Counting geodesics on prime-order $k$-differentials

TL;DR

This paper analyzes the asymptotic behavior of counting functions for geodesics on generic surfaces with prime-order $k$-differentials, classifying orbit closures and applying advanced ergodic theory techniques.

Contribution

It provides the first classification of $GL^+(2,R)$-orbit closures for holonomy covers of $k$-differential surfaces when $k$ is prime, extending ergodic theory methods.

Findings

Determined weak asymptotics of counting functions for geodesics.

Classified $GL^+(2,R)$-orbit closures for holonomy covers.

Showed orbit closures are either stratum components or hyperelliptic loci.

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.