Livsic theorems,maximizing measures and the stable norm

In this paper we consider two results motivated by Livsic's well-known theorem that, for a hyperbolic system, a Hölder continuous function is determined, up to a coboundary, by its values around closed orbits. The first result relates to negative values around orbits and the second result to values for finitely many orbits. We also present some new results on maximizing measures and the boundary of the unit ball in the stable norm for a surface.