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Annales scientifiques de l'ENS - Parutions - série 4, 47 (2014)

Parutions < série 4, 47

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 47, fascicule 5 (2014)

Alexander Bufetov, Giovanni Forni
Limit Theorems for Horocycle Flows
Annales scientifiques de l'ENS 47, fascicule 5 (2014), 851-903

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Résumé :
Théorèmes limites pour les flots horocycliques
Le but de cet article est de démontrer des théorèmes limites pour les flots horocycliques sur les surfaces compactes à courbure négative constante. Notre principal outil est une famille particulière de mesures additives, höldériennes, invariantes par le flot horocyclique sur les arcs rectifiables. Une formule asymptotique pour les intégrales ergodiques des flots horocycliques est ensuite établie en termes des mesures additives, et les théorèmes limites en découlent. Les constructions et les résultats ici développés ressemblent à ceux de [15], [16], [5] et [6] qui traitent des flots de translation sur les surfaces plates. Les preuves sont basées sur des méthodes de la théorie des répresentations développées en [12] pour l'étude de l'équation cohomologique et du comportement asymptotique des moyennes ergodiques des flots horocycliques.

Mots-clefs : Flots horocycliques, moyennes ergodiques, distributions limites.

Abstract:
The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely additive Hölder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula. The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on the representation theory methods developed in [12] for the classification of invariant distributions, the solution of the cohomological equation and the asymptotics of ergodic averages of horocycle flows.

Keywords: Horocycle flows, ergodic averages, limit distributions.

Class. math. : 37A17, 60F99


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

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