Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

Rosenberg, Maor

doi:10.1007/s11856-010-0099-5

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…In [11], Py constructed a remarkable functional on the set of smooth functions on , which, as was shown by Rosenberg [12], is (the restriction of) a quasi-state. We call it the Py-Rosenberg quasi-state.…”

Section: Introduction and Resultsmentioning

confidence: 89%

“…Let F ∈ C ∞ (M) be a generic Morse function. There is a notion of an essential critical point of F. Instead of giving its formal definition, let us note that from the work of Rosenberg [12] it follows that α is the critical value of F corresponding to an essential critical point if and only if for all ε > 0 small enough ({F ≤ α + ε}) differs from ({F ≤ α − ε}) by one pair-of-pants. There are 2g − 2 essential critical points.…”

Section: Py-rosenberg Topological Measuresmentioning

confidence: 99%

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Isotopy-invariant topological measures on closed orientable surfaces of higher genus

Zapolsky

2010

Math. Z.

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Given a closed orientable surface of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on and the convex compact set of additive functions on the set of isotopy classes of certain subsurfaces of . We then construct such additive functions, and thus isotopy-invariant topological measures, from probability measures on together with some additional data. The map associating topological measures to probability measures is affine and continuous. Certain Dirac measures map to simple topological measures, while the topological measures due to Py and Rosenberg arise from the normalized Euler characteristic.

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Section: Introduction and Resultsmentioning

confidence: 89%

Section: Py-rosenberg Topological Measuresmentioning

confidence: 99%

Isotopy-invariant topological measures on closed orientable surfaces of higher genus

Zapolsky

2010

Math. Z.

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“…Let us also note that in dimension dim M = 2 there exist alternative constructions of symplectic quasi-states (see e.g. [1], [4], [19], [20], [17]) which do not involve Floer homology. None of those quasi-states is known to be induced by a stable quasi-morphism.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

“…None of those quasi-states is known to be induced by a stable quasi-morphism. For instance, it is shown in [17] that Py's quasi-morphism [15] gives rise to a quasi-state, but it is unknown whether this quasi-morphism is stable. On the other hand, Zapolsky [19,20] proved that for a wide class of quasi-states ζ on surfaces one has inequality…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

Poisson brackets, quasi-states and symplectic integrators

Entov¹,

Polterovich²,

Rosen³

2010

Discrete &Amp; Continuous Dynamical Systems - A

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This paper is a fusion of a survey and a research article. We focus on certain rigidity phenomena in function spaces associated to a symplectic manifold. Our starting point is a lower bound obtained in an earlier paper with Zapolsky for the uniform norm of the Poisson bracket of a pair of functions in terms of symplectic quasi-states. After a short review of the theory of symplectic quasi-states, we extend this bound to the case of iterated Poisson brackets. A new technical ingredient is the use of symplectic integrators. In addition, we discuss some applications to symplectic approximation theory and present a number of open problems.

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“…ζ(f • φ) = ζ(f ) for any diffeomorphism φ. Building on the works of Py [37,38], Zapolsky (in [50]) and Rosenberg (in [39]) constructed genuine (and not partial) quasistates on the torus and surfaces of genus higher than one, respectively. Like ζ, both of these quasi-states can be described by simple formulas which are different than the formula for ζ.…”

Section: Further Consequencesmentioning

confidence: 99%

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

2016

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Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces [25,26], we introduce a dynamical invariant, denoted by N , for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces.Along the way, we obtain several results of independent interest: We show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with N on autonomous Hamiltonians thus establishing a certain uniqueness result for spectral invariants, we obtain a "Max Formula" for spectral invariants on aspherical manifolds, give a very simple description of the Entov-Polterovich quasi-state on aspherical surfaces and characterize the heavy and super-heavy subsets of such surfaces.Formal Spectral Invariants: Although the following definition makes sense on any symplectic manifold we will restrict our attention here to the case of a surface Σ which is either the plane R 2 or is closed and aspherical. Definition 3. A function c : C ∞ ([0, 1] × Σ) → R is a formal spectral invariant if it satisfies the following four axioms: 1. (Spectrality) c(H) ∈ spec(H) for all H ∈ C ∞ ([0, 1]×Σ), where spec(H), the spectrum of H, is the set of critical values of the Hamiltonian action, that is, the set of actions of fixed points of φ 1 H .

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Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

Abstract: We show that Py-Calabi quasi-morphism on the group of Hamiltonian diffeomorphisms of surfaces of higher genus gives rise to a quasistate.

Cited by 7 publications

References 21 publications

Isotopy-invariant topological measures on closed orientable surfaces of higher genus

Isotopy-invariant topological measures on closed orientable surfaces of higher genus

Poisson brackets, quasi-states and symplectic integrators

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

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