  We study the 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. Under a mild and necessary condition that we
call non weak modularity, $f$ admits two canonical probability measures $\mu^+$
and $\mu^-$ which are invariant by $f^*$ and $f_*$ respectively. If the
critical values of $f$ (resp. $f^{-1}$) are not periodic, the backward (resp.
forward) orbit of any point $a \in X$ equidistributes towards $\mu^+$ (resp.
$\mu^-$), uniformly in $a$ and exponentially fast.
