Symplectic rigidity, symplectic fixed points, and global perturbations of Hamiltonian systems

Dragnev, Dragomir L.

doi:10.1002/cpa.20203

Cited by 29 publications

(25 citation statements)

References 24 publications

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“…Since then the problem of existence of leafwise intersections has been extensively investigated, and Hofer's theorem has been extended to coisotropic submanifolds and to other ambient symplectic manifolds; see, e.g., [AF10,AF12,AMc,AMo,Dr,Gi07,Gü,Ka,Zi09] for an admittedly incomplete but representative list of results on leafwise intersections. A common feature of these results is that, in contrast with Moser's theorem, to ensure the existence of leafwise intersections one has to impose some additional requirements on the hypersurface or the coisotropic submanifold.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Fragility and persistence of leafwise intersections

Ginzburg

Gürel

2015

Math. Z.

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In this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds. Namely, we construct a closed hypersurface and a sequence of Hamiltonians C 0 -converging to zero such that the hypersurface and its images have no leafwise intersections, showing that some form of the contact type condition on the hypersurface is necessary in several persistence results. In connection with recent results in continuous symplectic topology, we also show that C 0 -convergence of hypersurfaces, Hamiltonian diffeomorphic to each other, does not in general force C 0 -convergence of the characteristic foliations.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Fragility and persistence of leafwise intersections

Ginzburg

Gürel

2015

Math. Z.

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“…Ekeland and Hofer [EH89,Hof90] replaced, for hypersurfaces of restricted contact type in R 2n , the C 1 -small condition by a much weaker smallness assumption, namely that the Hofer norm of the Hamiltonian symplectomorphism should be smaller than a certain capacity of the region bounded by the hypersurface. This result was then recently extended by Dragnev [Dra08] to coisotropic submanifolds of R 2n with higher codimension, and by Ginzburg [Gin07] [Zil10]. Translated points of contactomorphisms are a special case of leafwise coisotropic intersections.…”

Section: Introductionmentioning

confidence: 84%

ON ITERATED TRANSLATED POINTS FOR CONTACTOMORPHISMS OF ℝ²ⁿ⁺¹ AND ℝ²ⁿ × S¹

Sandon

2012

Int. J. Math.

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Abstract. A point q in a contact manifold is called a translated point for a contactomorphism φ with respect to some fixed contact form if φ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points has an interpretation in terms of Reeb chords between Legendrian submanifolds, and can be seen as a special case of the problem of leafwise coisotropic intersections. For a compactly supported contactomorphism φ of R 2n+1 or R 2n × S 1 contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if φ is positive then there are infinitely many non-trivial geometrically distinct iterated translated points, i.e. translated points of some iteration φ k . This result can be seen as a (partial) contact analogue of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of R 2n , and is obtained with generating functions techniques in the setting of [S11].

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“…We say that a manifold with corners Ξ ⊆ Gr ℓ (t * ), with facets Ξ s : s ∈ S, is • polyhedral, if for all s ∈ S, there is a 1-dimensional subspace E s ≤ t such that the facet Ξ s lies in the subgrassmannian Gr ℓ (E 0 s ) ⊆ Gr ℓ (t * ); • labelled by e = (e s ) s∈S if for all s ∈ S, e s ∈ t \ 0 and Ξ s ⊆ Gr ℓ (e 0 s ). A labelling e of Ξ determines a (possibly empty) cone (11) C e := {x ∈ t * | ∀ s ∈ S, x, e s ≥ 0 and S x ∈ Φ Ξ },…”

Section: Karshon and Lermanmentioning

confidence: 99%

“…The authors are grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, the London Mathematical Society, and the Labex CIMI (Toulouse) for hospitality and financial support. They would also like to thank D. Dragnev for pointing out to them the references [6,11], and R. Ponge for drawing their attention to fat distributions in subriemannian geometry.…”

Section: Introductionmentioning

confidence: 99%