Poisson brackets and symplectic invariants

“…The motivation for this function comes from Aubry-Mather theory where it was discovered by Contreras-Iturriaga-Paternain [19, Corollary 1] that the Mather -function, associated to a convex Hamiltonian on the cotangent bundle of a closed manifold, can be defined in a similar way. The proof of Theorem 27 relies heavily on beautiful ideas due to Buhovsky-Entov-Polterovich [15], Entov-Polterovich [22] and Polterovich [40]. Of course this result is only useful if we can find reasonable lower bounds for ∶ .…”

Section: S "M "mentioning

confidence: 99%

Mather theory and symplectic rigidity

Bisgaard¹

2019

Journal of Modern Dynamics

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A. Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow of a Hamiltonian ∈ ∞ ( ) on a symplectic manifold ( , ). These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when fails to be convex, even for nearly integrable . Parts of these results extend work by Viterbo [54], and Vichery [52].Using ideas due to , [40] we also detect interesting invariant measures for by studying a generalization of the symplectic shape of sublevel sets of . This approach differs from the first one in that it works also for ( , ) in which every compact subset can be displaced. We present applications to Hamiltonian systems on ℝ 2 and twisted cotangent bundles.

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“…Let us assume α = 0. We will prove the inequality bp L (α) ≤ def L (α) by a method developed in [4] (cf. [14]).…”

Section: The Invariant Bp L For General Lagrangian Submanifoldsmentioning

confidence: 99%

“…Cut S 1 into four consecutive arcs, denote their preimages under f by X 0 , Y 1 , X 1 , Y 0 (so that X 0 ∩ X 1 = Y 0 ∩ Y 1 = ∅, X 0 ∪ Y 1 ∪ X 1 ∪ X 0 = L) and set bp L (a) := 1/pb + 4 (X 0 , X 1 , Y 0 , Y 1 ). Here pb + 4 is the Poisson-bracket invariant of a quadruple of sets defined in [14] -it is a refined version of the pb 4 -invariant introduced in [4] and it admits a dynamical interpretation in terms of the existence of connecting trajectories of sufficiently small time-length between X 0 and X 1 for certain Hamiltonian flows -see Section 3 for details. The relation between bp L and def L is given by the following inequality (which will be proved in a stronger form in Theorem 3.5).…”

Section: Introductionmentioning

confidence: 99%

Lagrangian isotopies and symplectic function theory

Entov

Ganor

Membrez

2018

Comment. Math. Helv.

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We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus.The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy.The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation.We partially compute these invariants for certain Lagrangian tori.

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Poisson brackets and symplectic invariants

Cited by 28 publications

References 49 publications

Lagrangian tetragons and instabilities in Hamiltonian dynamics

Lagrangian tetragons and instabilities in Hamiltonian dynamics

Mather theory and symplectic rigidity

Lagrangian isotopies and symplectic function theory

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