2009/03/20 by Abate, Marco, Tovena, Francesca
#32H50 (Primary) #32S65 #34M35 (secondary) #37F75 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.0903.3485
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 ¶1(\C). We then show how to associate to a homogeneous vector field Q in \Bbb Cn a rank 1 singular holomorphic foliation \cal F of ¶n-1(\C) and a (partial) meromorphic connection ∇o along \ca F so that integral curves of Q are described by the geodesic flow of ∇o along the leaves of \ca 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 neighbourhood 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 neighbourhood of the origin for a substantial class of 2-dimensional 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 \C2.