Density of mild mixing property for vertical flows of Abelian differentials

TL;DR

This paper proves that for genus g ≥ 2, the set of Abelian differentials with mildly mixing vertical flows is dense in all moduli space strata, using conditions on special flows over irrational rotations.

Contribution

It establishes the density of Abelian differentials with mildly mixing vertical flows in all strata of the moduli space for genus g ≥ 2, introducing new criteria for mild mixing.

Findings

Density of such differentials in all strata for g ≥ 2

A new sufficient condition for mild mixing in special flows

Application of these conditions to Abelian differentials

Abstract

We prove that if $g\geq 2$ then the set of all Abelian differentials $(M,\omega)$ for which the vertical flow is mildly mixing is dense in every stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient condition for special flows over irrational rotations and under piecewise constant roof functions to be mildly mixing.