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

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