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 --> 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 --> 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.