Strict sub-solutions and Ma\~ne potential in discrete weak KAM theory

TL;DR

This paper explores the discrete weak KAM theory, focusing on the Ma e potential and critical sub-solutions, highlighting their properties and differences from the continuous case, especially regarding continuity and the Aubry set.

Contribution

It introduces a definition of Ma e potential in the discrete setting and analyzes its properties, including the lack of continuity and its relation to the Aubry set.

Findings

Ma e potential is not continuous in the discrete case

Critical sub-solutions can be discontinuous

Aubry set can be recovered from the Ma e potential's continuity points

Abstract

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 Ma\~ne 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 Ma\~ne potential, and also from critical sub-solutions.