On Limit Sets for Geodesics of Meromorphic Connections

Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behavior of geodesics of such connections has been studied by, e.g., Abate and Bianchi (Math Z 282:247–272, 2016) and Abate and Tovena (J Differ Equ 251(9):2612–2684, 2011) in relation with generalized Poincaré-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviors 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 ω-limit sets.