Most Fatou and Julia components are small for polynomials

The paper explicitly states and proves the target theorem: for any polynomial without irrationally neutral cycles, only finitely many Fatou components can have Euclidean diameter exceeding a given ε>0 (Theorem 1.2). The proof leverages Branner–Hubbard–Yoccoz puzzles, periodic “ends,” and hyperbolic expansion on annuli that avoid a periodic end, with the crucial input that bounded Fatou component boundaries are disjoint from non-trivial ends (Corollary 2.7). This yields uniform bounds on hyperbolic diameters and hence shrinking Euclidean diameters of components in sequences, completing the argument . By contrast, the candidate solution contains two critical gaps. First, it selects a Jordan domain V⊂⊂U disjoint from the postcritical set P(f) and then asserts all forward images fnk(V) avoid P(f). This does not follow: P(f) is forward invariant, so forward avoidance does not propagate from V; moreover, in attracting or parabolic basins P(f) typically accumulates at the cycle and meets every neighborhood, so such an orbit-avoiding V cannot exist as required for the “Shrinking Lemma.” Second, the claimed uniform inequality diam(Ω) ≤ C·diam(W) from a fixed-modulus collar does not hold in general without stronger geometric control; an annular separation inside Ω does not, by itself, uniformly control the global diameter of Ω across all preimages. The paper’s approach sidesteps these pitfalls via ends and expansion on C\e, thereby closing precisely the gaps where the model’s proof fails .