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

Abstract

Livšic theorem for flows asserts that a Lipschitz observable that has zero mean average along every periodic orbit is necessarily a coboundary, that is the Lie derivative of a Lipschitz function smooth along the flow direction. The positive Livšic theorem bounds from below the observable by such a coboundary as soon as the mean average along every periodic orbit is non negative. Previous proofs give a Hölder coboundary. Assuming that the dynamics is given by a locally maximal hyperbolic flow, we show that the coboundary can be Lipschitz. We introduce a new tool: the Lax-Oleinik semigroup, inspired by Fathi's weak KAM theory.

32 pages,2 picture, this is a flow version of the paper "Lipschitz sub-actions for locally maximal hyperbolic sets of a C1 map" published at Discrete and Continuous Dynamical Systems working with (Anosov) maps