Hölder regularity for the spectrum of translation flows
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 279-310.

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus g2. 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 2.

Cet article est consacré aux flots de translation génériques correspondant à des différentielles abéliennes sur des surfaces plates de genre arbitraire g2. 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 2.

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

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