Genericity in equivariant dynamical systems and equivariant Fuller index theory

TL;DR

This paper introduces a new concept of equivariant non-degeneracy for flows on manifolds with Lie group actions, proves its genericity, and develops an equivariant Fuller index to detect periodic orbits distinguished by isotropy.

Contribution

It defines equivariant non-degeneracy, proves its genericity, and constructs an equivariant Fuller index for analyzing periodic orbits in symmetric dynamical systems.

Findings

Equivariant non-degeneracy is a generic property for flows.

An equivariant Fuller index is constructed to detect group orbit periodic points.

The index distinguishes orbits based on their isotropy groups.

Abstract

We define a notion of equivariant non-degeneracy of $G$-maps to introduce the class of equivariantly non-degenerate flows on smooth compact manifolds with compact Lie group action. We prove genericity of this class and use this result to construct an equivariant version of the Fuller index, which detects group orbits of periodic orbits of the flow, distinguished by their isotropy.