Hölder regularity for the spectrum of translation flows

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