Symmetric $\omega$-limit sets for smooth $\Gamma$-equivariant dynamical systems with $\Gamma⁰$ abelian

Nonlinearity 10 (1997) 1551-1567

Ian Melbourne and Ian Stewart

Abstract

The symmetry groups of attractors for smooth equivariant dynamical systems have been classified when the underlying group of symmetries $\Gamma$ is finite. The problems that arise when $\Gamma$ is compact but infinite are of a completely different nature. We investigate the case when the connected component of the identity $\Gamma⁰$ is abelian and show that under fairly mild assumptions on the dynamics, it is typically the case that the symmetry of an $\omega$-limit set contains the continuous symmetries $\Gamma⁰$. Here, typicality is interpreted in both a topological and probabilistic sense (genericity and prevalence).