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.

中文翻译：

---

奥布里设置无限循环覆盖

在本文中，我们研究了定义在紧流形\(M\)的切丛上的托内利拉格朗日\(L \) 升力到\(M\)的无限循环覆盖的投影 Aubry 集。大多数弱KAM 和奥布里-马瑟理论都可以在这种设置下完成。我们给出了提升拉格朗日的投影 Aubry 集为空的充分必要条件，涉及 Mather 最小化测度和\(L\)的 Mather 类。最后，我们给出了二维环面的 Mañè 示例，表明当覆盖层不是无限循环时，我们的结果不一定成立。