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

Gutt, Jean; Kang, Jungsoo

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2485-2505.

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 étoil�e 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.

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.

Classification : 53D10, 37J55
Keywords: Reeb dynamics, Equivariant symplectic homology, Index jump
Mot clés : Dynamique de Reeb, Homologie symplectique équivariante, Saut d’indice

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