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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 \(f\).
58D19 Group actions and symmetry properties
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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