Partial section I: $\hat{I}\pm$-recurrence and equivariant Lyapunov maps

Abstract

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 $\hat{I}\pm$ of rank 1, we give a criterion for the existence of an $\hat{I}\pm$-equivariant Lyapunov map on an Abelian covering of $M$ associated to $\hat{I}\pm$. 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 $\hat{I}\pm$. We describe how that set depends on $\hat{I}\pm$.

57 pages, 1 figure