SYMMETRY BREAKING FOR COMPACT LIE GROUPS 7
the orbit strata. As part of the proof, we also need results on the persistence and
stability of branches of normally hyperbolic relative equilibria. We present the quite
delicate technical details on the proofs of these results in the Appendix.
In section 11, we show how the strong determinacy theorem may be used to
justify the use of Birkhoff normal form arguments in the study of the equivariant
Hopf bifurcation. In particular, we are able to remove the p-determined stability
assumption of Stewart [71, Definition 7.1] (see also Taliaferro [73]). Thus, suppose
that (V,r) is irreducible over C and d 0. We recall (see [43, Chapter XVI])
that if / is a normalized family and Di/(0,0) has a complex conjugate pair of
eigenvalues ±20;, u ^ 0, then it is possible to make local equivariant changes of
coordinates near zero so that / ^ is T x Sl-equivariant. Using a relative version
of the strong determinacy theorem (breaking symmetry from T x S1 to T), we are
able to show that branches of limit cycles obtained by analyzing the normal form
of / , will in general persist for / (for a slightly different approach, avoiding the
use of results from the Appendix, see [34]). We conclude by proving a variant of
Fiedler's theorem [24] on the existence of branches of limit cycles with maximal
isotropy type.
1.1. Notes for the reader. This work is primarily directed towards equivariant
bifurcation theory. It is not assumed that the reader is an expert in techniques
from real algebraic geometry, blowing-up and Newton-Puiseux series. Consequently,
reasonably complete - or at least, introductory - details are given on some of these
topics in the hope that they may find further applications in bifurcation theory. On
the other hand, reasons of space dictated that some of the material on resolution of
singularities, sheaves and algebraic groups used in sections 9 and 10 be abbreviated.
As justification, it can be argued that the results which use these concepts are
geometrically transparent.
In general terms, sections 2, 3, 5 and 6 are elementary, section 4 is a straightfor-
ward application of basic results from equivariant transversality theory. Sections 7
- 10, in particular section 9, are less elementary and depend on results from alge-
braic geometry. Section 11 is a straightforward application of the main results of
sections 7 - 10. The Appendix, though technical, is a natural extension of ideas
from the theory of normally hyperbolic sets to branches of relative equilibria. The
articles [32, 33] may provide an introduction to some of the topics of this work.
1.2. Acknowledgements. Much of the preliminary work for this paper was done
during a very pleasant year at Cornell in 1989/90 and the author gratefully ac-
knowledges the hospitality and support of John Guckenheimer and the Center for
Applied Mathematics, Cornell. Special thanks are due to Gerry Schwarz for his
generous help and for providing a proof of the crucial Lemma 9.6.1; to Tzee-Char
Kuo, for teaching me about Newton-Puiseux series and real singularities; and to Ed
Bierstone for his patient help in explaining details about resolution of singularities
and for pointing out that coherence was the key to proving Theorem 9.7.2. Thanks
also to Marty Golubitsky, for encouragement and many helpful and enlightening
discussions. Finally, this research was supported in part by NSF Grant DMS-
9403624, Texas Advanced Research Program award 003652026 and ONR Grant
N00014-94-1-0317.
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