Symmetric ω-limit sets for smooth Γ-equivariant dynamical systems with Γ⁰ 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 Γ is compact but infinite are of a completely different nature. We investigate the case when the connected component of the identity Γ⁰ is abelian and show that under fairly mild assumptions on the dynamics, it is typically the case that the symmetry of an ω-limit set contains the continuous symmetries Γ⁰. Here, typicality is interpreted in both a topological and probabilistic sense (genericity and prevalence).