Symplectic and contact properties of the Mañé critical value of the universal cover

We discuss several symplectic aspects related to the MaA (c) critical value c (u) of the universal cover of a Tonelli Hamiltonian. In particular we show that the critical energy level is never of virtual contact type for manifolds of dimension greater than or equal to three. We also show the symplectic invariance of the finiteness of the Peierls barrier and the Aubry set of the universal cover. We also provide an example where c (u) coincides with the infimum of Mather's alpha function but the Aubry set of the universal cover is empty and the Peierls barrier is finite. A second example exhibits all the ergodic invariant minimizing measures with zero homotopy, showing, quite surprinsingly, that the union of their supports is not a graph, in contrast with Mather's celebrated graph theorem.