##
Symmetry Groups of Attractors

* Arch. Rat. Mech. Anal. * **126** (1994) 59-78

** Peter Ashwin and Ian Melbourne **

** Abstract **

For maps equivariant under the action of a finite
group $\Gamma$ on **R**^{n}, the possible symmetries of fixed points are known
and correspond to the isotropy subgroups. This paper investigates the
possible symmetries of arbitrary, possibly chaotic, attractors and
finds that the necessary conditions of
Melbourne, Dellnitz and
Golubitsky are also sufficient, at least for continuous
maps.

The result shows that the reflection hyperplanes are important in
determining those groups which are admissible;
more precisely a subgroup $\Sigma$ of $\Gamma$ is admissible as the
symmetry group of an attractor if there exists a $\Delta$ with
$\Sigma/\Delta$ cyclic such that $\Delta$ fixes a connected component
of the complement of the set of reflection hyperplanes of reflections
in $\Gamma$ but not in $\Delta$. For finite reflection groups this
condition on $\Delta$ reduces to the condition that $\Delta$ is an
isotropy subgroup. Our results are illustrated for finite subgroups of
**O**(3).

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