A Markov model for factorisation of iterated cubic polynomials

TL;DR

This paper introduces a Markov model to analyze the factorization patterns of post-critically finite cubic polynomials, leveraging critical orbit data to understand Galois groups and polynomial dynamics.

Contribution

It develops a novel Markov model for PCF cubic polynomials with specific critical orbit structures and constructs groups that are conjectured to contain their Galois groups.

Findings

Constructed groups $M_n$ follow the Markov model

Model aligns with known cases of colliding critical orbits

Groups $M_n$ are conjectured to contain $ ext{Gal}(f^n)$

Abstract

Motivated by the work of Boston, Jones and Goksel, we propose a Markov model for the factorisation of post-critically finite (PCF) cubic polynomials f. Using the information encoded in the critical orbits, we define a Markov model for PCF cubic polynomials with combined critical orbits of lengths one and two. Thanks to the work of Anderson et al., a complete list of PCF cubic polynomials over $\mathbb{Q}$ is available. Some of these polynomials have already been studied, such as those with colliding critical orbits analysed by Benedetto et al., which align with our model. We construct groups $M_n$ and prove that they follow our Markov model. These groups $M_n$ are conjectured to contain $\mathrm{Gal}(f^n)$.