A Closed Graph Theorem for Hyperbolic Iterated Function Systems

The candidate solution proves exactly Theorem 4.2 of the paper: when (f, α) is a morphism, the fibred IFS on X×Y has attractor Gr(f|A), and for arbitrary β the attractor is a graph over A if and only if there exists a continuous g: A→B intertwining Γ and Λ via β. The paper’s proof verifies invariance of Gr(f|A) under the fibred Hutchinson operator and then invokes uniqueness of the attractor, and in the converse direction uses the closed-graph theorem on compact Hausdorff spaces to deduce continuity and the morphism relation; see Definition 4.1 and Theorem 4.2 with its proof and remarks in the text . The model’s proof follows the same structure, adding a helpful projection identity π_X(T_α(K)) = Γ(π_X(K)) to deduce π_X(D)=A for arbitrary β (consistent with Hutchinson’s uniqueness on K(X) ). No logical gaps or missing hypotheses beyond those assumed in the paper (hyperbolicity, compactness) are found, and Lemma 3.8 used in the paper to ensure f(A) ⊆ B is compatible with the model’s argument .