##
The Structure of Symmetric Attractors

* Arch. Rat. Mech. Anal. * **123** (1993) 75-98

** Ian Melbourne, Michael Dellnitz and Martin Golubitsky **

** Abstract **

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.

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 one-dimensional mappings, to prove results on sensitive dependence
and the density of periodic points.

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