Aubry Set on Infinite Cyclic Coverings

In this paper, we study the projected Aubry set of a lift of a Tonelli Lagrangian $L$ defined on the tangent bundle of a compact manifold $M$ to an infinite cyclic covering of $M$. Most of weak KAM and Aubry – Mather theory can be done in this setting. We give a necessary and sufficient condition for the emptiness of the projected Aubry set of the lifted Lagrangian involving both Mather minimizing measures and Mather classes of $L$. Finally, we give Mañè examples on the two-dimensional torus showing that our results do not necessarily hold when the cover is not infinite cyclic.