Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $

Abstract

<p style='text-indent:20px;'>In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}P^{2n-1} $\end{document}</tex-math></inline-formula>, and if all the closed Reeb orbits are non-degenerate, then there are at least <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> closed Reeb orbits, where <inline-formula><tex-math id="M5">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in <inline-formula><tex-math id="M6">\begin{document}$ {{\bf R}}^{2n} $\end{document}</tex-math></inline-formula> to symmetric compact star-shaped hypersurfaces. In addition, we use Ekeland-Hofer theory to give a new proof of a theorem recently by M. Abreu and L. Macarini that every dynamically convex symmetric compact star-shaped hypersurface carries an elliptic symmetric closed characteristic.</p>