Michael Field, Ian Melbourne and Matthew Nicol.
Symmetric attractors for diffeomorphisms and flows
Let G be a finite group of orthogonal
transformations acting on n-dimensional Euclidean space.
In this work we describe the possible symmetry groups that
can occur for attractors of smooth (invertible) G-equivariant
dynamical systems. (For continuous non-invertible systems, see
P. Ashwin and I. Melbourne. Symmetry groups of
attractors. Arch. Rat. Mech. Anal. 126 (1994) 59-78.)
In case G contains no reflections,
and n is at least 3,
our results imply there are no restrictions on symmmetry groups. In case n
is at least 4 (diffeomorphisms)
or 5 (flows), we show that we may construct attractors
which are Axiom A. We also give a
complete description of what can happen in low dimensions.