HAMILTONIAN SYSTEMS WITH POSITIVE TOPOLOGICAL ENTROPY AND CONJUGATE POINTS

In this article, we consider the geodesic flows induced by the natural Hamiltonian systems H (x, p) = 1/2g(ij) (x)p(i)p(j) + V (x) defined on a smooth Riemannian manifold (M = S-1 x N, g), where S-1 is the one dimensional torus, N is a compact manifold, g is the Riemannian metric on M and V is a potential function satisfying V <= 0. We prove that under suitable conditions, if the fundamental group pi(1)(N) has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric (h - V)g, i.e., (M, (h - V)g), is a manifold with conjugate points for all h with 0 < h < delta, where (5 is a small number.