Poincaré–Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields

Abstract

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré–Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P 1 ( C ) . We then show how to associate to a homogeneous vector field Q in C n a rank 1 singular holomorphic foliation F of P n − 1 ( C ) and a (partial) meromorphic connection ∇ o along F so that integral curves of Q are described by the geodesic flow of ∇ o along the leaves of F , which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of Q, and of the behavior of the geodesic flow in a neighborhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighborhood of the origin for a substantial class of holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in C 2 .