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

Abstract

Livšic theorem asserts that, for Anosov diffeomorphisms, 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 Livšic 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 with a uniform control on the Lipschitz norm. In addition our result holds true for possibly non invertible and not transitive $ C^1 $ maps. We actually prove the main result in the setting of locally maximal hyperbolic sets for 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.