M. Golubitsky and M. Nicol
Symmetry Detectives for SBR Attractors
Nonlinearity 8 (1995) 1027-1037
Let Gamma be a finite group acting on R^n and let x_0 be an
initial point for a Gamma-equivariant map f:R^n \to R^n.
The question of determining the symmetries of the omega-limit set omega_f(x_0)
is discussed in Barany et. al and Dellnitz et. al.
These methods are based on the notion
of a symmetry detective. Detectives replace the question of determining
the symmetries of the set omega_f(x_0) by the easier question of
determining the symmetries of a point in an associated space W.
The detective theorem in Barany et. al has a limitation in that
its implementation tacitly assumes that omega_f(x_0) contains a point
of trivial isotropy; this assumption is explicit in Dellnitz et. al.
In this
paper we extend these ideas to present sufficient
conditions for an equivariant polynomial phi:R^n \to W to be a
detective, even when the omega-limit set is contained in a proper
fixed-point subspace. We show that W need only satisfy the conditions
given in Dellnitz et. al while the map phi has to satisfy certain
conditions in addition to the ones listed in Dellnitz et. al.
We also present a density theorem for such detectives and we show
that the detective for rings of p coupled cells (nearest neighbor
coupling) with D_p symmetry first given in Barany et. al is a
detective for all (SBR) attractors.