Cet article est consacré aux flots de translation génériques correspondant à des différentielles abéliennes sur des surfaces plates de genre arbitraire . Ces flots sont faiblement mélangeants, d’après le théorème d’Avila-Forni. En genre , 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 .
The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus . These flows are weakly mixing by the Avila-Forni theorem. In genus , 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 .
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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
@article{JEP_2021__8__279_0, author = {Bufetov, Alexander I. and Solomyak, Boris}, title = {H\"older regularity for the spectrum of translation flows}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {279--310}, publisher = {Ecole polytechnique}, volume = {8}, year = {2021}, doi = {10.5802/jep.146}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.146/} }
TY - JOUR AU - Bufetov, Alexander I. AU - Solomyak, Boris TI - Hölder regularity for the spectrum of translation flows JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 279 EP - 310 VL - 8 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.146/ DO - 10.5802/jep.146 LA - en ID - JEP_2021__8__279_0 ER -
%0 Journal Article %A Bufetov, Alexander I. %A Solomyak, Boris %T Hölder regularity for the spectrum of translation flows %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 279-310 %V 8 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.146/ %R 10.5802/jep.146 %G en %F JEP_2021__8__279_0
Bufetov, Alexander I.; Solomyak, Boris. 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. http://www.numdam.org/articles/10.5802/jep.146/
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