Action minimizing invariant measures for positive definite Lagrangian systems

In recent years, several authors have studied "minimal" orbits of Hamiltonian systems in two degrees of freedom and of area preserving monotone twist diffeomorphisms. Here, "minimal" means action minimizing. This class of orbits has many interesting properties, as may be seen in the survey article of Bangert [4]. It is natural to ask if there is any generalization of this class of orbits to Hamiltonian systems in more degrees of freedom. In this article, we propose a generalization to periodic Hamiltonian systems in more degrees of freedom. However, we generalize not the notion of minimal orbit, but the closely related notion of minimal measure, which we introduced in [18]. We obtain two basic results here: an existence theorem for minimal measures, and a regularity theorem which asserts that the minimal measures can be expressed as (partially defined) Lipschitz sections of the tangent bundle. In the sort of generalization that we do here, a major difficulty is finding the right setting. The setting which we propose here has two important features: the results are valid for periodic positive definite Lagrangian systems, and the results are formulated in terms of invariant measures. I am indebted to J. Moser for pointing out to me several years ago that periodic positive definite Lagrangian systems in one degree of freedom provide a setting in which it is possible to formulate results which generalize both the author's results [17] (and the closely related results of Aubry and Le Dacron [1]) and the results of Hedlund [12] concerning "class A " geodesics on a Riemannian manifold diffeomorphic to the 2-torus. Indeed, Moser has proved [20] that every twist diffeomorphism is the time one map associated to a suitable periodic positive definite Lagrangian system. Denzler [10] has carried out Moser's program in one degree of freedom. This remark of Moser suggested to me that periodic positive definite Lagrangian systems should provide the right setting in more degrees of freedom. There is some earlier work in the direction of this paper. Bernstein and Katok [6] obtained results concerning periodic orbits near invariant tori, using a variational method related to the variational method of this paper.