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

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.