New topological measures on the torus

Knudsen, Finn F.

doi:10.4064/fm185-3-6

Cited by 6 publications

(7 citation statements)

References 4 publications

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“…It would be interesting to describe all symplectic quasi-measures on the 2-torus; for more examples of such quasi-measures see a recent preprint [28] by Knudsen.…”

Section: Problem 82 Extend Identity (9) To Poisson-commuting Functimentioning

confidence: 99%

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Quasi-states and symplectic intersections

Entov¹,

Polterovich²

2006

Comment. Math. Helv.

136

257

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We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.

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“…It would be interesting to describe all symplectic quasi-measures on the 2-torus; for more examples of such quasi-measures see a recent preprint [28] by Knudsen.…”

Section: Problem 82 Extend Identity (9) To Poisson-commuting Functimentioning

confidence: 99%

“…Remark 8.6. It would be interesting to describe all symplectic quasi-measures on the 2-torus; for more examples of such quasi-measures see a recent preprint [28] by Knudsen. By Theorem 8.1 above, a symplectic quasi-measure on T 2 gives rise to a symplectic quasi-state.…”

Section: Symplectic Quasi-states On Surfacesmentioning

confidence: 99%

Quasi-states and symplectic intersections

Entov¹,

Polterovich²

2006

Comment. Math. Helv.

136

257

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“…The bulk of the constructed F. Zapolsky (B) Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany e-mail: zapolsky@mis.mpg.de topological measures is on spaces of Aarnes genus 0 (see [3] for the definition), which in the case of compact CW complexes is equivalent to vanishing first integral cohomology, and so the only orientable closed surface with this property is the sphere S 2 . Apart from that, there are the works of Grubb [7] for general spaces of Aarnes genus 1 and the works of Knudsen [9,10] concerning topological measures on the torus. In any case, there has been little to no work of constructing topological measures on surfaces of higher genus, and the present paper hopefully helps to fill the gap.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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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“…Example 2.4. Examples of simple quasi-measures on the 2-torus have been constructed by Knudsen [Kn1], [Kn2].…”

Section: Examples Of Simple Quasi-statesmentioning

confidence: 99%

Quasi-states and the Poisson bracket on surfaces

Zapolsky¹

2007

Journal of Modern Dynamics

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We present a convexity-type result concerning simple quasi-states on closed manifolds. As a corollary, an inequality emerges which relates the Poisson bracket and the measure of non-additivity of a simple quasi-state on a closed surface equipped with an area form. In addition, we prove that the uniform norm of the Poisson bracket of two functions on a surface is stable from below under C 0 -perturbations.

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New topological measures on the torus

Cited by 6 publications

References 4 publications

Quasi-states and symplectic intersections

Quasi-states and symplectic intersections

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

Quasi-states and the Poisson bracket on surfaces

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