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.
e-mail: email@example.com or Michael.J.Field@rice.edu Professor Mike Field
(Reprints no longer available.)
Department of Mathematics
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