Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps

Abstract

A rational function of degree at least $2$ with coefficients in an algebraically closed field is postcritically finite (PCF) if and only if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattès maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattès PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a nonarchimedean version of Fatou’s classical result that every attracting cycle of a rational function over $\mathbb{C}$ attracts a critical point.