SYMMETRICAL OMEGA-LIMIT SETS FOR SMOOTH GAMMA-EQUIVARIANT DYNAMICAL-SYSTEMS WITH GAMMA(0) ABELIAN

The symmetry groups of attractors for smooth equivariant dynamical sys tems have been classified when the underlying group of symmetries Gamm a is finite. The problems that arise when Gamma is compact but infinit e are of a completely different nature. We investigate: the case when the connected component of the identity Gamma(0) is Abelian and show t hat 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 topolog ical and probabilistic sense (genericity and prevalence).