# ANNALES DE L'INSTITUT FOURIER

On the minimal number of periodic orbits on some hypersurfaces in ${ℝ}^{2n}$
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, p. 2485-2505
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 [24], 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.
Nous étudions les orbites périodiques du champ de Reeb sur les hypersurfaces non-dégénérées et dynamiquement convexes de ${ℝ}^{2n}$ en suivant les travaux de Long et Zhu mais en utilisant l’homologie symplectique ${S}^{1}$-équivariante. Nous démontrons qu’il existe au moins $n$ orbites simples de Reeb sur toute hypersurface étoilï¿½e et non dégénérée de ${ℝ}^{2n}$ satisfaisant la condition que le plus petit indice de Conley–Zehnder est au moins $n-1$. Cette dernière condition est plus faible que celle de convexité dynamique.
Revised : 2016-02-04
Accepted : 2016-03-24
Published online : 2016-10-04
DOI : https://doi.org/10.5802/aif.3069
Classification:  53D10,  37J55
Keywords: Reeb dynamics, Equivariant symplectic homology, Index jump
@article{AIF_2016__66_6_2485_0,
author = {Gutt, Jean and Kang, Jungsoo},
title = {On the minimal number of periodic orbits on some hypersurfaces in $\mathbb{R}^{2n}$},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {66},
number = {6},
year = {2016},
pages = {2485-2505},
doi = {10.5802/aif.3069},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_2016__66_6_2485_0}
}

On the minimal number of periodic orbits on some hypersurfaces in $\mathbb{R}^{2n}$. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2485-2505. doi : 10.5802/aif.3069. https://aif.centre-mersenne.org/item/AIF_2016__66_6_2485_0/

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