Boundaries of the Bounded Hyperbolic Components of Polynomials

The uploaded paper explicitly states the main claim as Theorem 1.2: for any non disjoint‑type bounded hyperbolic component H ⊂ C_d and any 1 ≤ l ≤ d−1−m_H, one has H.dim(∂LC H ∩ M_l) = 2d−2; the paper also gives the stronger local version (Theorem 1.3) asserting full local Hausdorff dimension at a geometrically finite boundary point, from which Theorem 1.2 is immediate . The proof architecture is organized around a boundary extension of Milnor’s parameterization via Blaschke divisors (Theorem 1.4), local connectivity at H‑admissible boundary points (Theorem 1.5), a perturbation/holomorphic‑motion theorem on ∂H (Theorem 1.7), and a refined parabolic‑implosion construction yielding dynamic hyperbolic sets of dimension arbitrarily close to 2 on Fatou boundaries (Theorem 1.8), culminating in Section 15’s derivation of Theorem 1.3 via Theorem 15.1 . The candidate’s answer correctly identifies the statement and broadly lists the right ingredients, but it contains two substantive inaccuracies: (i) it asserts that ∂LC H contains a dense subset of admissible maps, whereas the paper explicitly notes that the image ∂A_H of admissible divisors is not dense in ∂H (even though the admissible divisors are dense in the model boundary) ; and (ii) it miscounts the dimension by attributing only 2m_H base directions, leading to 2m_H+2l=2(d−1) (which is only valid when l=d−1−m_H), while the paper’s quantitative local estimates near the constructed parameters show full 2(d−1) real dimension for all 1 ≤ l ≤ d−1−m_H, using in particular the remaining (N−l) free‑critical directions and weak transversality to keep codimension under control (see the inequalities around Theorem 12.2 and the conclusion of Theorem 15.1) . Net: the paper’s argument is coherent and proves the claim; the model’s final conclusion matches the theorem but its reasoning includes incorrect density and dimension accounting.