SYMMETRY GROUPS OF ATTRACTORS

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 & GOLUBITSKY [15] are sufficient, at least for continuous maps. This 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).