Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in $$\mathbf{R}^8$$ R 8

Liu, Hui; Ye, Long; Wang, Wei; Zhang, Ping’an

doi:10.1007/s40304-015-0047-0

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…invariant under the symmetry). In low dimensional cases, this result is proved in [22,30] for a particular symmetry, but without the nondegeneracy assumption.…”

Section: Introductionmentioning

confidence: 84%

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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“…invariant under the symmetry). In low dimensional cases, this result is proved in [22,30] for a particular symmetry, but without the nondegeneracy assumption.…”

Section: Introductionmentioning

confidence: 84%

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 all ∈ SH con (2n) with n = 2 or 3 provided # T ( ) = n. The second result is that (1.2) holds for all ∈ SH con (8) provided # T ( ) = 4 proved by Liu, Long, Wang and Zhang recently in [18]. But in general, whether (1.2) holds for every ∈ SH st (2n) is still open.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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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Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in $$\mathbf{R}^8$$ R 8

Abstract: Let be a C 3 compact symmetric convex hypersurface in R 8 . We prove that when carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric. Due to the example of weakly non-resonant ellipsoids, our result is sharp.

Cited by 2 publications

References 19 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

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