Symplectic Invariants and Hamiltonian Dynamics, 1994
Birkhäuser Advanced Texts Basler Lehrbücher Series

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.

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 least action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory and first order elliptic systems, the Arnold conjectures and a survey on Floer- and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.