JournalscmhVol. 87, No. 1pp. 1–39

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

  • Maxime Zavidovique

    Université de Lyon, France
Strict sub-solutions and Mañé potential in discrete  weak KAM  theory cover
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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×XRc\colon X\times X \rightarrow \mathbb{R}, on a “reasonable” space XX. 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.

Cite this article

Maxime Zavidovique, Strict sub-solutions and Mañé potential in discrete weak KAM theory. Comment. Math. Helv. 87 (2012), no. 1, pp. 1–39

DOI 10.4171/CMH/247