Coherent measures and the existence of smooth Lyapunov 1-forms for flows

Let a smooth vector field VV on a smooth closed manifold MM be given and let Z⊂MZ⊂M be an isolated invariant set for the flow of VV. In this situation, we give a necessary and sufficient condition for the existence of a Lyapunov 1-form for (V,Z)(V,Z) in terms of the relative asymptotic cycles associated with certain invariant measures of the flow.