In recent years, there has been growing interest in finding examples of vector fields having univalued solutions and exhibiting complicated dynamics. Though the phrase "complicated dynamics" is slightly vague, it is understood that any vector field displaying wild dynamical behavior should be such that the leaves of its underlying foliation must be uniformised, as Riemann surfaces, by the disc. After further detailing this relation between dynamics and the nature of leaves, I will present a result that dismisses a fair amount of potential examples. It can also be interpreted as a criterion for identifying foliations with parabolic leaves. This is part of a joint work with Helena Reis and Julio Rebelo.
