Geometry & Topology

Everything is illuminated

Samuel Lelièvre, Thierry Monteil, and Barak Weiss

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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, )–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.

Article information

Source
Geom. Topol., Volume 20, Number 3 (2016), 1737-1762.

Dates
Received: 4 February 2015
Revised: 2 June 2015
Accepted: 16 July 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859001

Digital Object Identifier
doi:10.2140/gt.2016.20.1737

Mathematical Reviews number (MathSciNet)
MR3523067

Zentralblatt MATH identifier
06624257

Subjects
Primary: 37E35: Flows on surfaces
Secondary: 53A99: None of the above, but in this section

Keywords
illumination translation surfaces billiards everything

Citation

Lelièvre, Samuel; Monteil, Thierry; Weiss, Barak. Everything is illuminated. Geom. Topol. 20 (2016), no. 3, 1737--1762. doi:10.2140/gt.2016.20.1737. https://projecteuclid.org/euclid.gt/1510859001


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References

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