Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules

Polterovich, Leonid; Shelukhin, Egor

doi:10.1007/s00029-015-0201-2

Cited by 88 publications

(187 citation statements)

References 59 publications

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“…An interesting consequence of Theorem 2.2 pointed out in Polterovich and Shelukhin (2014) is that non-autonomous Hamiltonian diffeomorphisms (i.e., Hamiltonian diffeomorphisms that cannot be generated by autonomous Hamiltonians) on a manifold meeting the conditions of the theorem form a C ∞ -residual subset in the space of all C ∞ -smooth Hamiltonians. Indeed, when k > 1, simple k-periodic orbits of an autonomous Hamiltonian diffeomorphism are never isolated, and hence, in particular, never non-degenerate.…”

Section: Resultsmentioning

confidence: 99%

The Conley Conjecture and Beyond

Ginzburg

Gürel

2015

Arnold Math J.

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This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.

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Section: Resultsmentioning

confidence: 99%

The Conley Conjecture and Beyond

Ginzburg

Gürel

2015

Arnold Math J.

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“…The aim of this paper is to prove Theorem 1.2. In the second section, we explain the definition of spectral spread which was introduced by Polterovich and Shelukhin ( [10]) and we prove a lemma (see Lemma 2.1 below) in the full generality in the third section. Our proof of Lemma 2.1 simplifies that of Polterovich and Shelukhin.…”

Section: Resultsmentioning

confidence: 99%

“…We first prove (2). See also Proposition 5.3 in [13] and [10] where similar arguments are used. Proof of Lemma 2.1 in convex case (∂M = ∅) is the same as in the closed case.…”

Section: This Claim Implies That We Can Define Hfmentioning

confidence: 91%

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Spectral spread and non-autonomous Hamiltonian diffeomorphisms

Sugimoto

2018

manuscripta math.

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For any symplectic manifold (M, ω), the set of Hamiltonian diffeomorphisms Ham c (M, ω) forms a group and Ham c (M, ω) contains an important subset Aut(M, ω) which consists of time one flows of autonomous(timeindependent) Hamiltonian vector fields on M . One might expect that Aut(M, ω) is a very small subset of Ham c (M, ω). In this paper, we estimate the size of the subset Aut(M, ω) in C ∞ -topology and Hofer's metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement Ham c \Aut(M, ω) is a dense subset of Ham c (M, ω) in C ∞ -topology and Hofer's metric if (M, ω) is a closed symplectically aspherical manifold where Conley conjecture is established ([10]). In this paper, we generalize above theorem to general closed symplectic manifolds and general convex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms Ham c \Aut(M, ω) is a dense subset of Ham c (M, ω) in C ∞ -topology and Hofer's metric if (M, ω) is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

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“…(2) Let S be a compact oriented surface. One can easily show that there is an infinite family of egg-beater Hamiltonian diffeomorphisms {f i } ∞ i=1 of S (for definition see [24]), and a family of linearly independent quasimorphisms Ψ i ∈ Q(Diff(S, area)) which are Lipschitz with respect to the topological entropy such that Ψ i (f i ) = 0. This implies that each f i has a positive topological entropy.…”

Section: Final Remarksmentioning

confidence: 99%

Entropy and quasimorphisms

Brandenbursky¹,

Marcinkowski²

2019

Journal of Modern Dynamics

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Let S be a compact oriented surface. We construct homogeneous quasimorphisms on Diff(S, area), on Diff0(S, area) and on Ham(S) generalizing the constructions of Gambaudo-Ghys and Polterovich.We prove that there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area), on Diff0(S, area) and on Ham(S) whose absolute values bound from below the topological entropy. In case when S has a positive genus, the quasimorphisms we construct on Ham(S) are C 0 -continuous.We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on Ham(S) is unbounded.

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Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules

Cited by 88 publications

References 59 publications

The Conley Conjecture and Beyond

The Conley Conjecture and Beyond

Spectral spread and non-autonomous Hamiltonian diffeomorphisms

Entropy and quasimorphisms

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