Dynamics of holomorphic correspondences on Riemann surfaces

We study the global dynamics of holomorphic correspondences f on a compact Riemann surface X in the case, so far not well understood, where f and f(-1) have the same topological degree. In the absence of a mild and necessary obstruction that we call weak modularity, f admits two canonical probability measures mu(+) and mu(-) which are invariant by f* and f(*), respectively. If the critical values of f (respectively, f(-1)) are not periodic, the backward (respectively, forward) orbit of any point a is an element of X equidistributes towards mu(+) (respectively, mu(-)), uniformly in a and exponentially fast.