Bounded turning as a criterion for admissibility

Determine whether, in dimension three, the composite branched cover h∘q_m associated with a uniformly porous embedded circle A=h(E) is admissible if and only if A has bounded turning.

Background

Here q_m is the standard degree-m winding branched cover of the 3-sphere, E is its standard equatorial branch locus, h is an orientation-preserving homeomorphism, and A=h(E) is the embedded branch-value circle. Lemma 9.2 shows that admissibility depends only on A and the degree m, while the general porosity theorem supplies uniform porosity as a necessary condition for admissibility.

The paper notes that the constructed porous cusp fails bounded turning, whereas the quasisymmetric examples satisfy bounded turning. It does not establish either implication for arbitrary h, because the essential-loop obstruction requires control over the topology of inverse-image neighborhoods.