Fragility and persistence of leafwise intersections

Ginzburg, Viktor L.; Gürel, Başak Z.

doi:10.1007/s00209-015-1459-y

Cited by 4 publications

(4 citation statements)

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“…Since then various forms or generalizations of it were studied. See [19,16,25], [1, §1.1] and [34, §1.4] and references therein for the brief history of these problems.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark 1.30. Recently, Ginzburg and Gürel [25] showed that there exists a closed, smooth hypersurface S ⊂ R 2n , 2n ≥ 4, and a sequence of C ∞ -smooth autonomous Hamiltonian F k → 0 in C 0 , supported in the same compact set, such that S and ϕ F k (S) have no leafwise intersections. It suggests that Theorem 1.29 is best possible in one sense.…”

Section: Applications To Hamiltonian Dynamicsmentioning

confidence: 99%

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Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

Jin¹,

Lu²

2019

Preprint

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This is the first installment in a series of papers aimed at generalizing symplectic capacities and homologies. The main purposes of this paper are to construct analogues of Ekeland-Hofer and Hofer-Zehnder symplectic capacities based on a class of Hamiltonian boundary value problems motivated by Clarke's and Ekeland's work, and to study generalizations of some important results about the original these two capacities (for example, the famous Weinstein conjecture, representation formula for c EH and c HZ , a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).

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“…Since then various forms or generalizations of it were studied. See [19,16,25], [1, §1.1] and [34, §1.4] and references therein for the brief history of these problems.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Applications To Hamiltonian Dynamicsmentioning

confidence: 99%

Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

Jin¹,

Lu²

2019

Preprint

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“…V. L. Ginzburg and B. Z. Gürel [GG,Theorem 1.1] recently proved that for n ≥ 2 there exists a closed smooth hypersurface N ⊆ R 2n and a compact subset K ⊆ R 2n , such that for every ε > 0 there exists a smooth function…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

Ziltener

2017

International Mathematics Research Notices

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Consider a closed coisotropic submanifold N of a symplectic manifold (M, ω) and a Hamiltonian diffeomorphism ϕ on M . The main result of this article states that ϕ has at least the cup-length of N many leafwise fixed points w.r.t. N , provided that it is the time-1-map of a global Hamiltonian flow whose restriction to N stays C 0 -close to the inclusion N → M . If (ϕ, N ) is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of N . The nondegeneracy condition is generically satisfied.This appears to be the first leafwise fixed point result in which neither ϕ N is assumed to be C 1 -close to the inclusion N → M , nor N to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the C 0condition on ϕ cannot be replaced by the assumption that ϕ is Hofer-small.

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“…The To see that this operator is well-defined, recall that it is defined on the direct sum of Z 2 's indexed by the set occurring on the right hand side of (6). 10 Using the 7 By definition, for every such point x 0 , the point x(1) = ϕ 1 (x 0 ) lies in the isotropic leaf of x 0 . Hence x 0 is a leafwise fixed point of ϕ 1 .…”

mentioning

confidence: 99%

Note on coisotropic Floer homology and leafwise fixed points

Ziltener

2021

era

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For an adiscal or monotone regular coisotropic submanifold N of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of N . Given a Hamiltonian isotopy ϕ = (ϕ t ) and a suitable almost complex structure, the corresponding Floer chain complex is generated by the (N, ϕ)-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.1. Introduction. Consider a symplectic manifold (M, ω), a coisotropic submanifold N ⊆ M , and a Hamiltonian diffeomorphism ψ : M → M . The isotropic (or characteristic) distribution T N ω on N gives rise to the isotropic foliation on N . A leafwise fixed point for ψ is a point x ∈ N for which ψ(x) lies in the leaf through x of this foliation. We denote by Fix(ψ, N ) the set of such points. A fundamental problem in symplectic geometry is the following: Problem. Find conditions under which Fix(ψ, N ) is non-empty and find lower bounds on its cardinality.This generalizes the problems of showing that a given Hamiltonian diffeomorphism has a fixed point and that a given Lagrangian submanifold intersects its image under a Hamiltonian diffeomorphism. References for solutions to the general problem are provided in [20,22].Example (translated points). As explained in [19, p. 97], translated points of the time-1-map of a contact isotopy starting at the identity are leafwise fixed points of the Hamiltonian lift of this map to the symplectization.

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Fragility and persistence of leafwise intersections

Cited by 4 publications

References 31 publications

Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

Note on coisotropic Floer homology and leafwise fixed points

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