H\"older regularity for the spectrum of translation flows

TL;DR

This paper proves H"older regularity for spectral measures of translation flows on flat surfaces of any genus, extending previous results and combining symbolic and geometric methods.

Contribution

It establishes H"older regularity for spectral measures of translation flows across all genera by integrating Forni's ideas with symbolic approaches.

Findings

H"older regularity proven for spectral measures in all genera

Extension of previous genus-specific results to arbitrary genus

Combines symbolic and geometric techniques for spectral analysis

Abstract

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 H\"older property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained H\"older estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove H\"older regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.