Dynamics of Fuchsian meromorphic connections with real periods

In this talk, we discuss 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 $-1$. Moreover, we give a relation between billiard dynamics and dynamics of meromorphic connections in Riemann surfaces.