On the Unique Ergodicity of Quadratic Differentials and the Orientation Double Cover

TL;DR

This paper constructs an example of a measured foliation with unique ergodicity on a surface, yet its orientation double cover's flow is only minimal, and provides a geometric criterion for unique ergodicity of quadratic differentials.

Contribution

It introduces a novel example of ergodic behavior and generalizes existing criteria for unique ergodicity in quadratic differentials.

Findings

Constructed a uniquely ergodic foliation with non-uniquely ergodic double cover

Proved a geometric criterion for unique ergodicity of quadratic differentials

Generalized Trevi o's and Masur's results on ergodicity

Abstract

We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second theorem generalizes a result of Trevi\~no for the horizontal flow on a translation surface, as well as Masur's criterion for unique ergodicity of the horizontal foliation.