Geometry & Topology Volume 20, issue 3 (2016) (original) (raw)

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Volume 29, 1 issue Volume 28, 9 issues Volume 28 Issue 9, 3973–4381 Issue 8, 3511–3972 Issue 7, 3001–3510 Issue 6, 2483–2999 Issue 5, 1995–2482 Issue 4, 1501–1993 Issue 3, 1005–1499 Issue 2, 497–1003 Issue 1, 1–496 Volume 27, 9 issues Volume 27 Issue 9, 3387–3831 Issue 8, 2937–3385 Issue 7, 2497–2936 Issue 6, 2049–2496 Issue 5, 1657–2048 Issue 4, 1273–1655 Issue 3, 823–1272 Issue 2, 417–821 Issue 1, 1–415 Volume 26, 8 issues Volume 26 Issue 8, 3307–3833 Issue 7, 2855–3306 Issue 6, 2405–2853 Issue 5, 1907–2404 Issue 4, 1435–1905 Issue 3, 937–1434 Issue 2, 477–936 Issue 1, 1–476 Volume 25, 7 issues Volume 25 Issue 7, 3257–3753 Issue 6, 2713–3256 Issue 5, 2167–2711 Issue 4, 1631–2166 Issue 3, 1087–1630 Issue 2, 547–1085 Issue 1, 1–546 Volume 24, 7 issues Volume 24 Issue 7, 3219–3748 Issue 6, 2675–3218 Issue 5, 2149–2674 Issue 4, 1615–2148 Issue 3, 1075–1614 Issue 2, 533–1073 Issue 1, 1–532 Volume 23, 7 issues Volume 23 Issue 7, 3233–3749 Issue 6, 2701–3231 Issue 5, 2165–2700 Issue 4, 1621–2164 Issue 3, 1085–1619 Issue 2, 541–1084 Issue 1, 1–540 Volume 22, 7 issues Volume 22 Issue 7, 3761–4380 Issue 6, 3145–3760 Issue 5, 2511–3144 Issue 4, 1893–2510 Issue 3, 1267–1891 Issue 2, 645–1266 Issue 1, 1–644 Volume 21, 6 issues Volume 21 Issue 6, 3191–3810 Issue 5, 2557–3190 Issue 4, 1931–2555 Issue 3, 1285–1930 Issue 2, 647–1283 Issue 1, 1–645 Volume 20, 6 issues Volume 20 Issue 6, 3057–3673 Issue 5, 2439–3056 Issue 4, 1807–2438 Issue 3, 1257–1806 Issue 2, 629–1255 Issue 1, 1–627 Volume 19, 6 issues Volume 19 Issue 6, 3031–3656 Issue 5, 2407–3030 Issue 4, 1777–2406 Issue 3, 1155–1775 Issue 2, 525–1154 Issue 1, 1–523 Volume 18, 5 issues Volume 18 Issue 5, 2487–3110 Issue 4, 1865–2486 Issue 3, 1245–1863 Issue 2, 617–1244 Issue 1, 1–616 Volume 17, 5 issues Volume 17 Issue 5, 2513–3134 Issue 4, 1877–2512 Issue 3, 1253–1876 Issue 2, 621–1252 Issue 1, 1–620 Volume 16, 4 issues Volume 16 Issue 4, 1881–2516 Issue 3, 1247–1880 Issue 2, 625–1246 Issue 1, 1–624 Volume 15, 4 issues Volume 15 Issue 4, 1843–2457 Issue 3, 1225–1842 Issue 2, 609–1224 Issue 1, 1–607 Volume 14, 5 issues Volume 14 Issue 5, 2497–3000 Issue 4, 1871–2496 Issue 3, 1243–1870 Issue 2, 627–1242 Issue 1, 1–626 Volume 13, 5 issues Volume 13 Issue 5, 2427–3054 Issue 4, 1835–2425 Issue 3, 1229–1833 Issue 2, 623–1227 Issue 1, 1–621 Volume 12, 5 issues Volume 12 Issue 5, 2517–2855 Issue 4, 1883–2515 Issue 3, 1265–1882 Issue 2, 639–1263 Issue 1, 1–637 Volume 11, 4 issues Volume 11 Issue 4, 1855–2440 Issue 3, 1255–1854 Issue 2, 643–1254 Issue 1, 1–642 Volume 10, 4 issues Volume 10 Issue 4, 1855–2504 Issue 3, 1239–1853 Issue 2, 619–1238 Issue 1, 1–617 Volume 9, 4 issues Volume 9 Issue 4, 1775–2415 Issue 3, 1187–1774 Issue 2, 571–1185 Issue 1, 1–569 Volume 8, 3 issues Volume 8 Issue 3, 1013–1499 Issue 2, 511–1012 Issue 1, 1–509 Volume 7, 2 issues Volume 7 Issue 2, 569–1073 Issue 1, 1–568 Volume 6, 2 issues Volume 6 Issue 2, 495–990 Issue 1, 1–494 Volume 5, 2 issues Volume 5 Issue 2, 441–945 Issue 1, 1–440 Volume 4, 1 issue Volume 3, 1 issue Volume 2, 1 issue Volume 1, 1 issue

Abstract
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(2, ℝ)–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.

Abstract

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(2, ℝ)–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.

Keywords

illumination, translation surfaces, billiards, everything

Mathematical Subject Classification 2010

Primary: 37E35

Secondary: 53A99

Publication

Received: 4 February 2015

Revised: 2 June 2015

Accepted: 16 July 2015

Published: 4 July 2016

Proposed: Jean-Pierre Otal

Seconded: Ian Agol, Bruce Kleiner

Authors

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