Journal ArticleDOI
On the topological properties of symplectic maps
TLDR
In this paper, it was shown that the topological properties of the symplectic diffeomorphism groups are related to the topology of the phase space geometry of the manifold.Abstract:
In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.read more
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Symplectic Invariants and Hamiltonian Dynamics
Helmut Hofer,Eduard Zehnder +1 more
TL;DR: In this article, the authors consider the class of all symplectic manifolds (M, ω) possibly with boundary, but of fixed dimension 2n, where ω is a symplectic structure, i.e. a two-form on M which is closed and nondegenerate.
Journal ArticleDOI
The geometry of symplectic energy
François Lalonde,Dusa McDuff +1 more
TL;DR: In this paper, it was shown that if there were a symplectomorphism of M which had "too little" energy, one could embed a large ball into a thin cylinder M x B2, where B2 is a 2-disc.
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Calabi quasimorphism and quantum homology
Michael Entov,Leonid Polterovich +1 more
TL;DR: In this paper, it was shown that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property.
Journal ArticleDOI
Lagrangian Floer theory on compact toric manifolds, I
TL;DR: In this article, the authors carried out explicit calculations involving PO on toric manifolds and studied the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [G1], which advocates that the quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential.
References
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Journal ArticleDOI
Pseudo holomorphic curves in symplectic manifolds
TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).
Journal ArticleDOI
Symplectic topology and Hamiltonian dynamics
Ivar Ekeland,Helmut Hofer +1 more
TL;DR: On etudie des applications symplectiques non lineaires as mentioned in this paper, construction d'une capacite symplectique, Problemes de plongement, and Probleme de rigidite
Journal ArticleDOI
A proof of Weinstein’s conjecture in ℝ2n
Claude Viterbo,Claude Viterbo +1 more
TL;DR: In this article, the Clarke-Ekeland-Lasry dual action functional was used to prove that a hypersurface of contact type in ( ℝ 2 n, ∑ d x i ∧ d y i ) has a closed characteristic.
Journal ArticleDOI
Periodic solutions on hypersurfaces and a result by C. Viterbo
Helmut Hofer,Eduard Zehnder +1 more
TL;DR: On considere un champ vectoriel hamiltonien x˙=J⊇H(x)=:X H (x) sur x∈R 2n, H etant une fonction lisse dont le gradient ⊆H est defini par rapport a la metrique euclidienne On cherche des solutions periodiques sur une surface d'energie donnee S:={x∈ R 2n /H (x)=const} supposee reguliere