The reduction of equivariant dynamics to the orbit space for compact group actions
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.
scientific article; zbMATH DE number 1747775