Let c be an isolated closed geodesic of length L on a compact Riemannian manifold M which is homologically visible in the dimension of its index, and for which the index of the iterates has the maximal possible growth rate. We show that M has a sequence {cn}, n E 2+, of prime closed geodesics of length mnL en where mn E Z and En J 0. The hypotheses hold in particular when M is a two-sphere and the shortest Lusternik-Schnirelmann closed geodesic c is isolated and Let M be a compact Riemannian manifold of dimension n. For a C R let Aa be the space of H1 closed curves on M of energy mA + (m-1)(n-1) for m > 1. Then there is an mO C 2+ and a sequence am I 0 so that if m > mo, and if m is odd (or if n and A have different parity), M has a closed geodesic Ym with length C (mL am, mL). It follows that M has infinitely many closed geodesics. Corollary. Suppose M is a two-sphere; suppose the shortest Lusternik-Schnirelmann closed geodesic c is isolated and Then the above conclusion holds (with A = 1). Background. See [2, 3, 9] for a more complete discussion. When M is a twosphere, the theorem of Lusternik and Schnirelmann [6, 11] gives three simple (i.e. embedded) closed geodesics. If e.g. M has positive curvature, then associated to each simple closed geodesic c is the Birkhoff map B. This is an area-preserving self-map of the closed annulus which describes the geodesic flow on M [3]. Interior periodic points of B give closed geodesics on M. An elementary and well-known argument, which some say is due to Birkhoff, shows that B has infinitely many periodic orbits unless c is nonrotating, i.e. unless there is a point on c whose second conjugate point along c occurs after exactly one circuit about c [3, 9, 13]. For each simple closed geodesic c on a two-sphere M we have three cases: (i) B is defined, c not nonrotating. Received by the editors April 2, 1996. 1991 Mathematics Subject Classification. Primary 58E10; Secondary 53C22. (?)1997 Arrierican Matherrmatical Society