M. Field, M. Golubitsky and M. Nicol
A note on symmetries of invariant sets with compact group actions
In: Equadiff 8. Tatra Mountains Math. Publ. 4 , 1994, 93-104.
We investigate the symmetries of the asymptotic dynamics of a map equivariant under a compact Lie group Gamma. Let Gamma^0 denote the connected component of the identity in Gamma and let omega_f(x_0) denote the omega-limit set of the point x_0 under the map f. Assume that omega_f(x_0) contains a point of trivial isotropy and is not a relative periodic orbit (these are mild assumptions on the dynamics). Melbourne [14] shows that under these assumptions and when Gamma^0 is abelian, then generically (in the C^infty topology) the symmetry group of omega_f(x_0) contains Gamma^0. We show under the same assumptions on the dynamics but without the assumption that Gamma^0 is abelian that it is possible to construct a family of perturbations such that for a residual subset of perturbations (in the C^0 topology) the resulting omega-limit point set of x_0 has at least Gamma_0 symmetry. Our argument does not extend directly to the C^1 topology.