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.
International audienceWe consider a continuous coercive Hamiltonian H on the cotangent bundle of the compact connected manifold M which is convex in the momentum. If uλ:M→ℝ is the viscosity solution of the discounted equationλuλ(x)+H(x,dxuλ)=c(H),where c(H) is the critical value, we prove that uλ converges uniformly, as λ→0, to a specific solution u0:M→ℝ of the critical equationH(x,dxu)=c(H).We characterize u0in terms of Peierls barrier and projected Mather measures. As a corollary, we infer that the ergodic approximation, as introduced by Lions, Papanicolaou and Varadhan in 1987 in their seminal paper on homogenization of Hamilton–Jacobi equations, selects a specific corrector in the limit
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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