Minimal measures beyond Mather

For a positive definite Lagrangian, the minimal measure was defined in terms of first homology or cohomology class. For a configuration manifold that has a larger fundamental group than its first homology group, it makes a difference to define minimal measure in terms of path in fundamental group. Unlike Mather measures that are supported only on the level set not below the Mañé critical value in autonomous case, it is found in this paper that newly defined minimal measures are supported on the level sets not only above but also below the Mañé critical value. In particular, the support of the measure for a commutator looks like a figure of four petals that persists when the energy crosses the critical value.