Positive fixed-point proportion conjecture for exceptional maps

Prove that for every tamely ramified rational function of degree at least two whose geometric iterated Galois group is a martingale group, the fixed-point proportion is positive if and only if the rational function is exceptional and not linearly conjugate to $x^{\pm d}$.

Background

The paper proves that the fixed-point proportion is zero for rational functions that are not dynamical pullbacks. It then distinguishes exceptional maps from the broader class of dynamical pullbacks and proposes a precise characterization of when positive fixed-point proportion can occur.

The conjecture excludes the power maps x±dx^{\pm d} and predicts that all remaining exceptional maps have positive fixed-point proportion. This remains an explicitly stated conjecture rather than a theorem in the paper.

References

\begin{Conjecture}[{Conjecture 7] Let $f$ be a tamely ramified rational function of degree $d \geq 2$ defined over a field $K$ such that $G_\infty(,f,t)$ is a martingale group. Then $\FPP(G_\infty(,f,t)) > 0$ if and only if $f$ is exceptional and not linearly conjugate to $x{\pm d}$. \end{Conjecture}

— The inverse Galois problem of iterated Galois groups and their fixed-point proportion  (2608.14524 - Radi, 14 Aug 2026) in Conjecture 7, Section 1, Introduction