Dynamics of holomorphic correspondences on Riemann Surfaces

Abstract

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 $f^{\ast }$ and $f_{\ast }$ 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.