From one Reeb orbit to two

Cristofaro-Gardiner, Daniel; Hutchings, Michael

doi:10.4310/jdg/1452002876

Cited by 59 publications

(52 citation statements)

References 48 publications

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“…This question has been studied in depth in the lowest dimensional case, in which the question is nontrivial, i.e. for a hypersurface Σ ⊂ R 4 in [8,12,13,17,18,19,21]. It turns out that, in this case, (Σ, α) carries at least two simple periodic orbits and if there are more than two simple periodic orbits, infinitely many of them are guaranteed generically.…”

Section: Introductionmentioning

confidence: 99%

On the minimal number of periodic orbits on some hypersurfaces in \mathbb{R}^{2n}

Gutt¹,

Kang²

2016

Ann. inst. Fourier

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We study periodic orbits of the Reeb vector field on a nondegenerate dynamically convex starshaped hypersurface in R 2n along the lines of Long and Zhu [LZ02], but using properties of the S 1 -equivariant symplectic homology. We prove that there exist at least n distinct simple periodic orbits on any nondegenerate starshaped hypersurface in R 2n satisfying the condition that the minimal Conley-Zehnder index is at least n − 1. The condition is weaker than dynamical convexity. * This paper was written while JG was a postdoc at UC Berkeley under the supervision of Michael Hutchings.

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Section: Introductionmentioning

confidence: 99%

On the minimal number of periodic orbits on some hypersurfaces in \mathbb{R}^{2n}

Gutt¹,

Kang²

2016

Ann. inst. Fourier

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“…For further results on the multiplicity of closed characteristics on ∈ H con (2n) or H st (2n), we refer to [7,8,13,14,23,26,30,32] as well as [22]. Recently # T ( ) ≥ 2 was first proved for every ∈ H st (4) by CristofaroGardiner and Hutchings [3] without any pinching or non-degeneracy conditions. Different proofs of this result can be found in [9,10,16].…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

Liu

2015

Calc. Var.

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So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface in R 2n are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface in R 2n when there exist only finitely many geometrically distinct symmetric closed characteristics on , which extend the identity established by Liu and Long (J Differ Equ 255:2952-2980, 2013) of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu et al. (J Funct Anal 166:5598-5638, 2014) for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in R 4 , then both of them must be elliptic.

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“…Another application of Theorem 4.2 considered in Ginzburg et al (2013) (and also in Ginzburg and Gören 2015;Liu and Long 2013) is the following result originally proved in Cristofaro-Gardiner and Hutchings (2012).…”

Section: Then the Reeb Flow Of α Has Infinitely Many Simple Periodic mentioning

confidence: 95%

“…Theorem 4.2 can also be applied to give a simple proof, based on the same idea, of the result from Bangert and Long (2010) that any Finsler geodesic flow on S 2 has at least two closed geodesics; see Ginzburg and Gören (2015). [Of course, this fact also immediately follows from Cristofaro-Gardiner and Hutchings (2012).] Interestingly, no multiplicity results along the lines of Theorem 4.3 have been proved in higher dimensions without restrictive additional assumptions on the contact form.…”

Section: Theorem 43mentioning

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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From one Reeb orbit to two

Cited by 59 publications

References 48 publications

On the minimal number of periodic orbits on some hypersurfaces in \mathbb{R}^{2n}

On the minimal number of periodic orbits on some hypersurfaces in \mathbb{R}^{2n}

Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

The Conley Conjecture and Beyond

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