Critical dynamics of variable-separated affine correspondences

TL;DR

This paper studies affine correspondences defined by polynomial pairs with degree constraints, focusing on those with critical points having finite forward orbits, extending finiteness results and analyzing their parameter space.

Contribution

It extends finiteness results for critical orbit properties to affine correspondences and characterizes their parameter space, showing no non-trivial holomorphic families exist.

Findings

Correspondences with critical points having finite orbits form a bounded subset in parameter space.

No non-trivial holomorphic families of such correspondences exist.

The collection of these correspondences has bounded Weil height.

Abstract

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.