Symmetry Breaking for Compact Lie Groups.

Intro -- Contents -- 1. Introduction -- 1.1. Notes for the reader -- 1.2. Acknowledgements -- 2. Technical Preliminaries and Basic Notations -- 2.1. Γ-sets and isotropy types -- 2.2. Representations -- 2.3. Isotropy types for representations -- 2.4. Polynomial Invariants and Equivariants -- 2.5. Smooth families of equivariant maps -- 2.6. Normalized families -- 3. Branching and invariant group orbits -- 3.1. Relative equilibria and normal hyperbolicity -- 3.2. Branches of relative equilibria -- 3.3. The branching pattern -- 3.4. Stabilities -- 3.5. Branching conditions -- 3.6. The signed indexed branching pattern -- 3.7. Stable families -- 3.8. Determinacy -- 3.9. Strong determinacy -- 4. Genericity theorems -- 4.1. Semi-algebraic and semi-analytic sets -- 4.2. Invariant and equi variant generators -- 4.3. The variety Σ -- 4.4. Stability theorems I: Weak regularity -- 4.5. Stability theorems II: Regular families -- 4.6. Determinacy -- 4.7. Examples related to finite reflection groups -- 5. Finitely determined bifurcation problems I -- 5.1. The phase vector field -- 5.2. The spaces A[sub(h)](Γ,V), B[sub(h)](Γ,V) -- 5.3. Strong determinacy -- 6. Finitely-determined bifurcation problems II -- 6.1. Statement of the main theorem -- 6.2. 2-stable relative equilibria -- 7. Strong determinacy: Technical preliminaries -- 7.1. Introduction -- 7.2. Notational conventions -- 7.3. Local geometry -- 7.4. Weakly regular families -- 7.5. Analytic families and solution branches -- 7.6. Compatible parametrizations and initial exponents -- 7.7. Remarks on the set ξ(f) -- 7.8. The parametrization theorem -- 7.9. The space R[sup(2)] -- 7.10. Initial exponents and the space R[sup(3)] -- 8. Strong determinacy: Γ finite -- 8.1. Analytic parametrizations -- 8.2. Estimates on eigenvalues -- 8.3. Fractional power series -- 8.4. Eigenvalue estimates: Analytic case.

8.5. Eigenvalue estimates: Smooth case -- 8.6. Proof of Theorem 8.2.6 -- 8.7. Strong determinacy: Γ finite -- 8.8. Formation of new branches under perturbation -- 9. Strong determinacy: Γ compact, non-finite -- 9.1. Polar blowing-up: Local theory -- 9.2. Polar blowing-up: Global theory -- 9.3. Polar blowing-up a Γ-manifold -- 9.4. Blowing- up -- 9.4.1. Blowing-up along a linear subspace -- 9.4.2. Blowing-up analytic varieties -- 9.4.3. Blowing-up algebraic varieties -- 9.5. Conical sets -- 9.6. Algebraic and analytic structure of the orbit strata -- 9.7. Blowing-up representations -- 9.7.1. Analytic theory -- 9.7.2. Algebraic theory -- 9.8. A tangent and normal decomposition -- 9.9. Blowing-up arcs -- 9.10. Analytic parametrizations of solution branches -- 9.11. Lifting analytic parametrizations -- 9.12. Controlling the lifts of analytic parametrizations -- 9.13. Symmetric structure of parametrizations -- 9.14. An alternative proof of Proposition 9.13.2 -- 9.15. Extensions of Proposition 9.13.2 -- 9.16. Review and Summary of Notations -- 9.17. The exponent iΔ( ) -- 9.18. Estimates on eigenvalues -- 10. Proofs of the parametrization theorems -- 10.1. Resolution of singularities -- 10.2. Blowing-up -- 10.3. Singular sets of real algebraic varieties -- 10.4. Embedded resolution of singularities -- 10.5. Blowing-up embeddings -- 10.6. Blowing up Σ -- 10.7. Reduction to a subspace of V[sub(ω)](Γ,V)[sub(N)] -- 10.8. Families of subspaces of V[sub(ω)](Γ,V)[sub(NR)] -- 10.9. Blowing- up the maps G[sup(q)] -- 10.10. Proof of Theorem 7.8.5 -- 11. An application to the equivariant Hopf bifurcation -- 11.1. Limit cycles for equivariant flows -- 11.2. The equivariant Hopf bifurcation &amp -- Fiedler's theorem -- 11.3. The equivariant Hopf bifurcation for Γ x S[sup(1)-equivariant families -- 11.4. Completion of the proof of Theorem 11.2.1.

A. Branches of relative equilibria -- A.l. Introduction -- A.2. Background on normal hyperbolicity -- A.3. Branches of relative equilibria I -- A.4. Horn neighborhoods I -- A.5. Statement of the main Theorem -- A.6. Lipschitz maps -- A.7. Norm equivalences -- A.8. Diffeomorphisms -- A.9. Coordinates near α -- A.10. Lipschitz sections of E -- A.11. The graph transform -- A.11.1. The map f in p[sub(p)]-coordinates -- A.11.2. Invertibility of g -- A.11.3. Contractivity of f # -- A.11.4. Perturbation theory -- A.12. Horn neighborhoods II -- A.13. Vector space theory: Norm estimates in terms of eigenvalues -- A. 13.1. Hyperbolic linear maps -- A. 13.2. Relatively hyperbolic families -- A.14. Conditions (U) and (V) -- A.15. Branches of relative equilibria II -- A. 15.1. Standing assumptions -- A.16. Proof of Theorem A.5.1 -- References.

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