The reduction of equivariant dynamics to the orbit space for compact group actions
type: Item
instance of: scholarly article
title: The reduction of equivariant dynamics to the orbit space for compact group actions
zbMATH Open document ID: 0998.37012
DOI: 10.1023/A:1013970014204
author: Pascal Chossat
published in: Acta Applicandae Mathematicae
publication date: 2002-05-30T00:00:00Z
review text: The aim of this expository paper is to describe the methods that have been developed to describe the dynamics of smooth vector fields \(X\) defined on \({\mathbb R}^n\) that are equivariant under the action of a compact subgroup \( G \subset O(n) \) (the group of real orthogonal \(n \times n \) matrices), i.e. \( X(g x) = g X(x) \) for all \(x \in {\mathbb R}^n \) and \( g \in G \). Such symmetry induces degeneracies which makes the study of the dynamics more difficult. An obvious idea to remove such degeneracies would be to project the dynamics onto the orbit space, i.e. the quotient space obtained by identifying points in phase space which lie in the same group orbit. However, in general the orbit-space is not a manifold and it is not clear how to compute and analyze the projected dynamics on it. In this paper a different general method to achieve the same task is described and illustrated on some classical examples. This review paper is clear and well-written.
reviewed by: Massimiliano Berti
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37-02
zbMATH DE Number: 1747775
zbMATH Keywords: equivariant dynamics
zbMATH Keywords: group action
zbMATH Keywords: symmetry
zbMATH Keywords: orbit space
zbMATH Keywords: expository paper
MaRDI profile type: Publication
full work available at URL: a:1013970014204
OpenAlex ID: W195098268
description: scientific article; zbMATH DE number 1747775