Densely the 2-sphere Has an Elliptic Closed Geodesic

We prove that a riemannian metric on the 2-sphere or the projective plane can be C 2 approximated by a C ∞ metric whose geodesic flow has an elliptic closed geodesic. In this paper we show how to overcome a difficulty presented by Henri Poincaré in the C 2 generic case. In 1905, H. Poincaré [34, p. 259] claimed that any convex surface in R 3 should have an elliptic or degenerate non self-intersecting closed geodesic. This is, the linearized Poincaré map of the geodesic flow at the closed geodesic has an eigenvalue of modulus 1. In 1980, A. Grjuntal [13] showed a counterexample to Poincaré's claim. Victor Donnay [9], [10] constructs an example of a C ∞ riemannian metric on the 2-sphere S 2 which has positive metric entropy and all whose closed geodesics but a finite number (which are degenerate) are hyperbolic. Donnay's theorem is not known in positive curvature. It is also not known if there exists a C ∞ riemannian metric on S 2 all of whose closed geodesics are hyperbolic. Here we prove Theorem A. A riemannian metric on the 2-sphere or the projective plane can be C 2 approximated by a C ∞ metric with an elliptic closed geodesic. In August 2000 at Rio de Janeiro's dynamical systems conference, Michel Herman [15] conjectured the above result and announced a proof of it in the case of positive curvature. His proof used, as we shall also do, Ricardo Mañé's theory on dominated splittings adapted to the geodesic flow and also an equivariant version of Brouwer's translation theorem. We shall see that in the positive curvature case one can replace the use of Brouwer's translation theorem by the intermediate value theorem on the interval. For the non convex case we use Hofer, Wysocki, Zehnder theory on Reeb flows for generic tight contact forms on the 3-sphere S 3. Theorem A is a version for geodesic flows of a theorem by Sheldon Newhouse [31]. In 1977, Newhouse proved that if H is a smooth hamiltonian on a symplectic manifold, 0 is a regular value for H and the energy level H −1 {0} is compact, then there is a C 2 perturbation H 1 of H such that the hamiltonian flow on H −1 1 {0} is either Anosov or it has an elliptic closed orbit. But Newhouse's arguments heavily rely on a C 2 closing lemma …