Topological conjugacy of singularities of vector fields

Abstract

ROUGHLY speaking, a smooth vector field is finitely determined for topological conjugacy at a singularity if its conjugacy class contains some finite Taylor approximation. Finite determinacy is one criterion for the existence of conjugacies between smooth planar vector fields. This criterion can fail at a singularity only if the vector field is either P-flat or not sufficiently smooth. In both situations we construct specific examples of singularities. not rotation points, that are topologically equivalent but not conjugate. In addition, we determine a useful necessary and sufficient condition for the conjugacy of a large class of singularities. This allows for a simple characterization of the resulting conjugacy classes.