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

TL;DR

This paper proves that non-Lattes post-critically finite rational functions over number fields form a bounded set in moduli space, leading to finiteness results, by extending Fatou's classical attracting cycle theorem to non-archimedean fields.

Contribution

It establishes height bounds for non-Lattes PCF rational functions over number fields, excluding a well-understood family, and proves finiteness of conjugacy classes.

Findings

Bounded height of non-Lattes PCF rational functions in moduli space

Finiteness of conjugacy classes over number fields for given degree

Extension of Fatou's attracting cycle theorem to non-archimedean fields

Abstract

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) 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 Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.