##
Symmetric $\omega$-limit sets for smooth $\Gamma$-equivariant
dynamical systems
with $\Gamma^{0}$ 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 $\Gamma^{0}$ 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 $\Gamma^{0}$. Here, typicality is interpreted in both a
topological and probabilistic sense (genericity and prevalence).

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