| Description |
1 online resource |
| Contents |
Cover -- 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 breaking8212;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)-deter |
| Summary |
This book contains the first systematic exposition of the global and local theory of dynamics equivariant with respect to a (compact) Lie group. Aside from general genericity and normal form theorems on equivariant bifurcation, it describes many general families of examples of equivariant bifurcation and includes a number of novel geometric techniques, in particular, equivariant transversality. This important book forms a theoretical basis of future work on equivariant reversible and Hamiltonian systems. This book also provides a general and comprehensive introduction to codimension one equivariant bifurcation theory. In particular, it includes the bifurcation theory developed with Roger Richardson on subgroups of reflection groups and the Maximal Isotropy Subgroup Conjecture. A number of general results are also given on the global theory. Introductory material on groups, representations and G-manifolds are covered in the first three chapters of the book. In addition, a self-contained introduction of equivariant transversality is given, including necessary results on stratifications as well as results on equivariant jet transversality developed by Edward Bierstone |
| Subject |
Bifurcation theory.
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Hamiltonian systems.
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Lie groups.
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Symmetry (Mathematics)
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Topological dynamics.
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Bifurcation theory.
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Hamiltonian systems.
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Lie groups.
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Symmetry (Mathematics)
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Topological dynamics.
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| Form |
Electronic book
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| ISBN |
128186756X |
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9781281867568 |
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9781860948541 |
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1860948545 |
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