Geodesic
Right-angled billiards and volumes of moduli spaces of quadratic differentials on ℂP1
[Billards à angles droits et volumes des espaces de modules des différentielles quadratiques sur ℂP1 ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 6, pp. 1311-1386.

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We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on ℂP1 and counting functions of the number of (bands of) simple closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago.

Applying ergodic techniques to the Teichmüller geodesic flow we obtain quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards.

Nous utilisons le lien entre les volumes des strates de différentielles méromorphes quadratiques avec des pôles simples sur ℂP1 et les fonctions de comptage du nombre de (cylindres de) géodésiques fermées simples pour la métrique plate associée afin de démontrer une formule très explicite pour le volume des strates, conjecturée par M. Kontsevich il y a une décennie.

En appliquant des techniques ergodiques au flot géodésique de Teichmüller nous obtenons une asymptotique quadratique pour le nombre de (bandes de) trajectoires fermées et le nombre de diagonales généralisées dans presque tout billard à angles « droits ».

Publié le :
DOI : 10.24033/asens.2310
Classification : 32G15, 37E35, 37C27.
Keywords: Billiards, meromorphic differentials, moduli spaces, Siegel-Veech constants.
Mots-clés : Billards, différentielles méromorphes, espaces de modules, constantes de Siegel-Veech. 
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     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1311--1386},
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Athreya, Jayadev S.; Eskin, Alex; Zorich, Anton. Right-angled billiards  and volumes of moduli spaces  of quadratic differentials on ${\mathbb {C}}\!\operatorname{P}^1$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 6, pp. 1311-1386. doi : 10.24033/asens.2310. http://geodesic.mathdoc.fr/articles/10.24033/asens.2310/

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