Abstract. We consider the set of maps f ∈ F α+ = ∪ β>α C 1+β of the circle which are covering maps of degree D, expanding, min x∈S 1 f (x) > 1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f -invariant probability measure µ minimizing ln f dµ over all f -invariant probability measures. We show there exists a set G + ⊂ F α+ , open and dense in the C 1+α -topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C 1+α -topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8].We will also present some results on the set of f -invariant measures µ maximizing A dµ for a fixed C 1 -expanding map f and a general potential A, not necessarily equal to −ln f .

Let L be a convex superlinear Lagrangian on a closed connected manifold N . We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value of the lift of L to a covering of N equals the infimum of the values of k such that the energy level k bounds an exact Lagrangian graph in the cotangent bundle of the covering. As a consequence, we show that up to reparametrization, the dynamics of the Euler-Lagrange flow of L on an energy level that contains supports of minimizing measures with non-zero rotation vector can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of L on the energy level k is Anosov, then k must be strictly bigger than the critical value c u (L) of the lift of L to the universal covering of N . It follows that given k < c u (L), there exists a potential ψ with arbitrarily small C 2 -norm such that the energy level k of L + ψ possesses conjugate points. Finally we show the existence of weak KAM solutions for coverings of N and we explain the relationship between Fathi's results in [F1,2] and Mañé's critical values and action potentials.

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Mañé's critical value of the lift of the Lagrangian to the universal cover, c u (L), almost all energy levels have conjugate points.We in addition prove that if an energy level is of contact type, projects onto M and M = T 2 , then the free time action functional of L + k satisfies the PalaisSmale condition.

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.

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