离散弱KAM理论中的严格子解和Mañe势
Strict sub-solutions and Mañe potential in discrete weak KAM theory
摘要：
在本文中，我们解释了弱KAM理论的离散情况下的一些事实。在该设置中，拉格朗日被替换为成本$c:X\times X \to \mathbb{R}$，在“合理”空间$X$上。这包括例如周期性的时间依赖拉格朗日的情况。众所周知，在这种情况下，可以适应大多数弱KAM理论。一个主要的区别是临界子解不一定是连续的。我们将展示如何定义Mañe势。与拉格朗日情形相反，这种势是不连续的。我们将从Mañe势的连续点集合以及临界子解中恢复Aubry集。
英文摘要：
In this paper, we explain some facts on the discrete case of weak KAM theory. In that setting, the Lagrangian is replaced by a cost $c:X\times X \to \mathbb{R}$, on a "reasonable" space $X$. This covers for example the case of periodic time-dependent Lagrangians. As is well known, it is possible in that case to adapt most of weak KAM theory. A major difference is that critical sub-solutions are not necessarily continuous. We will show how to define a Mañe potential. In contrast to the Lagrangian case, this potential is not continuous. We will recover the Aubry set from the set of continuity points of the Mañe potential, and also from critical sub-solutions.
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