  We examine affine correspondences of the form g(y)=f(x), for f and g
polynomials satisfying deg(g) < deg(f), with the property that every critical
point of the correspondence admits at least one finite forward orbit. In the
case g(y)=y, this reduces to the study of post-critically finite polynomials,
and our main result extends earlier finiteness results of the author.
Specifically, we show that the collection of such correspondences of a given
bidegree coincides with a subset of the parameter space of bounded Weil height.
We also show that there are no non-trivial holomorphic families of
correspondences with the above-described property.
