On the postcritical set of a rational map

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.