Hölder regularity for the spectrum of translation flows

Alexander I. Bufetov; Boris Solomyak

doi:10.5802/jep.146

Hölder regularity for the spectrum of translation flows
[Régularité Hölder pour le spectre des flots de translation]

Alexander I. Bufetov ¹ ; Boris Solomyak ²

¹ Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 rue F. Joliot Curie, Marseille, France Steklov Mathematical Institute of RAS, Moscow, Russia Institute for Information Transmission Problems, Moscow, Russia
² Department of Mathematics, Bar-Ilan University Ramat-Gan, Israel

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 279-310.

corrigé par Corrigendum to “Hölder regularity for the spectrum of translation flows”
[Correction à “Régularité Hölder pour le spectre des flots de translation”]

Résumé
Abstract

Cet article est consacré aux flots de translation génériques correspondant à des différentielles abéliennes sur des surfaces plates de genre arbitraire $g \geq 2$ . Ces flots sont faiblement mélangeants, d’après le théorème d’Avila-Forni. En genre $2$ , la propriété de Hölder pour les mesures spectrales de ces flots a été établie dans [12, 14]. Récemment, Forni [18], motivé par [12], a obtenu des estimées Hölder pour les mesures spectrales dans le cas des surfaces de genre arbitraire. Ici, nous combinons l’idée de Forni avec l’approche symbolique de [12] et nous démontrons la régularité Hölder pour les mesures spectrales des flots sur des « compacta » de Markov aléatoires, et en particulier pour des flots de translation pour un genre arbitraire $\geq 2$ .

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g \geq 2$ . These flows are weakly mixing by the Avila-Forni theorem. In genus $2$ , the Hölder property for the spectral measures of these flows was established in [12, 14]. Recently, Forni [18], motivated by [12], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni’s idea with the symbolic approach of [12] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows for an arbitrary genus $\geq 2$ .

Reçu le : 2019-09-12
Accepté le : 2021-01-13
Publié le : 2021-02-05

DOI : 10.5802/jep.146

Classification : 37A30, 37B1, 37E35, 28A78
Keywords: Translation flows, spectral measures, matrix Riesz products, upper Lyapunov exponents, Erdős-Kahane argument, Bratteli-Vershik automorphisms, renormalization cocycle
Mot clés : Flots de translation, mesures spectrales, produits de Riesz matriciels, exposants de Liapounoff supérieurs, l’argument d’Erdős-Kahane, automorphismes de Bratteli-Vershik, cocycle de renormalisation

Affiliations des auteurs :

Alexander I. Bufetov ¹ ; Boris Solomyak ²

¹ Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 rue F. Joliot Curie, Marseille, France Steklov Mathematical Institute of RAS, Moscow, Russia Institute for Information Transmission Problems, Moscow, Russia
² Department of Mathematics, Bar-Ilan University Ramat-Gan, Israel

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Alexander I. Bufetov; Boris Solomyak. Hölder regularity for the spectrum of translation flows. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 279-310. doi : 10.5802/jep.146. https://jep.centre-mersenne.org/articles/10.5802/jep.146/

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