Structure of Julia Sets for Post-Critically Finite Endomorphisms on P2

Ji’s paper proves the two desired statements for PCF endomorphisms on P^2 (Theorem 1.1) and does so by a two-step argument: (i) every x in J1 \ J2 admits a Fatou disk (Theorem 4.5), and (ii) any point of J1 lying on a Fatou disk must be in a basin of a critical component cycle or in the stable manifold of a sporadic super‑saddle cycle (Theorem 4.8). These steps rely on Ueda’s covering/normal-family machinery (Corollary 2.6 and Lemma 2.7) and an exclusion of Siegel varieties, together with standard facts about attracting critical components and basins (Section 1.2) and the location/type of periodic points. The paper also notes (as a corollary) laminarity of T in J1 \ J2, originally due to de Thélin. By contrast, the model’s proof substantially overstates what laminarity yields: it asserts a Fatou disk through every point of J1 \ J2 from laminarity of T, while laminarity only guarantees plaques for σ_T-almost every point (see the background paragraph immediately before Corollary 1.5). It also sketches a Weierstrass-based global contraction for periodic critical components without invoking the precise Fornæss–Sibony trapping-region estimate, invokes a nonstandard graph-transform/stable-manifold argument at a super-saddle with a zero eigenvalue, and relies on an unsubstantiated exclusion of invariant curves disjoint from PC(f) to conclude that limits of fn|_D must be constant. The paper’s argument is complete and correct; the model’s proof contains gaps and overclaims. See Theorem 1.1, Theorem 4.5, Theorem 4.8, the laminarity background and corollary, and the facts on attracting critical components cited in Section 1.2.