BIFURCATIONS FOR FAMILIES OF AHLFORS ISLAND MAPS

The paper proves that, for natural families of Ahlfors island maps (not automorphisms), passivity of all singular values is equivalent to J-stability (Theorem 1.9), by building a holomorphic motion of the full backward orbit of a repelling cycle under the hypothesis that there are no non‑persistent Misiurewicz relations (Proposition 3.13), extending it to the Julia set via the λ-lemma (Corollary 3.14), and using J-stability ⇒ passivity (Proposition 3.15). These steps match the model’s outline. In the finite-type case, the paper adds the uniform bound on attracting periods as a third equivalent condition (Theorem 1.10) and derives openness and density of stability (Corollary 1.11). The model’s argument mirrors the paper’s: it invokes density of Misiurewicz parameters in any activity locus (Propositions 2.23–2.24) and the shooting lemma (Proposition 2.17), and it uses the same “backward-orbit holomorphic motion + λ-lemma” mechanism. No substantive logical gap is introduced by the model; any omitted technicalities (e.g., treatment of exceptional maps, compatibility condition in the definition of J-stability) are handled in the paper and are consistent with the model’s assumptions. Thus, both are correct with substantially the same proof structure. See Theorem 1.9 and its proof (holomorphic motion of backward orbits, density of backward orbits in J) and Proposition 3.15 for J-stability ⇒ passivity . For the finite-type enhancement and density of stability, see Theorem 1.10 and Corollary 1.11, with the creation of high-period attracting cycles near active singular values given in Propositions 4.10–4.11 and Theorem 4.4, and the density results in §2.4 (Propositions 2.23–2.24) and the Shooting Lemma (Proposition 2.17) .