Abstract
In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a $C^{1,1}$ 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.
Highlights
In the past twenty years, new techniques have been developed in order to study time-periodic or autonomous Lagrangian dynamical systems
A quite similar formalism was used in the study of time periodic Lagrangians, for example in ([CISM00] or [Mas07])
We study, following ideas of [FM07] the case of invariant cost functions and we apply this study in section 6 to symmetries coming from deck transformations of a cover
Summary
In the past twenty years, new techniques have been developed in order to study time-periodic or autonomous Lagrangian dynamical systems. Following [FS04], [Ber07] and [FFR09] we will study the existence of more regular strict subsolutions. Under these hypothesis, we prove the following theorem: Theorem 0.1. The first two sections, 1 and 2, are devoted to recalling some results proved in [Zav08] and to introducing the notion of twist condition,. In the third section, 3, we study the particular case of cost coming from Tonelli Lagrangians and we prove that they fit into our framework,. Following ideas of Mather ([Mat91]), we introduce Mather’s α function on the cohomology
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