Partial section I: $\alpha$-recurrence and equivariant Lyapunov maps

This is the first article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we deal with the dynamical aspect of the question. Given a flow on a compact manifold $M$ and a cohomology class $\alpha$ of rank 1, we give a criterion for the existence of an $\alpha$-equivariant Lyapunov map on an Abelian covering of $M$ associated to $\alpha$. One important aspect of the existence of such Lyapunov maps, and of the classification of partial sections, is a type of recurrence set relative to $\alpha$. We describe how that set depends on $\alpha$.