Connectedness of independence attractors of graphs with independence number three

The paper defines the independence attractor as A(G) = lim_m Roots(IG^m), with IG^m(z) = P_G^m(z) + 1 and hence Roots(IG^m) = {z : P_G^m(z) = −1} (Definition 2.1 and (2.1) ). In the unicritical 3-vertex case a1=3, P(z) = (1+z)^3−1, so P^m(z) = (1+z)^{3^m}−1 and Roots(IG^m) = {−1} for all m; the Hausdorff limit is therefore {−1}. However, the paper asserts (Theorem 1.2) that A(G) equals {−1} ∪ {|z+1|=1} for a1=3, appealing to Theorem 1.5 which claims that, in an exceptional family (including a1=3), A(G) is the disjoint union of J(P_G) and ⋃_{k≥1} Roots(IG^k) . This contradicts the literal limit: in the a1=3 case, ⋃_{k≥1} Roots(IG^k) = {−1}, so the Hausdorff limit cannot suddenly include the circle. The model flags exactly this definitional inconsistency and shows that the intended classification holds if one instead takes A0(G) := lim_m Roots(IG^m−1) = lim_m Roots(P_G^m), i.e., preimages of 0, which equals J(P_G) by standard Fatou–Julia theory. Apart from this definitional error, the paper’s connectivity sub-classification of J(P) along the curves a2^2 = 3a1a3, 4a3(a1−1), 4a1a3 and the finer thresholds matches the model’s dynamics-based derivations (e.g., Lemma 3.7 and Lemma 3.8) .