snurnikov, “Polygonal billiards with small obstacles”, J. Statist. Phys. 90 (1998), p. 453-466 MR 1611100 | Zbl 0995.37019
[Kl] V. Klee, “Is every polygonal region illuminable from some point?”, Amer. Math. Monthly 76 (1969) MR 1535289
[KW] V. Klee & S. Wagon, Old and new unsolved problems in plane geometry and number theory, Math. Assoc. America, 1991 MR 1133201 | Zbl 0784.51002
[KMS] S. Kerckhoff, H. Masur & J. Smillie, “Ergodicity of billiard flows and quadratic differentials”, Ann. Math. 124 (1986), p. 293-311 MR 855297 | Zbl 0637.58010
[Ko] M. Kontsevich, Lyapunov exponents and Hodge theory, Adv. Ser. Math. Phys., World Sci. Publishing, 1997, p. 318-332 Zbl 1058.37508
[KZ] M. Kontsevich & A. Zorich, “Connected components of the moduli spaces of Abelian differentials with prescribed singularities”, Invent. Math. 153 (2003), p. 631-678 MR 2000471 | Zbl 02001031
[MT] H. Masur & S. Tabachnikov, Rational billiards and flat structures, Handbook on dynamical systems, North-Holland, 2002, p. 1015-1089 Zbl 1057.37034
[Mc] C. McMullen, “Billiards and Teichmüller curves on Hilbert modular surfaces”, J. Amer. Math. Soc. 16 (2003), p. 857-885 MR 1992827 | Zbl 1030.32012
[Mo] T. Monteil, “A counter-example to the theorem of Hiemer and Snurnikov”, J. Statist. Phys. 114 (2004), p. 1619-1623 MR 2039490 | Zbl 1058.37040
[Mo2] T. Monteil, “Finite blocking property versus pure periodicity”, Preprint
arXiv
[ST] J. Schmeling & S. Troubetzkoy, “Inhomogeneous Diophantine Approximation and Angular Recurrence for Polygonal Billiards”, Mat. Sb. 194 (2003), p. 129-144 MR 1992153 | Zbl 1043.37028
[To] G. Tokarsky, “Polygonal rooms not illuminable from every point”, Amer. Math. Monthly 102 (1995), p. 867-879 MR 1366048 | Zbl 0849.52007
[Ve] W. Veech, “Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards”, Invent. Math. 97 (1989), p. 553-583
Article | MR 1005006 | Zbl 0676.32006
[Vo] Y. Vorobets, “Planar structures and billiards in rational polygons: the Veech alternative”, Russian Math. Surveys 51 (1996), p. 779-817 MR 1436653 | Zbl 0897.58029
[ZK] A. Zemljakov & A. Katok, “Topological transitivity of billiards in polygons”, Mat. Zametki 18 (1975), p. 291-300 MR 399423 | Zbl 0323.58012
[Zo] A. Zorich, “Private communication”, , 2003
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310