  Let $p(z,w)$ be a polynomial in two variables. We call the solution of the
algebraic equation $p(z,w) = 0$ the algebraic correspondence. We regard it as
the graph of the multivalued function $z \mapsto w$ defined implicitly by
$p(z,w) = 0$. Algebraic correspondences on the Riemann sphere $\hat{\mathbb C}$
give a generalization of dynamical systems of Klein groups and rational
functions. We introduce $C^*$-algebras associated with algebraic
correspondences on the Riemann sphere. We show that if an algebraic
correspondence is free and expansive on a closed $p$-invariant subset $J$ of
$\hat{\mathbb C}$, then the associated $C^*$-algebra ${\mathcal O}_p(J)$ is
simple and purely infinite.
