On the postcritical set of a rational map

TL;DR

This paper demonstrates that any finite set of algebraic points can be realized as the postcritical set of a rational map, and any map on such a set can be approximated by a rational map with a similar postcritical structure.

Contribution

It establishes the realization of arbitrary finite sets and maps as postcritical sets of rational maps, using Belyi's theorem and Teichmüller space techniques.

Findings

Any finite algebraic set can be realized as a postcritical set.

Any finite set map can be approximated by a rational map with similar postcritical behavior.

The proofs utilize Belyi's theorem and Teichmüller space iteration.

Abstract

The postcritical set $P(f)$ of a rational map $f:\mathbb P^1\to \mathbb P^1$ is the smallest forward invariant subset of $\mathbb P^1$ that contains the critical values of $f$. In this paper we show that every finite set $X\subset \mathbb P^1(\overline{\mathbb Q})$ can be realized as the postcritical set of a rational map. We also show that every map $F:X\to X$ defined on a finite set $X\subset \mathbb P^1(\mathbb C)$ can be realized by a rational map $f:P(f)\to P(f)$, provided we allow small perturbations of the set $X$. The proofs involve Belyi's theorem and iteration on Teichm\"uller space.