Symmetries of center singularities of plane vector fields

TL;DR

This paper investigates the symmetry groups of plane vector fields with a center singularity, establishing conditions for the smooth extension of the period function and describing the homotopy type of the symmetry group.

Contribution

It characterizes when the period function extends smoothly at the center and describes the homotopy type of the orbit-preserving diffeomorphism group.

Findings

Period function extends smoothly iff the 1-jet at the origin is non-degenerate.

The symmetry group is homotopy equivalent to a circle under certain conditions.

Provides criteria linking local linearization to global symmetry properties.

Abstract

Let D be a closed unit 2-disk on the plane centered at the origin 0, and F be a smooth vector field on D such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a ``center'' singularity. Let p:D\0-->(0,+\infty) be the function associating to each $z\not=O$ its period with respect to F. This function can be discontinuous at O. Let Diff(F) be the group of all diffeomorphisms of D which preserve orientation and orbits of F. We prove that p smoothly extends to all of D if and only if the 1-jet of F at the origin is a non-degenerate linear map, and that in this case Diff(F) is homotopy equivalent to the circle.