General determination of the field of constants

Determine the field of constants $\overline{K}\cap K_\infty(f,t)$ and characterize the corresponding extension over the base field $K$ for general tamely ramified rational functions.

Background

The arithmetic and geometric iterated Galois groups are related by an exact sequence whose quotient is the Galois group of the field of constants $\overline{K}\cap K_\infty(f,t)$ over $K$. Understanding this extension is important for transferring structural results from geometric to arithmetic iterated Galois groups.

The paper explains that this extension is understood only in particular families, while its general behavior remains unresolved. The later arithmetic virtually-mixing result therefore assumes that the field-of-constants extension is finite.