On a family of self-affine IFS whose attractors have a non-fractal top

The paper proves that for parameters (λ,µ) in a computably large region G, the top boundary ∂top(Aλ,µ) is the graph of a continuous, strictly increasing function by constructing a barrier attractor Bλ,µ for gated maps S0,S1, and then iterating R(A)=∂top(T0(A)∪T1(A)) starting from Bλ,µ to obtain an increasing sequence Rn that converges to ∂top(Aλ,µ); this yields continuity and strict monotonicity (Lemma 2.1 and Lemma 2.3) and explicitly places Bλ,µ below ∂top(Aλ,µ) (B bounds ∂top from below) . The candidate solution’s core claim W(b)≤b (and hence f≤b) contradicts the paper’s monotonicity Rn−1≤Rn and the fact that Bλ,µ≤∂top(Aλ,µ); in the paper R(B)≥B and Rn↗∂top(Aλ,µ) (not the other way around) . The model also assumes unproven endpoint equalities and a unique crossing on the overlap, which are not established in the paper and are used critically in its continuity and strict-increase arguments.