Local Connectivity of the Julia Sets with Bounded Type Siegel Disks

The paper establishes Theorems A and B via a new Main Lemma based on quasi-Blaschke models and concludes local connectivity using the Shrinking Lemma and Whyburn’s criterion, explicitly noting that a global puzzle construction may not be available for the maps under consideration . By contrast, the candidate relies on building a single global forward-invariant puzzle/Markov partition for both rational (non-polynomial) and transcendental entire cases, invoking external or (pre)periodic rays and uniform a priori bounds around Siegel boundaries. This is unsupported: external (or “rational external”) rays used in their Step A3 generally do not exist for non-polynomial rational maps, and the paper replaces this very obstacle with a different method. For the transcendental case, the model again requires a global puzzle and does not justify the Whyburn-type compactness/local finiteness step that the paper proves carefully . Hence, while the paper’s results and logic hold, the model’s approach is not valid as stated.