Hölder regularity for the spectrum of translation flows

Bufetov, Alexander I.; Solomyak, Boris

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

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

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. 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 $\ge 2$.

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

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