On self-correspondences on curves

We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve $C$ over an algebraically closed field is the data of another curve $D$ and two non-constant separable morphisms $\pi_1$ and $\pi_2$ from $D$ to $C$. A subset $S$ of $C$ is complete if $\pi_1^{-1}(S)=\pi_2^{-1}(S)$. We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which $C$ is a union of finite complete sets. The latter ones are called finitary and have a trivial dynamics. For a non-finitary self-correspondence in characteristic zero, we give a sharp bound for the number of \'etale finite complete sets.