On the minimal number of periodic orbits on some hypersurfaces in 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 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 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 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 étoil�e et non dégénérée de 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.

Received : 2015-09-01
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
     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 = {http://www.numdam.org/item/AIF_2016__66_6_2485_0}
Gutt, Jean; Kang, Jungsoo. 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. http://www.numdam.org/item/AIF_2016__66_6_2485_0/

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