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

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, )–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.

illumination, translation surfaces, billiards, everything
Mathematical Subject Classification 2010
Primary: 37E35
Secondary: 53A99
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
Samuel Lelièvre
Laboratoire de mathématique d’Orsay
Université Paris-Sud 11
Bâtiment 425, campus Orsay-vallée
91405 Orsay Cedex
Thierry Monteil
Laboratoire d’Informatique de Paris Nord
Université Paris 13
F-93430 Villetaneuse
Barak Weiss
School of Mathematical Sciences
Tel Aviv University
Tel Aviv 69978