Dynamics of Fuchsian meromorphic connections with real periods

TL;DR

This paper investigates the behavior of geodesics in Fuchsian meromorphic connections with real periods, providing a detailed classification of their limit sets using flat metric analysis and generalizing classical Teichmüller results.

Contribution

It offers a new characterization of geodesic dynamics in Fuchsian meromorphic connections with real periods, extending classical Teichmüller theory to this setting.

Findings

Characterization of possible $oldsymbol{ m ext{ extomega}}$-limit sets for simple geodesics.

Explicit description of geodesics near Fuchsian poles with large real residues.

Generalization of the Teichmüller lemma for quadratic differentials to meromorphic connections.

Abstract

In this paper, we study the dynamics of geodesics of Fuchsian meromorphic connections with real periods, giving a precise characterization of the possible $\omega$-limit sets of simple geodesics in this case. The main tools are the study of the singular flat metric associated to the meromorphic connection, an explicit description of the geodesics nearby a Fuchsian pole with real residue larger than $-1$ and a far-reaching generalization to our case of the classical Teichm\"uller lemma for quadratic differentials.