In this work we state and prove a number of foundational results in the local bifurcation theory of smooth equivariant maps. In particular, we show that stable one-parameter families of maps are generic and that stability is characterised by semi-algebraic conditions on the finite jet of the family at the bifurcation point. We also prove strong determinacy theorems that allow for high order forced symmetry breaking. We give a number of examples, related to earlier work of Field & Richardson, that show that even for finite groups we can expect branches of fixed or prime period two points with submaximal isotropy type. Finally, we provide a simplified proof of a result that justifies the use of normal forms in the analysis of the equivariant Hopf bifurcation.

For preprint, e-mail: mikefield@gmail.com
(Reprints no longer available.)

Professor Mike Field
Department of Mathematics
Imperial College
London SW7 2AZ