Symmetry Groups of Attractors

For maps equivariant under the action of a nite group ? on R n , the possible symmetries of xed points are known and correspond to the isotropy subgroups. This paper investigates the possible symmetries of arbitrary, possibly chaotic, attractors and nds that the necessary conditions of Melbourne, Dellnitz and Golubitsky 15] are also suucient, at least for continuous maps. The result shows that the reeection hyperplanes are important in determining those groups which are admissible; more precisely a subgroup of ? is admissible as the symmetry group of an attractor if there exists a with = cyclic such that xes a connected component of the complement of the set of reeection hyperplanes of reeections in ? but not in. For nite reeection groups this condition on reduces to the condition that is an isotropy subgroup. Our results are illustrated for nite subgroups of O(3).