Twisted translation flows and effective weak mixing

Abstract

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a "spectral gap" for its Lyapunov spectrum with respect to the Masur-Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.

Spectral cocycle for substitution tilings

“…The tools of spectral cocycle (without calling it such) provided a framework for the proof of almost sure Hölder regularity of translation flows on higher genus flat surfaces, first in [9] for genus 2 and then in [12] for an arbitrary genus greater than 1, including many surfaces of infinite genus and finite area. (The last paper appeared after the preprint of, now published, [16] who used a different technique. )…”