Printed in the United Kingdom c © 1999 Cambridge University Press Convex Hamiltonians without conjugate points

Abstract. We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit. We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps. Let M be a closed connected Riemannian manifold and T M its cotangent bundle. By a convex Hamiltonian on T M we shall mean a C3 function H V T M! R satisfying the following conditions