A note on symmetries of invariant sets with compact group actions

Rok, strany: 1994, 93 - 104

We investigate the symmetries of the asymptotic dynamics of a map equivariant under a compact Lie group $Γ$. Let $Γ⁰$ denote the connected component of the identity in $Γ$ and let $ω_f (x₀)$ denote the $ω$-limit set of the point $x₀$ under the map $f$. Assume that $ω_f(x₀)$ contains a point of trivial isotropy and is not a relative periodic orbit (these are mild assumptions on the dynamics). Melbourne [I. Melbourne: Generalizations of a result on symmetry groups of attractors, in: Pattern Formation: Symmetry, Methods and Applications (J. Chadam & W. F. Langford, eds.), Fields Institute Communications, AMS, Providence (to appear)] shows that under these assumptions and when $Γ⁰$ is abelian, then generically (in the $C^∞$ topology) the symmetry group of $ω_f (x₀)$ contains $Γ⁰$. We show under the same assumptions on the dynamics but without the assumption that $Γ⁰$ is abelian that it is possible to construct a family of perturbations such that for a residual subset of perturbations (in the $C⁰$ topology) the resulting $ω$-limit point set of $x₀$ has at least $Γ⁰$ symmetry. Our argument does not extend directly to the $C¹$ topology.

ISO 690:

Field, M., Golubitsky, M., Matthew, N. 1994. A note on symmetries of invariant sets with compact group actions. In Tatra Mountains Mathematical Publications, vol. 4, no.1, pp. 93-104. 1210-3195.

APA:

Field, M., Golubitsky, M., Matthew, N. (1994). A note on symmetries of invariant sets with compact group actions. Tatra Mountains Mathematical Publications, 4(1), 93-104. 1210-3195.