A note on the factorization of iterated quadratics over finite fields

Let $f$ be a monic quadratic polynomial over a finite field of odd characteristic. In 2012, Boston and Jones constructed a Markov process based on the post-critical orbit of $f$, and conjectured that its limiting distribution explains the factorization of large iterates of $f$. Later on, Xia, Boston, and the author did extensive Magma computations and found some exceptional families of quadratics that do not seem to follow the original Markov model conjectured by Boston and Jones. They did this by empirically observing that certain factorization patterns predicted by the Boston-Jones model never seem to occur for these polynomials, and suggested a multi-step Markov model which takes these missing factorization patterns into account. In this note, we provide proofs for all these missing factorization patterns. These are the first provable results that explain why the original conjecture of Boston and Jones does not hold for all monic quadratic polynomials.