Counting geodesics on prime-order $k$-differentials

Abstract

We determine weak asymptotics of counting functions on generic surfaces in a component of a stratum of $k$-differentials when $k$ is prime and genus is greater than $2$. In order to do so, we classify the $GL^+(2,\mathbb{R})$-orbit closure of holonomy covers of components and apply Eskin-Mirzakhani-Mohammadi generalized to translation surfaces. We show that the $GL^+(2,\mathbb{R})$-orbit closure of these holonomy covers is generically a component of a stratum of translation surfaces or a hyperelliptic locus therein.