Non-Martingale Fixed-Point Processes for Iterated Monodromy Groups

We construct families of rational functions $f\text{\hspace{0.17em}⁣}:{\text{bP}}_k^1\to {\text{bP}}_k^1$ of degree $d\ge 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X\subset {\text{bP}}_{\bar{k}}^N$ and a finite, generically étale morphism $f\text{\hspace{0.17em}⁣}:X\to X$, we establish geometric conditions on the critical orbits of $f$ that guarantee the fixed-point process is a martingale. Our constructions answer a question of Bridy, Jones, Kelsey, and Lodge \cite{iterated} regarding the existence of non-martingale behaviour in arboreal Galois representations, and extend their martingale criteria to higher-dimensional dynamical systems. In particular, we exhibit infinitely many postcritically finite maps with non-martingale fixed-point processes and characterize the group-theoretic obstructions to the martingale property in the genus-zero case. Furthermore, we prove that despite the failure of the martingale property, the fixed-point proportion still vanishes with a quantifiable convergence rate.