On limit sets for geodesics of meromorphic connections

TL;DR

This paper explores the asymptotic behavior of geodesics in meromorphic connections on Riemann surfaces, providing new examples with complex limit sets to address gaps in existing theoretical understanding.

Contribution

It introduces explicit examples of geodesics with intricate limit behaviors using branched affine structures from Fuchsian meromorphic connections.

Findings

Existence of geodesics with infinitely many self-intersections

Examples of geodesics with complex omega-limit sets

Filling gaps in the understanding of asymptotic behaviors

Abstract

Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behaviour of geodesics of such connections has been studied by e.g. Abate, Bianchi and Tovena in relation with generalized Poincar\'{e}-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviours of such geodesics really exist. In order to fill the gap, we use the branched affine structure induced by a Fuchsian meromorphic connection to present several examples with geodesics having infinitely many self-intersections and quite peculiar omega-limit sets.