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

In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex \begin{document}$ \mathbb{R}P^{2n-1} $\end{document} , and if all the closed Reeb orbits are non-degenerate, then there are at least \begin{document}$ n $\end{document} closed Reeb orbits, where \begin{document}$ n\geq2 $\end{document} , the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in \begin{document}$ {{\bf R}}^{2n} $\end{document} 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.