Invariant vector fields and groupoids

TL;DR

This paper introduces a new perspective on linearizing invariant vector fields near relative equilibria by using isomorphisms between such fields, connecting stack theory with dynamical systems.

Contribution

It proposes a novel approach based on isomorphisms of invariant vector fields to better understand the transition from tubular neighborhoods to slices in group actions.

Findings

Provides a new framework for linearization near relative equilibria.

Connects stack theory with the analysis of invariant vector fields.

Enhances understanding of the structure of invariant vector fields in symmetry settings.

Abstract

We use the notion of isomorphism between two invariant vector fields to shed new light on the issue of linearization of an invariant vector field near a relative equilibrium. We argue that the notion is useful in understanding the passage from the space of invariant vector fields in a tube around a group orbit to the space invariant vector fields on a slice to the orbit. The notion comes from Hepworth's study of vector fields on stacks.