CONTENTS vii

11. An application to the equivariant Hopf bifurcation 135

11.1. Limit cycles for equivariant flows 135

11.2. The equivariant Hopf bifurcation & Fiedler's theorem 136

11.3. The equivariant Hopf bifurcation for T x Sl-equivariant families 136

11.4. Completion of the proof of Theorem 11.2.1 138

A. Branches of relative equilibria 139

A.l. Introduction 139

A.2. Background on normal hyperbolicity 140

A.3. Branches of relative equilibria I 142

A.4. Horn neighborhoods I 143

A.5. Statement of the main Theorem 145

A.6. Lipschitz maps 146

A.7. Norm equivalences 146

A.8. Diffeomorphisms 147

A.9. Coordinates near a 148

A. 10. Lipschitz sections of E 150

A. 11. The graph transform 151

A. 11.1. The map / in pp-coordinates

A. 11.2. Invertibility of g

A. 11.3. Contractivity of / #

A. 11.4. Perturbation theory

A. 12. Horn neighborhoods II 155

A. 13. Vector space theory: Norm estimates in terms of eigenvalues 157

A. 13.1. Hyperbolic linear maps

A. 13.2. Relatively hyperbolic families

A. 14. Conditions (U) and (V) 162

A.15. Branches of relative equilibria II 163

A. 15.1. Standing assumptions

A. 16. Proof of Theorem A.5.1 165

References 168