Symmetric Attractors for Diffeomorphisms and Flows

Proc. London Math. Soc. 72 (1996) 657-696

Mike Field, Ian Melbourne and Matthew Nicol

Abstract

Let $\Gamma\subset O(n)$ be a finite group acting on Rⁿ. In this work we describe the possible symmetry groups that can occur for attractors of smooth (invertible) $\Gamma$-equivariant dynamical systems. In case Rⁿ contains no reflection planes and n>4, our results imply there are no restrictions on symmmetry groups. In case n>5 (diffeomorphisms) and n>6 (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.