Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner (with an appendix by Emilio Corso)

Ravotti, Davide

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 305-334

We apply a method inspired by Ratner’s work on quantitative mixing for the geodesic flow [29] and developed by Burger [11] to study ergodic integrals for horocycle flows. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni [15], and we show that the coefficients in the asymptotic expansion are Hölder continuous with respect to the base point. Furthermore, we provide short and streamlined proofs of the spatial limit theorems of Bufetov and Forni [10] and, in an appendix by Emilio Corso, of a temporal limit theorem by Dolgopyat and Sarig [12].

Nous appliquons une méthode inspirée du travail de Ratner sur le mélange quantitatif pour le flot géodésique [29] et développée par Burger [11] pour étudier les intégrales ergodiques pour les flots horocycliques. Nous en déduisons un développement asymptotique explicite pour les moyennes horocycliques, retrouvant ainsi un résultat célèbre de Flaminio et Forni [15], et nous montrons que les coefficients dans le développement asymptotique sont Hölder continus par rapport au point de base. En outre, nous fournissons des preuves courtes et simplifiées des théorèmes limites spatiaux de Bufetov et Forni [10] et, dans un appendice d’Emilio Corso, d’un théorème limite temporel de Dolgopyat et Sarig [12].

Classification : 37D40, 37A17, 37A25, 37A50
Keywords: Horocycle flow, ergodic averages, distributional limit theorems
Mots-clés : Flot horocyclique, moyennes ergodiques, théorèmes limites distributionnels

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