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 nonconstant 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 happen only when $\mathrm{deg}<!--FUNCTION\; APPLICATION-->{\pi }_1=\mathrm{deg}<!--FUNCTION\; APPLICATION-->{\pi }_2$ and have a trivial dynamics. For a nonfinitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets.

我们研究曲线上自对应的代数动力学。（适当且平滑的）曲线上的自对应$C$代数闭域上是另一条曲线的数据$D$和两个非常数可分离态射${\pi }_1$和${\pi }_2$从$D$到$C$。一个子集$S$的$C$完成如果_${\pi }_1^{-1}（S）={\pi }_2^{-1}（S）$。我们证明自对应分为两类：那些只有有限多个有限完整集的类，以及那些具有有限完整集的类。$C$是有限完全集的并集。后者称为有限的，并且仅在以下情况下发生：$度<!--FUNCTION\; APPLICATION-->{\pi }_1=度<!--FUNCTION\; APPLICATION-->{\pi }_2$并且有微不足道的动力。对于特征零的非有限自对应，我们给出了étale有限完整集的数量的尖锐界限。