Symmetry Groups of Attractors

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

Peter Ashwin and Ian Melbourne


For maps equivariant under the action of a finite group $\Gamma$ on Rn, 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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