Quasi-states and the Poisson bracket on surfaces

Zapolsky, Frol

doi:10.3934/jmd.2007.1.465

Cited by 21 publications

(27 citation statements)

References 11 publications

(12 reference statements)

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“…In a recent work by one of the authors [19] this question is answered in the positive for two-dimensional symplectic manifolds using methods of the topology of surfaces.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

“…The answer is so far unknown even for the case of the quasi-state ζ from Example 1.2 on the 2-sphere. In this case Floer-theoretical Proposition 1.6 above yields C ≤ 2, while the topological argument from [19] improves this to C ≤ 1/2. …”

mentioning

confidence: 98%

“…This somewhat paradoxical situation was partially resolved in [19]: For the sphere, Theorem 1.4 is proved in [19] by "soft" methods. Moreover, an analogous theorem is proved with the sphere replaced by an arbitrary closed symplectic surface Σ, and with the quasi-state of Theorem 1.4 replaced by an arbitrary simple quasi-state (that is a quasi-state which is multiplicative on each singly generated closed subalgebra of C(Σ)).…”

mentioning

confidence: 99%

“…For instance, Zapolsky's result mentioned in the beginning of this section shows that inequality (8) stating that Υ x 2 ,y 2 ( ) ≥ (1 − 2 ) 2 /4 is not asymptotically (as → 0) sharp: indeed one readily computes that ||{x 2 , y 2 }|| ≈ 9.7 > 0.25. It would be interesting to understand whether the approach of [19] gives rise to sharp lower bounds for Υ x 2 ,y 2 ( ) on surfaces. The answer is so far unknown even for the case of the quasi-state ζ from Example 1.2 on the 2-sphere.…”

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confidence: 99%

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Quasi-morphisms and the Poisson Bracket

Entov¹,

Polterovich²,

Zapolsky³

2007

Pure and Applied Mathematics Quarterly

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For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with respect to the uniform norm. On the other hand, it serves as a measure of non-commutativity of functions in the sense of the Poisson bracket, the operation which involves first derivatives of the functions. Furthermore, the same functional gives rise to a non-trivial lower bound for the error of the simultaneous measurement of a pair of non-commuting Hamiltonians. These results manifest a link between the algebraic structure of the group of Hamiltonian diffeomorphisms and the function theory on a symplectic manifold. The above-mentioned functional comes from a special homogeneous quasi-morphism on the universal cover of the group, which is rooted in the Floer theory.

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“…In a recent work by one of the authors [19] this question is answered in the positive for two-dimensional symplectic manifolds using methods of the topology of surfaces.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Quasi-morphisms and the Poisson Bracket

Entov¹,

Polterovich²,

Zapolsky³

2007

Pure and Applied Mathematics Quarterly

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“…Entov, Polterovich and Zapolsky have improved this result by giving explicit lower bounds on Υ(F, G), in terms of quasi-states (see [2] and [16]). We may wonder whether there exist similar inequalities in the non abelian case.…”

Section: Remarkmentioning

confidence: 99%

Hamiltonian pseudo-representations

Humilière¹

2009

Comment. Math. Helv.

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The question studied here is the behavior of the Poisson bracket under C^0 -perturbations. For this purpose we introduce the notion of pseudo-representation and prove that the limit of a converging pseudo-representation of any normed Lie algebra is a representation. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C^{−1} bi-invariant metric. Our methods also provide a new proof of the Gromov–Eliashberg Theorem, which says that the group of symplectic diffeomorphisms is C^0 -closed in the group of all diffeomorphisms.

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Poisson brackets and symplectic invariants

Buhovsky

Entov

Polterovich

2011

Sel. Math. New Ser.

100

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We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementarylooking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the µ 3 -operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.MSC classes: 53Dxx, 37J05

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Quasi-states and the Poisson bracket on surfaces

Cited by 21 publications

References 11 publications

Quasi-morphisms and the Poisson Bracket

Quasi-morphisms and the Poisson Bracket

Hamiltonian pseudo-representations

Poisson brackets and symplectic invariants

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