ISBN-10:
3764366478
ISBN-13:
9783764366476
Pub. Date:
04/29/2002
Publisher:
Birkh�user Basel
Index Theory for Symplectic Paths with Applications / Edition 1

Index Theory for Symplectic Paths with Applications / Edition 1

by Yiming Long

Hardcover

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Product Details

ISBN-13: 9783764366476
Publisher: Birkh�user Basel
Publication date: 04/29/2002
Series: Progress in Mathematics , #207
Edition description: 2002
Pages: 380
Product dimensions: 6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

I The Symplectic Group Sp(2n).- 1 Algebraic Aspects.- 1.1 Symplectic matrices.- 1.2 Symplectic spaces.- 1.3 Eigenvalues of symplectic matrices.- 1.4 Normal forms for the eigenvalue 1.- 1.5 Normal forms for the eigenvalue ?1.- 1.6 Normal forms for eigenvalues in U?R.- 1.7 Normal forms for eigenvalues outside U.- 1.8 Basic normal forms.- 1.9 Perturbations basic normal forms.- 2 Topological Aspects.- 2.1 Structures of Sp(2) and its subsets.- 2.2 The global structure of Sp(2n,R).- 2.3 Hyperbolic symplectic matrix set.- 2.4 Structure of regular sets.- 2.5 Structures of singular sets.- 2.6 Transversality of rotation paths.- 2.7 Orientability of M?,(2n) in Sp(2n).- II The Variational Method.- 3 Hamiltonian Systems and Canonical Transformations.- 3.1 Canonical transformations.- 3.2 Generating functions.- 4 The Variational Functional.- 4.1 The Galerkin approximation.- 4.2 The L2-Variational Structure.- 4.3 The saddle point reduction.- 4.4 The dimension theorem on kernels.- 4.5 Certain estimates.- III Index Theory.- 5 Index Functions for Symplectic Paths.- 5.1 Paths in Sp(2).- 5.2 Non-degenerate paths in Sp(2n).- 5.3 Index properties of non-degenerate paths.- 5.4 Perturbations of degenerate paths.- 6 Properties of Index Functions.- 6.1 Index functions and Morse indices.- 6.2 An axiom approach and further properties.- 7 Relations with other Morse Indices.- 7.1 The Galerkin approximation.- 7.2 Second order Hamiltonian systems.- 7.3 Lagrangian systems.- IV Iteration Theory.- 8 Precise Iteration Formulae.- 8.1 Paths in Sp(2).- 8.2 Hyperbolic and elliptic paths.- 8.3 General symplectic paths.- 9 Bott-type Iteration Formulae.- 9.1 Splitting numbers.- 9.2 Bott-type formulae.- 9.3 Abstract precise iteration formulae.- 10 Iteration Inequalities.- 10.1 Estimates via mean index and initial index.- 10.2 Successive estimates.- 10.3 Controlling iteration numbers via indices.- 11 The Common Index Jump Theorem.- 11.1 A common selection theorem.- 11.2 The common index jump theorem.- 12 Index Iteration Theory for Closed Geodesics.- 12.1 Morse index theory.- 12.2 Splitting numbers.- V Applications.- 13 The Rabinowitz Conjecture.- 13.1 Minimax principle preparations.- 13.2 Controlling the minimal period via indices.- 13.3 Asymptotically linear Hamiltonian systems.- 13.4 Superquadratic Hamiltonian systems.- 13.5 Second order systems.- 13.6 Subharmonics.- 13.7 Notes and comments.- 14 Periodic Lagrangian Orbits on Tori.- 14.1 Critical module preparations.- 14.2 The finite energy homology theory.- 14.3 Critical modules and isomorphisms.- 14.4 Global homological injectivity.- 14.5 Global homological vanishing.- 14.6 Notes and comments.- 15 Closed Characteristics on Convex Hypersurfaces.- 15.1 Index theorem for dual action principle.- 15.2 Variational properties.- 15.3 Critical orbits and index jumps.- 15.4 Existence and multiplicity.- 15.5 Stability results.- 15.6 Symmetric hypersurfaces.- 15.7 Notes and comments.

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