Analysis of an old variational principal in classical mechanics has established global periodic phenomena in Hamiltonian systems. One of the links is a class of sympletic invariants, called sympletic capacities, and these invariants are the main theme of this book. Topics covered include basic sympletic geometry, sympletic capacities and rigidity, sympletic fixed point theory, and a survey on Floer homology and sympletic homology.

Contenuti:

1 Introduction.- 1.1 Symplectic vector spaces.- 1.2 Symplectic diffeomorphisms and Hamiltonian vector fields.- 1.3 Hamiltonian vector fields and symplectic manifolds.- 1.4 Periodic orbits on energy surfaces.- 1.5 Existence of a periodic orbit on a convex energy surface.- 1.6 The problem of symplectic embeddings.- 1.7 Symplectic classification of positive definite quadratic forms.- 1.8 The orbit structure near an equilibrium, Birkhoff normal form.- 2 Symplectic capacities.- 2.1 Definition and application to embeddings.- 2.2 Rigidity of symplectic diffeomorphisms.- 3 Existence of a capacity.- 3.1 Definition of the capacity c0.- 3.2 The minimax idea.- 3.3 The analytical setting.- 3.4 The existence of a critical point.- 3.5 Examples and illustrations.- 4 Existence of closed characteristics.- 4.1 Periodic solutions on energy surfaces.- 4.2 The characteristic line bundle of a hypersurface.- 4.3 Hypersurfaces of contact type, the Weinstein conjecture.- 4.4 “Classical” Hamiltonian systems.- 4.5 The torus and Herman’s Non-Closing Lemma.- 5 Compactly supported symplectic mappings in ?2n.- 5.1 A special metric d for a group D of Hamiltonian diffeomorphisms.- 5.2 The action spectrum of a Hamiltonian map.- 5.3 A “universal” variational principle.- 5.4 A continuous section of the action spectrum bundle.- 5.5 An inequality between the displacement energy and the capacity.- 5.6 Comparison of the metric d on D with the C0-metric.- 5.7 Fixed points and geodesics on D.- 6 The Arnold conjecture, Floer homology and symplectic homology.- 6.1 The Arnold conjecture on symplectic fixed points.- 6.2 The model case of the torus.- 6.3 Gradient-like flows on compact spaces.- 6.4 Elliptic methods and symplectic fixed points.- 6.5 Floer’s appraoch to Morse theory for the action functional.- 6.6 Symplectic homology.- A.2 Action-angle coordinates, the Theorem of Arnold and Jost.- A.4 The Cauchy-Riemann operator on the sphere.- A.5 Elliptic estimates near the boundary and an application.- A.6 The generalized similarity principle.- A.7 The Brouwer degree.- A.8 Continuity property of the Alexander-Spanier cohomology.

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