Local Connectivity of Julia Sets of Some Transcendental Entire Functions with Siegel Disks

The paper’s Theorem 1.1 proves local connectivity under assumptions (a)–(c) for f in class B with AV(f)=∅, by combining: (i) a tailored orbifold metric that is uniformly expanding on a domain W away from Siegel boundaries (Lemma 3.3, building on [Mih12] and [Par22]) , (ii) a proof that all Fatou components are bounded Jordan disks (Lemma 3.5) , and (iii) a key weak expansion/shrinking theorem for pullbacks near bounded-type Siegel boundaries (Theorem 2.1) , then applying Whyburn’s criterion (Lemma 3.6) to conclude the proof (Section 3.4) . The model’s solution mirrors the high-level strategy but omits the crucial weak expansion near Siegel boundaries (the role of Theorem 2.1), instead invoking generic “collar annuli” and “puzzles” without establishing the needed uniform shrinking for pullbacks approaching ∂Δ. The paper explicitly supplies this missing ingredient and completes the argument; hence the model solution is incomplete while the paper’s is correct. See Theorem 1.1 for the precise statement of assumptions (a)–(c) and conclusion ; the final shrinking and finiteness steps are carried out in the proof of Theorem 1.1 using Theorem 2.1 and bounded degree on Fatou components .