Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ maps

Liv\v{s}ic theorem asserts that, for Anosov diffeomorphisms/flows, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Liv\v{s}ic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz. The map is only assumed to be $C^1$ and hyperbolic, but not necessarily bijective nor transitive. We actually prove our main result in the setting of locally maximal hyperbolic sets for not general $C^1$ map. The construction of the coboundary uses a new notion of the Lax-Oleinik operator that is a standard tool in the discrete Aubry-Mather theory.