Critical dynamics of variable-separated affine correspondences

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.