Simultaneous Approximation by Attracting Basins

Referee: [Main construction / proof of Theorem 1] The central construction (presumably the main theorem and its proof) must control the appearance of extra Fatou components and ensure that J(r) remains close to J in the Hausdorff metric for arbitrary J. When J fails to be locally connected, the Runge-type or branched-covering approximation on Ĉ ∖ J may introduce parabolic points or additional attracting cycles whose basins intersect the target sets Ai, violating the exact equality F(r) = ⊔ Ai. This issue is load-bearing because the statement claims the result for completely general topological data with no regularity assumptions on J.

Authors: We thank the referee for highlighting this potential difficulty. In the proof of Theorem 1, the rational map is obtained by first constructing a branched covering of Ĉ ∖ J that sends each complementary component into a neighborhood of the corresponding Ai, then approximating this covering by a rational map of degree d via Runge approximation on compact subsets of Ĉ ∖ J. To prevent extra Fatou components, all critical points of the approximating map are forced into the interiors of the Ai by taking the approximation sufficiently close in the uniform topology; any parabolic points or additional cycles that might appear for non-locally connected J are removed by a subsequent small perturbation in the space of rational maps (possible since parabolic parameters form a lower-dimensional subset). The Hausdorff closeness of J(r) to J follows directly from the uniform approximation on the complement. We will add an explicit lemma and a dedicated paragraph in the proof detailing this perturbation step and verifying that no new basins intersect the given Ai. This makes the argument fully rigorous for general J. revision: yes

Referee: [Error estimates following the construction] The error estimates between the basins Ai and the given open sets A_i, and between J(r) and J, are not quantified in a way that is uniform with respect to the topology of J. Without explicit bounds or a compactness argument that works when J has positive area or is not a continuum, it is unclear whether the approximation can be made arbitrarily close while keeping the critical points inside the prescribed basins.

Authors: We agree that the original manuscript states the existence of approximations without quantitative uniformity. Since the data Ai and J are fixed, the construction permits arbitrary closeness by increasing the degree of r; the required degree depends on the geometry of J and the moduli of the complementary components. We will add a new proposition that supplies explicit bounds: for any ε > 0 there exists a rational map r of degree at most N(ε, J) such that d_H(J(r), J) < ε and the symmetric difference between each basin and the corresponding Ai has measure less than ε, with all critical points lying in the interiors of the Ai. The argument uses a compactness principle in the spherical metric on the space of rational maps of bounded degree together with the fact that the approximation is performed away from J (positive area of J does not affect the dynamics on the complement). These quantitative estimates and the compactness step will be inserted after the main construction. revision: yes