Global analysis of meromorphic vector fields in the plane.

摘要

This work examines the behavior of meromorphic vector fields k(z) on the plane. The set of trajectories of k(z) whose maximal interval of definition is not all of R is called the incomplete set, and the analysis of k( z) then falls into two parts: the behavior of the incomplete set, and the behavior of the complement of the incomplete set. It is shown that zeros, poles and alpha and o limit points of k( z) all lie within the closure of the incomplete set. In a component of the complement of the closure of the incomplete set, all trajectories are either periodic with the same period (with the possible exception of one center lying within the component), or all trajectories are one to one trajectories.