Everything is illuminated

Lelièvre, Samuel¹ ; Monteil, Thierry² ; Weiss, Barak³

¹ Laboratoire de mathématique d’Orsay, UMR 8628 CNRS, Université Paris-Sud 11, Bâtiment 425, campus Orsay-vallée, 91405 Orsay Cedex, France
² Laboratoire d’Informatique de Paris Nord, UMR 7030 CNRS, Université Paris 13, F-93430 Villetaneuse, France
³ School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Geometry & topology, Tome 20 (2016) no. 3, pp. 1737-1762.

Résumé

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of the $SL\left(2,ℝ\right)$–action on the moduli space of translation surfaces. We characterize surfaces with the finite blocking property and bounded blocking property, completing work of the second-named author. Concerning the illumination problem, we also extend results of Hubert, Schmoll and Troubetzkoy, removing the hypothesis that the surface in question is a lattice surface, thus settling a conjecture of theirs. Our results crucially rely on the recent breakthrough results of Eskin and Mirzakhani and of Eskin, Mirzakhani and Mohammadi, and on related results of Wright.

Détail

DOI : 10.2140/gt.2016.20.1737

Classification : 37E35, 53A99
Keywords: illumination, translation surfaces, billiards, everything

Bibliographie

[1] A Eskin, M Mirzakhani, Invariant and stationary measures for the SL(2, R) action on moduli space, preprint (2016)

[2] A Eskin, M Mirzakhani, A Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2, R) action on moduli space, Ann. of Math. 182 (2015) 673

[3] P Hubert, M Schmoll, S Troubetzkoy, Modular fibers and illumination problems, Int. Math. Res. Not. 2008 (2008)

[4] S Lelièvre, B Weiss, Translation surfaces with no convex presentation, Geom. Funct. Anal. 25 (2015) 1902

[5] H Masur, S Tabachnikov, Rational billiards and flat structures, from: "Handbook of dynamical systems, Volume 1A" (editors B Hasselblatt, A Katok), North-Holland (2002) 1015

[6] T Monteil, On the finite blocking property, Ann. Inst. Fourier (Grenoble) 55 (2005) 1195

[7] T Monteil, Finite blocking property versus pure periodicity, Ergodic Theory Dynam. Systems 29 (2009) 983

[8] J O’Rourke, O Petrovici, Narrowing light rays with mirrors, from: "Proceedings of the 13th Canadian Conference on Computational Geometry" (2001) 137

[9] J Smillie, B Weiss, Finiteness results for flat surfaces: a survey and problem list, from: "Partially hyperbolic dynamics, laminations, and Teichmüller flow" (editors G Forni, M Lyubich, C Pugh, M Shub), Fields Inst. Commun. 51, Amer. Math. Soc. (2007) 125

[10] J Smillie, B Weiss, Characterizations of lattice surfaces, Invent. Math. 180 (2010) 535

[11] G W Tokarsky, Polygonal rooms not illuminable from every point, Amer. Math. Monthly 102 (1995) 867

[12] W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553

[13] W A Veech, Geometric realizations of hyperelliptic curves, from: "Algorithms, fractals, and dynamics" (editor Y Takahashi), Plenum (1995) 217

[14] Y B Vorobets, Plane structures and billiards in rational polygons : the Veech alternative, Uspekhi Mat. Nauk 51 (1996) 3

[15] A Wright, The field of definition of affine invariant submanifolds of the moduli space of abelian differentials, Geom. Topol. 18 (2014) 1323

[16] A Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015) 413

[17] J C Yoccoz, Interval exchange maps and translation surfaces, from: "Homogeneous flows, moduli spaces and arithmetic" (editors M L Einsiedler, D A Ellwood, A Eskin, D Kleinbock, E Lindenstrauss, G Margulis, S Marmi, J C Yoccoz), Clay Math. Proc. 10, Amer. Math. Soc. (2010) 1

[18] A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry, I" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437