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Annales scientifiques de l'ENS - Parutions - série 4, 49 (2016)

Parutions < série 4, 49

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 49, fascicule 6 (2016)

Jayadev S. Athreya, Alex Eskin, Anton Zorich
Right-angled billiards and volumes of moduli spaces of quadratic differentials on CP^1
Annales scientifiques de l'ENS 49, fascicule 6 (2016), 1311-1386

Télécharger cet article : Fichier PDF

Résumé :
Billards à angles droits et volumes des espaces de modules des différentielles quadratiques sur CP^1
Nous utilisons le lien entre les volumes des strates de différentielles méromorphes quadratiques avec des pôles simples sur CP^1 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.

Mots-clefs : Billards, différentielles méromorphes, espaces de modules, constantes de Siegel-Veech.

Abstract:
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on CP^1 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.

Keywords: Billiards, meromorphic differentials, moduli spaces, Siegel-Veech constants.

Class. math. : 32G15, 37E35, 37C27.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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