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| In this article, we consider the natural Hamiltonian systems $H(x,p)= rac{1}{2}g^{ij}(x)p_{i}p_{j}+V(x)$ defined on a smooth Riemannian manifold $(M = S^{1} imes N, g)$, 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 $V leq 0$. We prove that under suitable conditions, if the system has positive topological entropy and 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 $delta$ is a small number. |
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| Keywords:Dynamical systems; topological entropy; fundamental group; conjugate points |
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