A topological bound on the Cantor-Bendixson rank of meromorphic differentials

TL;DR

This paper establishes a genus-dependent upper bound on the Cantor-Bendixson rank of saddle connection directions in meromorphic differentials, revealing complexity distinctions from holomorphic cases.

Contribution

It introduces a sharp topological bound on the Cantor-Bendixson rank for meromorphic differentials, based on a novel geometric lemma involving nested invariant subsurfaces.

Findings

Bound depends only on the genus of the surface

Saddle connection directions are less dense in meromorphic cases

New geometric lemma relating genus of nested subsurfaces

Abstract

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three nested invariant subsurfaces the genus of the third one is always bigger than the genus of the first one.