Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ${\bf R}^{2n}$

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface $\Sigma\subset{\bf R}^{2n}$, there exist at least $n$ non-hyperbolic closed characteristics with even Maslov-type indices on $\Sigma$ when $n$ is even. When $n$ is odd, there exist at least $n$ closed characteristics with odd Maslov-type indices on $\Sigma$ and at least $(n-1)$ of them are non-hyperbolic. Here we call a compact star-shaped hypersurface $\Sigma\subset {\bf R}^{2n}$ {\rm index perfect} if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic $(\tau,y)$ on $\Sigma$ possesses positive mean index and whose Maslov-type index $i(y, m)$ of its $m$-th iterate satisfies $i(y, m)\not= -1$ when $n$ is even, and $i(y, m)\not\in \{-2,-1,0\}$ when $n$ is odd for all $m\in {\bf N}$.