Symplectic Invariants and Hamiltonian Dynamics

The discoveries of the last decades have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.

Introduction
Symplectic vector spaces	p. 1
Symplectic diffeomorphisms and Hamiltonian vector fields	p. 6
Hamiltonian vector fields and symplectic manifolds	p. 9
Periodic orbits on energy surfaces	p. 18
Existence of a periodic orbit on a convex energy surface	p. 23
The problem of symplectic embeddings	p. 31
Symplectic classification of positive definite quadratic forms	p. 35
The orbit structure near equilibrium, Birkhoff normal form	p. 42
Symplectic capacities
Definition and application to embeddings	p. 51
Rigidity of symplectic diffeomorphisms	p. 58
Existence of a capacity
Definition of the capacity co	p. 69
The minimax idea	p. 77
The analytical setting	p. 82
The existence of a critical point	p. 91
Examples and illustrations	p. 98
Existence of closed characteristics
Periodic solutions on energy surfaces	p. 105
The characteristic line bundle of a hypersurface	p. 113
Hypersurfaces of contact type, the Weinstein conjecture	p. 119
"Classical" Hamiltonian systems	p. 127
The torus and Herman's Non-Closing Lemma	p. 137
Geometry of compactly supported symplectic mappings in R²ⁿ
A special metric d for a group V of Hamiltonian diffeomorphisms	p. 143
The action spectrum of a Hamiltonian map	p. 151
A "universal" variational principle	p. 154
A continuous section of the action spectrum bundle	p. 161
An inequality between the displacement energy and the capacity	p. 165
Comparison of the metric d on D with the C°-metric	p. 173
Fixed points and geodesics on D	p. 182
The Arnold conjecture, Floer homology and symplectic homology
The Arnold conjecture on symplectic fixed points	p. 194
The model case of the torus	p. 202
Gradient-like flows on compact spaces	p. 217
Elliptic methods and symplectic fixed points	p. 222
Floer's approach to Morse theory for the action functional	p. 250
Symplectic homology	p. 265
Appendix
Generating functions of symplectic mappings in R²ⁿ	p. 273
Action-angle coordinates, the Theorem of Arnold and Jost	p. 278
Embeddings of H^1/2(S¹) and smoothness of the action	p. 286
The Cauchy-Riemann operator on the sphere	p. 291
Elliptic estimates near the boundary and an application	p. 298
The generalized similarity principle	p. 302
The Brouwer degree	p. 305
Continuity property of the Alexander-Spanier cohomology	p. 314
ndex	p. 321
Bibliography	p. 327
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Symplectic Invariants and Hamiltonian Dynamics

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