The Structure of Symmetric Attractors

We consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems. We show, for example, that the symmetry of an attractor cannot, in general, be an arbitrary subgroup of the group of symmetries. In addition, there are group-theoretic restrictions on the symmetry of connected components of a symmetric attractor. The symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time-average of a chaotic dynamical system. Our methods are topological in nature and exploit connectedness properties of the ambient space. In particular, we prove a general lemma about connected components of the complement of preimage sets and how they are permuted by the mapping. These methods do not themselves depend on equivariance. For example, we use them to prove that the presence of periodic points in the dynamics limits the number of ~connected components of an attractor, and, for onedimensional mappings, to prove results on sensitive dependence and the density of periodic points.