Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Usher, Michael

doi:10.2140/gt.2011.15.1313

Cited by 31 publications

(25 citation statements)

References 51 publications

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“…Thus, |u ⊗ e A | = |u| + I c1 (A), where u ∈ H * (M ) and A ∈ Γ. The isomorphism between HF * (H) and HQ * (M ) is defined via the PSS-homomorphism; see [PSS] or [MS,U2]. Alternatively, it can be obtained from a homotopy of H to an autonomous C 2 -small Hamiltonian (under slightly more restrictive conditions than weak monotonicity, [HS]) or with a somewhat different definition of the total Floer homology (as the limit of HF (a, b) * (H) as a → −∞ and b → ∞, [On]).…”

Section: We Set Hfmentioning

confidence: 99%

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Action–index relations for perfect Hamiltonian diffeomorphisms

Chance

Ginzburg

Gürel

2013

Journal of Symplectic Geometry

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We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic requirement satisfied for projective spaces, Grassmannians and many other manifolds. We also refine a previous result on the Conley conjecture for negative monotone symplectic manifolds, due to the second and third authors, and show that a Hamiltonian diffeomorphism of such a manifold must have simple periodic orbits of arbitrarily large period whenever its fixed points are isolated.

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Section: We Set Hfmentioning

confidence: 99%

“…Finally note that for the triangle inequality to hold one has to work with a suitable definition of the pair-of-pants product in Floer homology; cf. [AS,U2]. We refer the reader to [U2] for a very detailed treatment of action selectors.…”

Section: We Set Hfmentioning

confidence: 99%

Action–index relations for perfect Hamiltonian diffeomorphisms

Chance

Ginzburg

Gürel

2013

Journal of Symplectic Geometry

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“…The invariantĉ ∞ (H) is closely related to Calabi quasi-morphisms; see [EP,McD10,Os,U11]. For us, however,ĉ ∞ is of interest because of its role in the proofs of the main theorems.…”

Section: Local Floer Homologymentioning

confidence: 99%

Conley Conjecture Revisited

Ginzburg

Gürel

2017

International Mathematics Research Notices

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Abstract. We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class).The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also show that for the iterations of a Hamiltonian diffeomorphism with finitely many periodic orbits the sequence of action gaps between the "largest" and the "smallest" spectral invariants remains bounded and, as a consequence, establish some new cases of the C ∞ -generic existence of infinitely many simple periodic orbits.

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“…Starting from [12], this "anti-Gleason phenomenon" in classical mechanics has been established for various manifolds, including complex projective spaces and their products, toric manifolds, blow ups and coadjoint orbits [32,40,15,6].…”

Section: From Quantum Indeterminism To Quasi-statesmentioning

confidence: 99%

Quantum Footprints of Symplectic Rigidity

Polterovich¹

2016

EMS Newsl.

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In this note, we discuss an interaction between symplectic topology and quantum mechanics. The interaction goes in both directions. On one hand, ideas from quantum mechanics give rise to new notions and structures on the symplectic side and, furthermore, quantum mechanical insights lead to useful symplectic predictions when topological intuition fails. On the other hand, some phenomena discovered within symplectic topology admit a meaningful translation into the language of quantum mechanics, thus revealing quantum footprints of symplectic rigidity. What is . . . symplectic?A symplectic manifold is an even-dimensional manifold M 2n equipped with a closed differential 2-form ω that can be written as n i=1 dp i ∧ dq i in appropriate local coordinates (p, q). For an oriented surface Σ ⊂ M, the integral Σ ω plays the role of a generalised area, which, in contrast to the Riemannian area, can be negative or vanish.To have some interesting examples in mind, think of surfaces with an area form and their products, as well as complex projective spaces equipped with the Fubini-Study form, and their complex submanifolds.Symplectic manifolds model the phase spaces of systems of classical mechanics. Observables (i.e. physical quantities such as energy, momentum, etc.) are represented by functions on M. The states of the system are encoded by Borel probability measures µ on M. The simplest states are given by the Dirac measure δ z concentrated at a point z ∈ M.The laws of motion are governed by the Poisson bracket, a canonical operation on smooth functions on M, given by. The evolution of the system is determined by its energy, a time-dependent function f t : M → R called its Hamiltonian. Hamilton's famous equation describing the motion of a system is given, in the Heisenberg picture, byġ t = { f t , g t }, where g t = g • φ −1 t stands for the time evolution of an observable function g on M under the Hamiltonian flow φ t . The maps φ t are called Hamiltonian diffeomorphisms. They preserve the symplectic form ω and constitute a group with respect to composition.In the 1980s, new methods, such as Gromov's theory of pseudo-holomorphic curves and the Floer-Morse theory on loop spaces, gave birth to "hard" symplectic topology. It detected surprising symplectic rigidity phenomena involving symplectic manifolds, their subsets and diffeomorphisms. A number of recent advances show that there is yet another manifestation of symplectic rigidity taking place in function spaces associated to a symplectic manifold. Its study forms the subject of function theory on symplectic manifolds, a rapidly evolving area whose development has led to the interactions with quantum mechanics described below. 2The non-displaceable fiber theoremIn 1990, Hofer [21] introduced an intrinsic "small scale" on a symplectic manifold: a subset X ⊂ M is called displaceable if there exists a Hamiltonian diffeomorphism φ such that φ(X) ∩ X = ∅.

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Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Cited by 31 publications

References 51 publications

Action–index relations for perfect Hamiltonian diffeomorphisms

Action–index relations for perfect Hamiltonian diffeomorphisms

Conley Conjecture Revisited

Quantum Footprints of Symplectic Rigidity

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