Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces

We prove that every smooth Riemannian metric on the underlying smooth manifold of a simply connected compact rank-one symmetric space, namely $S^n$ $(n\ge2)$, $\CP^m$ $(m\ge1)$, $\HP^m$ $(m\ge1)$, or $\OP^2$ admits infinitely many geometrically distinct prime closed geodesics. Here a prime closed ge