AUBRY-MATHER MEASURES IN THE NON CONVEX SETTING F. CAGNETTI, D. GOMES, AND H.V. TRAN Abstract. The adjoint method, introduced in [Eva] and [Tra], is used to construct analogs to the Aubry-Mather measures for non convex Hamiltonians. More precisely, a general construction of probability measures, that in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semi- definite matrix of Borel measures. However, in the important case of uniformly qua- siconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed. Keywords: Aubry-Mather theory, weak KAM, non convex Hamiltonians, adjoint method. 1. Introduction Let us consider a periodic Hamiltonian system whose energy is described by a smooth Hamil- tonian H : T n × R n → R. Here T n denotes the n-dimensional torus, n ∈ N. It is well known that the time evolution t → (x(t), p(t)) of the system is obtained by solving the Hamilton’s ODE        ˙ x = −D p H (x, p), ˙ p = D x H (x, p). (1.1) Assume now that, for each P ∈ R n , there exists a constant H (P ) and a periodic function u(·,P ) solving the following time independent Hamilton-Jacobi equation H (x,P + D x u(x,P )) = H (P ). (1.2) Suppose, in addition, that both u(x,P ) and H (P ) are smooth functions. Then, if the following relations X = x + D P u(x,P ), p = P + D x u(x,P ), (1.3) define a smooth change of coordinates X (x,p) and P (x,p), the ODE (1.1) can be rewritten as        ˙ X = −D P H (P), ˙ P =0. (1.4) 1