A Markov model for factorization of iterated cubic polynomials

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 $\mathrm{Gal}(f^n)$.