A Markov model for factorization of iterated cubic polynomials

TL;DR

This work extends the Markov-model approach from quadratic to cubic PCF polynomials by encoding factorization data of iterates into labeled states tied to the combined critical orbit. The authors construct self-similar Markov groups $M_n$ (and related groups $L_n,H_n$) that realize the predicted splitting frequencies under a tree-automorphism framework, and they compute the associated Hausdorff dimensions to quantify their growth. They further extend the construction to combined critical orbit length $2$, obtaining richer Markov groups with analogous dimension results. The paper posits conjectural inclusions ${\\mathrm{Markov extbackslash groups}} \,\subseteq \,{\\mathrm{Gal}}(f_a^{n}-t)$ for suitable $t\\in\\mathbb{Q}(\\sqrt{-3})$, connecting the Markov model to known maximal groups and local-global rigidity phenomena, thus providing a concrete, testable framework for understanding arboreal Galois representations of cubic PCF maps.

Abstract

Motivated by Boston and Jones and Goksel, we propose a Markov model for the factorization of PCF cubic polynomials $f$. Using the information encoded in the critical orbits of cubic polynomials, we define a Markov model for PCF cubic polynomials with combined critical orbits of lengths one and two. A complete list of PCF cubic polynomials over $\mathbb{Q}$ is available thanks to the work of Anderson et al. Some of these polynomials have been previously studied -- for example, those with colliding critical orbits analyzed by Benedetto et al.; the results from these studies 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 $\mathrm{Gal}(f^n)$.

Theorems & Definitions (80)

• Definition 1
• Definition 1.1
• Example 1.2
• Example 1.3
• Example 1.4
• Definition 1.5
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3