Poincaré-Bendixson theorems for meromorphic connections and 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 $\omega$-limit sets of geodesics for a meromorphic connection on $¶^1(\C)$. We then show how to associate to a homogeneous vector field $Q$ in ${\mathbb{C}}^n$ a rank 1 singular holomorphic foliation $\mathcal{F}$ of $¶^{n-1}(\C)$ and a (partial) meromorphic connection ${\nabla }^o$ along $\ca F$ so that integral curves of $Q$ are described by the geodesic flow of ${\nabla }^o$ 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 $\C^2$.

We generalized some of the main ideas to homogeneous vector fields in ${\mathbb{C}}^n$ for any n, rewriting the introduction and adding a new section

References in corpus (1)

• Discrete holomorphic local dynamical systems

Cited by in corpus (4)

• Formal Poincare'-Dulac renormalization for holomorphic germs
• Flat structure of meromorphic connections on Riemann surfaces
• Domain of attraction for maps tangent to the identity in ${\mathbb{C}}^2$ with characteristic direction of higher degree
• A Poincaré-Bendixson theorem for meromorphic connections on Riemann surfaces