  We study the evolution, under convex Hamiltonian flows on cotangent bundles
of compact manifolds, of certain distinguished subsets of the phase space.
These subsets are generalizations of Lagrangian graphs, we call them
pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By
this method, we find various orbits which connect prescribed regions of the
phase space. Our study is inspired by works of John Mather. As an application,
we obtain the existence of diffusion in a large class of a priori unstable
systems and provide a solution to the large gap problem. We hope that our
method will have applications to more examples.
