Local Connectivity of Polynomial Julia Sets at Bounded Type Siegel Boundaries

Yang’s paper proves that if a polynomial f of degree ≥2 has connected Julia set and a Siegel disk with bounded-type rotation number, then J_f is locally connected at every boundary point of the Siegel disk; this is the Main Theorem stated at the outset and proved via a Blaschke-product model, puzzle partitions (external/bubble rays), real a priori bounds, and a Kahn–Lyubich covering lemma, culminating in trivial boundary fibers and hence local connectivity (see the Main Theorem statement and proof strategy, and the roles of Theorem 2.3, Theorem 4.3, and the final spreading arguments) . The candidate solution correctly states the same implication (citing Yang) and sketches essentially the same mechanism (puzzles + a priori bounds leading to shrinking neighborhoods). One extra assertion in the candidate solution—“no hedgehog containing the Siegel disk”—is not part of the paper’s statement as presented here and is not needed for the conclusion; otherwise, the approach aligns with the paper and is correct for the stated goal. The paper also explicitly notes the quasi-circle/critical-point boundary property (via Zhang) as background, consistent with the candidate’s remarks .