Non-Martingale Fixed-Point Processes for Iterated Monodromy Groups

We construct families of rational functions $f \colon \bP^1_k \to \bP^1_k$ of degree $d \geq 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X \subset \bP^N_{\overline{k}}$ and a finite, generically \'etale morphism $f \colon X \to X$, 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.