The Hoelder Property for the Spectrum of Translation Flows in Genus Two

TL;DR

This paper establishes the Hoelder regularity of spectral measures for generic translation flows on genus two surfaces, providing quantitative spectral estimates using symbolic and Diophantine techniques.

Contribution

It introduces the first quantitative Hoelder estimates for the spectrum of translation flows in genus two, combining symbolic dynamics and Diophantine analysis.

Findings

Spectral measures exhibit Hoelder continuity.

Quantitative bounds on spectral measures are derived.

The methods involve twisted Birkhoff integrals and symbolic dynamics.

Abstract

The paper is devoted to generic translation flows corresponding to Abelian differentials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by the Avila-Forni theorem. Our main result gives first quantitative estimates on their spectrum, establishing the Hoelder property for the spectral measures of Lipschitz functions. The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdos and Kahane.