Twisted Translation Flows and Effective Weak Mixing

TL;DR

This paper introduces a twisted cohomology cocycle over the Teichmüller flow, establishes a spectral gap in its Lyapunov spectrum, and derives estimates on spectral measures and weak mixing speeds for translation flows.

Contribution

It develops a new twisted cohomology framework and proves spectral gap results, providing quantitative bounds on mixing and ergodic averages for translation flows.

Findings

Spectral gap established for the Lyapunov spectrum of the cocycle.

Hölder estimates on spectral measures derived.

Bounds on weak mixing speeds and ergodic deviations obtained.

Abstract

We introduce a twisted cohomology cocycle over the Teichmueller flow and prove a "spectral gap" for its Lyapunov spectrum with respect to the Masur-Veech measures. We then derive Hoelder 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.