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

TL;DR

This paper proves that for locally maximal hyperbolic flows, Lipschitz observables with non-negative periodic averages are coboundaries, using a novel approach inspired by weak KAM theory.

Contribution

It extends Livšic's theorem to Lipschitz coboundaries in hyperbolic flows using the Lax-Oleinik semigroup, a new tool in this context.

Findings

Lipschitz coboundaries exist for hyperbolic flows with non-negative periodic averages.

Introduction of the Lax-Oleinik semigroup as a new analytical tool.

Extension of Livšic's theorem to Lipschitz regularity in hyperbolic dynamics.

Abstract

Liv\v{s}ic 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 Liv\v{s}ic 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 H\"older 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.