  In this paper, motivated by the classical notion of a Strebel qua- dratic
differential on a compact Riemann surface without boundary, we in- troduce
several classes of quadratic differentials (called non-chaotic, gradient, and
positive gradient) which possess some properties of Strebel differentials and
appear in applications. We discuss the relation between gradient differen-
tials and special signed measures supported on their set of critical
trajectories. We provide a characterisation of gradient differentials for which
there exists a positive measure in the latter class.
