Convex Hamiltonians without conjugate points

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 ent ropy 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 i f the Green bundles are transversal. Using the Clebsch transformation of th e index form we prove that if the unique minimizing measure of a generic La grangian 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 orb its without invariant transversal bundles, segments without conjugate point s but with crossing solutions and non-surjective exponential maps.