On symmetric attractors in reversible dynamical systems

Let ? O(n) be a nite group acting orthogonally on IR n. We say that ? is a reversing symmetry group of a homeomorphism, diieomorphism or ow f t : IR n 7 ! IR n (t 2 Z Z or t 2 IR) if ? has an index two subgroup ~ ? whose elements commute with f t and for all elements 2 ? ? ~ ? and all t , f t (x) = f ?t (x). We give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors (Liapunov stable !-limit sets). These conditions arise due to topological obstructions. In dimensions 1 and 2 we present a complete description of possible symmetry groups of asymptotically stable attractors for homeomorphisms and diieomorphisms (these at-tractors cannot possess reversing symmetries). We also have a fairly complete description in the context of subgroups which contain reversing symmetries. For all dimensions n we present complete results on the possible symmetry groups of connected asymptotically stable attractors. In addition, we summarize our results in the context of reversible equivariant ows, which are complete in dimensions 1 and 2 as well as for subgroups of ~ ? O(n) in any dimension n. A survey of the corresponding results in the equivariant context is also given.