#### Volume 20, issue 3 (2016)

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Everything is illuminated

### Samuel Lelièvre, Thierry Monteil and Barak Weiss

Geometry & Topology 20 (2016) 1737–1762
##### 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 $SL\left(2,ℝ\right)$–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.

##### Keywords
illumination, translation surfaces, billiards, everything
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
 Samuel Lelièvre 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