# zbMATH — the first resource for mathematics

A note on symmetries of invariant sets with compact group actions. (English) Zbl 0813.58008
Let $$\Gamma \subset O(n)$$ be a (nonfinite) compact Lie group acting on $$\mathbb{R}^ n$$ with the identity component $$\Gamma^ 0$$. For a subset $$X \subset \mathbb{R}^ n$$ the symmetry group $$\Sigma(X)$$ of $$X$$ is defined by $$\Sigma(X) = \{\sigma \in \Gamma \mid \sigma(X) = X\}$$. Let $$f : \mathbb{R}^ n \to \mathbb{R}^ n$$ be a continuous $$\Gamma$$-equivariant mapping. Given $$x_ 0 \in \mathbb{R}^ n$$ we denote by $$\omega_ f(x_ 0)$$ the $$\omega$$- limit set for $$f$$ with initial point $$x_ 0$$. The authors show that under appropriate conditions on $$f$$ and $$x_ 0$$, it is generically true that $$\Sigma(\omega_ g(x)) \supset \Gamma^ 0$$ for perturbations $$g$$ (in the $$C^ 0$$ topology) of $$f$$.
##### MSC:
 58D19 Group actions and symmetry properties 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
##### Keywords:
symmetry group; $$\omega$$-limit sets; perturbations
Full Text: