Contents -- Preface -- 1. Groups -- 1.1 Definition of a group and examples -- 1.2 Homomorphisms, subgroups and quotient groups -- 1.2.1 Generators and relations for .nite groups -- 1.3 Constructions -- 1.4 Topological groups -- 1.5 Lie groups -- 1.5.1 The Lie bracket of vector fields -- 1.5.2 The Lie algebra of G -- 1.5.3 The exponential map of g -- 1.5.4 Additional properties of brackets and exp -- 1.5.5 Closed subgroups of a Lie group -- 1.6 Haarmeasure -- 2. Group Actions and Representations -- 2.1 Introduction -- 2.2 Groups and G-spaces -- 2.2.1 Continuous actions and G-spaces -- 2.3 Orbit spaces and actions -- 2.4 Twisted products -- 2.4.1 Induced G-spaces -- 2.5 Isotropy type and stratification by isotropy type -- 2.6 Representations -- 2.6.1 Averaging over G -- 2.7 Irreducible representations and the isotypic decomposition -- 2.7.1 C-representations -- 2.7.2 Absolutely irreducible representations -- 2.8 Orbit structure for representations -- 2.9 Slices -- 2.9.1 Slices for linear finite group actions -- 2.10 Invariant and equivariant maps -- 2.10.1 Smooth invariant and equivariant maps on representations -- 2.10.2 Equivariant vector fields and flows -- 3. Smooth G-manifolds -- 3.1 Proper G-manifolds -- 3.1.1 Proper free actions -- 3.2 G-vector bundles -- 3.3 Infinitesimal theory -- 3.4 Riemannianmanifolds -- 3.4.1 Exponential map of a complete Riemannian manifold -- 3.4.2 The tubular neighbourhood theorem -- 3.4.3 Riemannian G-manifolds -- 3.5 The differentiable slice theorem -- 3.6 Equivariant isotopy extension theorem -- 3.7 Orbit structure for G-manifolds -- 3.7.1 Closed filtration of M by isotropy type -- 3.8 The stratification of M by normal isotropy type -- 3.9 Stratified sets -- 3.9.1 Transversality to a Whitney stratification -- 3.9.2 Regularity of stratification by normal isotropy type.

3.10 Invariant Riemannian metrics on a compact Lie group -- 3.10.1 The adjoint representations -- 3.10.2 The exponential map -- 3.10.3 Closed subgroups of a Lie group -- 4. Equivariant Bifurcation Theory: Steady State Bifurcation -- 4.1 Introduction and preliminaries -- 4.1.1 Normalized families -- 4.2 Solution branches and the branching pattern -- 4.2.1 Stability of branching patterns -- 4.3 Symmetry breaking-theMISC -- 4.3.1 Symmetry breaking isotropy types -- 4.3.2 Maximal isotropy subgroup conjecture -- 4.4 Determinacy -- 4.4.1 Polynomial maps -- 4.4.2 Finite determinacy -- 4.5 The hyperoctahedral family -- 4.5.1 The representations (Rk,Hk) -- 4.5.2 Invariants and equivariants for Hk -- 4.5.3 Cubic equivariants for Hk -- 4.5.4 Bifurcation for cubic families -- 4.5.5 Subgroups of Hk -- 4.5.6 Some subgroups of the symmetric group -- 4.5.7 A big family of counterexamples to the MISC -- 4.5.8 Examples where P3G (Rk, Rk) = P3H k (Rk, Rk) -- 4.5.9 Stable solution branches of maximal index and trivial isotropy -- 4.5.10 An example with applications to phase transitions -- 4.6 Phase vector field and maps of hyperbolic type -- 4.6.1 Cubic polynomial maps -- 4.6.2 Phase vector field -- 4.6.3 Normalized families -- 4.6.4 Maps of hyperbolic type -- 4.6.5 The branching pattern of JQ -- 4.7 Transforming to generalized spherical polar coordinates -- 4.7.1 Preliminaries -- 4.7.2 Polar blowing-up -- 4.8 d(V,G)-determinacy, d(V,G) = 2,3 -- 4.8.1 d(V,G) = 2 but (V,G) is not 2-determined -- 4.9 Counting branches and finding their location -- 4.10 The symmetric and alternating groups -- 4.10.1 Preliminaries on Sk+1 -- 4.10.2 Zeroes of the phase vector field -- 4.10.3 Subgroups of Sk+1 -- 4.10.4 The sign and index functions -- 4.10.5 The case k + 1 odd -- 4.10.6 The case k + 1 even -- 4.11 The groups Sk+1 × Z2 and Ak+1 × Z2 -- 4.11.1 The zeros of PC.

4.11.2 Applications -- 4.12 Appendix: Proof of theorem on hyperbolic elements -- 4.13 Notes on Chapter 4 -- 5. Equivariant Bifurcation Theory: Dynamics -- 5.1 The invariant sphere theorem -- 5.1.1 Extensions and generalizations -- 5.1.2 Applying the invariant sphere theorem -- 5.2 The examples of dos Reis and Guckenheimer & Holmes -- 5.3 Steady state bifurcation to limit cycles -- 5.3.1 Axes of symmetry in Λ3 -- 5.3.2 Cubic equivariants -- 5.3.3 Homoclinic cycles -- 5.3.4 Stabilities and a 'hidden Hopf bifurcation' -- 5.4 Bifurcation to complex dynamics in dimension four -- 5.4.1 A basis for the action of on C4 -- 5.4.2 The representation (V,G) -- 5.4.3 Geometry of the representation (V,G) -- 5.4.4 Equilibria of a normalized cubic family -- 5.4.5 Dynamics close to the plane S -- 5.4.6 Random switching and dynamics near a network -- 5.4.7 Random switching theorem -- 5.5 The converse to the MISC -- 5.6 Hopf bifurcation and the invariant sphere theorem -- 5.6.1 G × S1-equivariant families -- 5.6.2 (Complex) phase vector field -- 5.6.3 Projective space and the Hopf fibration -- 5.6.4 Phase blowing-up -- 5.6.5 Transforming a cubic normal form -- 5.6.6 Invariant sphere theorem for the Hopf bifurcation -- 5.6.7 The algebraic Hopf theorem -- 5.7 Notes on Chapter 5 -- 6. Equivariant Transversality -- 6.1 Introduction -- 6.2 C∞-topologies on function spaces -- 6.2.1 Jet bundles -- 6.2.2 Natural constructions of C∞-topologies -- 6.3 Transversality -- 6.3.1 Basic theorems on transversality -- 6.4 Stratumwise transversality and stability -- 6.5 Reduction to a problem about solving equations -- 6.6 Invariants and equivariants -- 6.6.1 Smooth equivariants -- 6.6.2 Generators for equivariants -- 6.6.3 Smooth invariant theory -- 6.6.4 Smooth invariants and the orbit space -- 6.7 The universal variety -- 6.7.1 Changing generators.

6.8 Stratifications and semialgebraic sets -- 6.8.1 Semialgebraic sets -- 6.8.2 Semialgebraic stratifications -- 6.8.3 The canonical stratification -- 6.9 Canonical stratification of the universal variety -- 6.9.1 Partition of Σ by isotropy type -- 6.9.2 A first (restricted) de.nition of G-transversality -- 6.10 Stratifying Στ and U -- 6.11 Equivariant coordinate changes on V × Rs and W -- 6.12 Symmetries of the stratification A -- 6.13 Openness of equivariant transversality -- 6.14 Global definitions and results -- 6.14.1 G-transversality on a manifold -- 6.15 Solutions with specific isotropy type -- 6.16 Notes on Chapter 6 -- 7. Applications of G-transversality to Bifurcation Theory I -- 7.1 Weak stability and determinacy -- 7.1.1 Generic 1-parameter steady-state bifurcation theory -- 7.1.2 Bifurcation on absolutely irreducible representations -- 7.1.3 Symmetry breaking isotropy types -- 7.1.4 Weak determinacy -- 7.1.5 Weak stability of equivariant reversible vector fields -- 7.2 Jet transversality -- 7.2.1 An equivariant Thom jet transversality theorem -- 7.2.2 Invariance lemmas -- 7.3 Equivariant jet transversality for families -- 7.3.1 Intrinsic formulation of jet transversality -- 7.4 Stability and determinacy -- 7.4.1 A reformulation of the stability criterion -- 7.4.2 Example: computations for (R2,D4) -- 7.5 Higher order versions of G-transversality -- 7.6 Extensions to the case of non-finite G -- 7.6.1 Equilibrium G-orbits -- 7.6.2 Branching and stability for compact Lie groups -- 7.7 Notes on Chapter 7 -- 8. Equivariant Dynamics -- 8.1 Invariant G-orbits -- 8.1.1 Vector fields and flows -- 8.2 Stabilities and normal hyperbolicity -- 8.2.1 Diffeomorphisms -- 8.2.2 Flows -- 8.2.3 Genericity -- 8.2.4 Stable and unstable manifolds -- 8.3 Relative fixed and periodic sets for diffeomorphisms -- 8.3.1 Cartan subgroups of a compact Lie group G.

8.3.2 Dynamics on relative fixed & periodic sets -- 8.3.3 Isotopy lemmas -- 8.3.4 Stabilities of relative .xed sets and periodic orbits -- 8.3.5 Perturbation theory -- 8.4 Genericity theorems for equivariant diffeomorphisms -- 8.5 Equivariant vector fields -- 8.5.1 Relative equilibria -- 8.5.2 Stability of relative equilibria -- 8.5.3 Classification of relative periodic orbits -- 8.5.4 Periodic orbits and symmetry -- 8.5.5 Poincaré map for a relative periodic orbit -- 8.5.6 Perturbation lemmas for relative periodic orbits -- 8.6 Genericity theorems for equivariant vector fields -- 8.7 Notes on Chapter 8 -- 9. Dynamical Systems on G-manifolds -- 9.1 Skew products -- 9.1.1 Skew extensions of diffeomorphisms -- 9.1.2 Principal G-extensions -- 9.1.3 Skew products for flows and vector fields. -- 9.2 Gradient dynamics -- 9.2.1 Handlebundle decompositions of a G-manifold -- 9.2.2 Triangulations and handlebundle decompositions -- 9.3 G-subshifts of finite type -- 9.3.1 Stability and the realization of G-subshifts of .nite type as basic sets of equivariant diffeomorphisms -- 9.4 Suspensions -- 9.5 The inverse limit: turning maps into homeomorphisms -- 9.6 Solenoidal attractors -- 9.6.1 Finite graphs -- 9.6.2 Branched 1-manifolds -- 9.6.3 Neighbourhoods of branched 1-manifolds -- 9.6.4 Smooth graphs -- 9.6.5 Group actions on graphs -- 9.6.6 Twisted products and embeddings -- 9.6.7 Smooth Eulerian paths -- 9.6.8 Condition (W) -- 9.6.9 Symmetric hyperbolic attractors - simply connected case -- 9.6.10 Symmetric hyperbolic attractors - general case -- 9.6.11 Examples in dimension 3 -- 9.6.12 Non-free finite group actions on attractors -- 9.6.13 Symmetric hyperbolic attractors for flows -- 9.6.14 A tubular neighbourhood of the embedded suspension -- 9.6.15 Extensions to skew and twisted products -- 9.7 Equivariant Anosov diffeomorphisms -- 9.8 Notes on Chapter 9.

10. Applications of G-transversality to Bifurcation Theory II.