Classification of fixed-point proportions for exceptional rational functions

Classify the possible values of the fixed-point proportion of the geometric iterated Galois group for every exceptional rational function, including exceptional maps beyond the power maps and Chebyshev polynomials.

Background

The paper conjectures that a geometric iterated Galois group has positive fixed-point proportion exactly when the underlying rational map is exceptional and is not linearly conjugate to a power map $x^{\pm d}$.

The fixed-point proportion has been computed for power maps and for maps linearly conjugate to Chebyshev polynomials, but the paper states that the possible values have not been classified for all exceptional rational functions. In particular, the unresolved cases include exceptional maps that are dynamical pullbacks but are not covered by the known explicit computations.