  In this paper, we study the dynamics of geodesics of Fuchsian meromorphic
connections with real periods, giving a precise characterization of the
possible $\omega$-limit sets of simple geodesics in this case. The main tools
are the study of the singular flat metric associated to the meromorphic
connection, an explicit description of the geodesics nearby a Fuchsian pole
with real residue larger than $-1$ and a far-reaching generalization to our
case of the classical Teichm\"uller lemma for quadratic differentials.
