CONTENTS
1. Introduction 1
1.1. Notes for the reader 7
1.2. Acknowledgements 7
2. Technical Preliminaries and Basic Notations 8
2.1. T-sets and isotropy types 8
2.2. Representations 8
2.3. Isotropy types for representations 10
2.4. Polynomial Invariants and Equivariants 10
2.5. Smooth families of equivariant maps 11
2.6. Normalized families 11
3. Branching and invariant group orbits 13
3.1. Relative equilibria and normal hyperbolicity 13
3.2. Branches of relative equilibria 17
3.3. The branching pattern 18
3.4. Stabilities 19
3.5. Branching conditions 19
3.6. The signed indexed branching pattern 20
3.7. Stable families 20
3.8. Determinacy 21
3.9. Strong determinacy 22
4. Genericity theorems 25
4.1. Semi-algebraic and semi-analytic sets 25
4.2. Invariant and equivariant generators 26
4.3. The variety £ 26
4.4. Stability theorems I: Weak regularity 30
4.5. Stability theorems II: Regular families 34
4.6. Determinacy 41
4.7. Examples related to finite reflection groups 45
5. Finitely determined bifurcation problems I 47
5.1. The phase vector field 47
5.2. The spaces Ah{T, V), Bh{T, V) 49
5.3. Strong determinacy 54
6. Finitely-determined bifurcation problems II 56
6.1. Statement of the main theorem 56
6.2. 2-stable relative equilibria 56
7. Strong determinacy: Technical preliminaries 62
7.1. Introduction 62
7.2. Notational conventions 62
7.3. Local geometry 63
7.4. Weakly regular families 65
7.5. Analytic families and solution branches 67
7.6. Compatible parametrizations and initial exponents 68
7.7. Remarks on the set £(/) 70
7.8. The parametrization theorem 70
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