Hyperbolic characteristics on star-shaped hypersurfaces

In this paper, we study the stability of closed characteristics on a starshaped compact smooth hypersurface Σ in ²ⁿ. We show that the Maslov-type mean index of such a closed characteristic is independent of the choice of the Hamiltonian functions, and prove that on Σ either there are infinitely many closed characteristics, or there exists at least one nonhyperbolic closed characteristic, provided every closed characteristic possesses its Maslov-type mean index greater than 2 when n is odd, and greater than 1 when n is even.