# Commentarii Mathematici Helvetici

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**Volume 87, Issue 1, 2012, pp. 1–39**

**DOI: 10.4171/CMH/247**

Published online: 2012-01-16

Strict sub-solutions and Mañé potential in discrete weak KAM theory

Maxime Zavidovique^{[1]}(1) Université de Lyon, France

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 $c\colon X\times X \rightarrow \mathbb{R}$, 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 Mañé 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 Mañé potential, and also from critical sub-solutions.

*Keywords: *Discrete weak KAM theory, Aubry–Mather theory, continuous and discontinuous critical sub-solutions

Zavidovique Maxime: Strict sub-solutions and Mañé potential in discrete weak KAM theory. *Comment. Math. Helv.* 87 (2012), 1-39. doi: 10.4171/CMH/247