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