Unique ergodicity of branched covers of translation surfaces

Abstract

Let $X$ be a finite-area translation surface whose vertical flow is uniquely ergodic. Given a slit joining two nonsingular points of $X$, one can form a branched cyclic cover by gluing $\mathrm{N}$ copies of $X$ crosswise along the slit. We study when the vertical flow on the resulting cover is uniquely ergodic. We first prove a geometric criterion for unique ergodicity of the branched cover. We show that if, for a sequence of times along the Teichmüller geodesic, one endpoint of the slit is contained in embedded Euclidean disks of uniformly positive radius that avoid the other endpoint, then the branched cover is uniquely ergodic. The proof uses the special symmetry of the cover together with an analysis of forward and backward generic points for the vertical flow. We then show that this criterion applies for Lebesgue-almost every choice of slit endpoint under a natural geometric hypothesis on the Teichmüller orbit of $X$, namely a uniform lower bound for the embedded radius along a subsequence. Finally, we give sufficient conditions for such a lower bound in terms of the cylinder geometry of $g_tX$, introducing the notion of pipe cylinders and proving that embedded disks of definite size must exist. As a consequence, for the class of uniquely ergodic translation surfaces, almost every slit produces a uniquely ergodic branched $\mathrm{N}$-cover.