Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in .2n

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface . . R2n, there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on . when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on . and at least (n.1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface . . R2n index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (., y) on . possesses positive mean index and whose Maslov-type index i(y,m) of its m-th iterate satisfies i(y,m) . .1 when n is even, and i(y,m) . {.2,.1, 0} when n is odd for all m . N.