EXISTENCE OF C1,1 CRITICAL SUBSOLUTIONS IN DISCRETE WEAK KAM THEORY

In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function c : M x M -> R defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a C-1,C-1 critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's alpha function on the cohomology.