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Symmetry Breaking for Compact Lie Groups
Michael Field, University of Houston, TX
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Memoirs of the American Mathematical Society
1996; 170 pp; softcover
Volume: 120
ISBN-10: 0-8218-0435-9
ISBN-13: 978-0-8218-0435-3
List Price: US$52 Individual Members: US$31.20
Institutional Members: US\$41.60
Order Code: MEMO/120/574

This work comprises a general study of symmetry breaking for compact Lie groups in the context of equivariant bifurcation theory. The author starts by extending the theory developed by Field and Richardson for absolutely irreducible representations of finite groups to general irreducible representations of compact Lie groups. In particular, the author allows for branches of relative equilibria and phenomena such as the Hopf bifurcation.

The author also presents a general theory of determinacy for irreducible Lie group actions along the lines previously described by Field in Equivariant Bifurcation Theory and Symmetry Breaking. In the main result of this work, it is shown that branching patterns for generic equivariant bifurcation problems defined on irreducible representations persist under perturbations by sufficiently high order non-equivariant terms.

The author gives applications of this result to normal form computations yielding, for example, equivariant Hopf bifurcations and shows how normal form computations of branching and stabilities are valid when taking account of the non-normalized tail.

Graduate students and research mathematicians specializing in equivariant bifurcation theory.

• Introduction
• Technical preliminaries and basic notations
• Branching and invariant group orbits
• Genericity theorems
• Finitely determined bifurcation problems I
• Finitely-determined bifurcation problems II
• Strong determinacy: Technical preliminaries
• Strong determinacy: $$\Gamma$$ finite
• Strong determinacy: $$\Gamma$$ compact, non-finite
• Proofs of the parametrization theorems
• An application to the equivariant Hopf bifurcation
• Branches of relative equilibria
• References