Symmetries of degenerate center singularities of plane vector fields

TL;DR

This paper investigates the structure of the group of orbit-preserving diffeomorphisms for plane vector fields with degenerate center singularities, extending previous work on non-degenerate cases.

Contribution

It characterizes the homotopy type of the diffeomorphism group for degenerate center singularities under certain non-degeneracy conditions.

Findings

Describes the homotopy type of Diff(F) in degenerate cases

Classifies path components of Diff(F) with respect to weak topologies

Extends previous results from non-degenerate to degenerate singularities

Abstract

Let $D$ be a closed unit $2$-disk on the plane centered at the origin $O$, and $F$ be a smooth vector field 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 also $\mathrm{Diff}(F)$ be the group of all diffeomorphisms of $D$ which preserve orientation and orbits of $F$. In arXiv:0907.0359 the author described the homotopy type of $\mathrm{Diff}(F)$ under assumption that the $1$-jet of $F$ at $O$ is non-degenerate. In this paper degenerate case is considered. Under additional "non-degeneracy assumptions" on $F$ the path components of $\mathrm{Diff}(F)$ with respect to distinct weak topologies are described.