A census of quadratic post-critically finite rational maps defined over Q

TL;DR

This paper classifies all quadratic post-critically finite rational maps over the rationals, providing an algorithm for their identification and describing their preperiodic structures.

Contribution

It introduces a new algorithm to identify quadratic PCF maps over Q and completes the classification of such maps.

Findings

Identified all 12 quadratic PCF rational maps over Q.

Developed an algorithm to search for PCF maps.

Described possible preperiodic structures for these maps.

Abstract

We find all quadratic post-critically finite (PCF) rational maps defined over the rationals. We describe an algorithm to search for possibly PCF maps. Using the algorithm, we eliminate all but twelve rational maps, all of which are verifiably PCF. We also give a complete description of possible rational preperiodic structures for quadratic PCF maps defined over Q.