Hamiltonian Systems With Positive Topological Entropy And Conjugate Points

In this article,we consider the natural Hamiltonian syetems H(x,p) = 1/2lij(x)pipj+V(x) defined on a smooth Riemannian manifold(M =T1×N,l), where T1 is the one dimensional torus,N is a compact manifold, g is a Riemannian metric on M and V is a potential function satisfying V?0. We prove that under suitable conditions, if the fundamental group ?1(N) has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric (h-V)l , i.e.,(M,(h-v)l), is maniflod with conjugate points for all h with 0<h<?,where ? is a small number.