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.
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Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Research
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ISBN-13
978-3-7643-5066-6 (9783764350666)
Schweitzer Klassifikation
Part 1 Introduction: symplectic vectro spaces; symplectic diffeomorphisms and Hamiltonian vector fields in (R2n, omega-0); Hamiltonian vector fields and symplectic manifolds; periodic orbits on energy surfaces; existence of a periodic orbit on a convex energy surface; the problem of symplectic embeddings; symplectic classification of positive definite quadratic forms; the orbit structure near an equilibrium, Birkhoff normal form. Part 2 Symplectic capacities: definition and application to embeddings; rigidity of symplectic diffeomorphisms. Part 3 Existence of a capacity: definition of the capacity c-0; the minimax idea; the anlytical setting; the existence of a critical point; examples and illustrations. Part 4 Existence of closed characteristics: periodic solutions in energy surfaces; the characterisrtic line bundle of a hypersurface; hypersurfaces of contact type, the Weinstein conjecture; "classical" Hamiltonian systems; the torus and Herman's non-closing Lemma. Part 5 Compactly supported symplectic mappings: a special metric "d" for a group "D"; the action spectrum of a Hamiltonian map; a "universal" variational principle; a continuous section of the action spectrum bundle; an inequality between the displacement energy; comparison of the metric "d" on "D" with the "C0-metric"; fixed points and geodesics on "D". Part 6 The Arnold conjecture, Floer homology: the Arnold conjecture on symplectic fixed points; the model case of the torus; gradient-like flows on compact spaces; elliptic methods and symplectic fixed points; Floer's approach to Morse theory for the action functional; symplectic homology; generating functions of symplectic mappings in R2n; action-angle coordinates, the theorem of Arnold and Jost; embeddings of "H1/2(S1)" and smoothness of the action; the Cauchy-Riemann operator on the sphere; ellpitic estimates near the boundary and an application; the generalized Carleman similarity principle; the Brouwer degree; continuity property of the Alexander-Spanier cohomology.
