On the Aubry-Mather theory in statistical mechanics

We generalize Aubry-Mather theory for configurations on the line to general sets with a group action. Cocycles on the group play the role of rotation numbers. The notion of Birkhoff configuration can be generalized to this setting. Under mild conditions on the group, we show how to find Birkhoff pound states for many-body interactions which are ferromagnetic, invariant under the group action and having periodic phase space.