Stability of closed characteristics on symmetric compact convex hypersurfaces in ${\mathbb{R}}^{2n}$

In this article, let $\mathrm{\Sigma }\subset {\mathbb{R}}^{2n}$ be a compact convex hypersurface which is symmetric with respect to the origin. We prove that if $\text{Sg}$ carries finitely many geometrically distinct closed characteristics, then at least $n-1$ of them must be non-hyperbolic; if $\text{Sg}$ carries exactly $n$ geometrically distinct closed characteristics, then at least two of them must be elliptic.