ON THE TREE MODELS OF MILDLY DISSIPATIVE MAPS

The paper’s Theorem 2 proves the existence of continuous surjective semiconjugacies ĥ1: Xf → Xg and ĥ2: Xg → Xf under stable efficiency when f|Λf and g|Λg are conjugate, but it does so via a careful construction using an iterate fN, a map H on the dense curve set Γf, and a uniform-continuity extension to Xf; this is necessary because h is only defined on Λf and h(γ) generally fails to land inside a single Γg-curve without first passing to a sufficiently deep iterate (Lemma 5.3) and then extending (Proof of Theorem 2) . By contrast, the model’s solution assumes one can define ĥ1(πf(x)) := πg(h(x)) on Λf and immediately conclude this is well-defined and constant along πf-fibers by appealing to “stable components.” The paper explicitly notes this naive inclusion fails in general when h is only defined on Λf, motivating Lemma 5.3 and the fN step (see the discussion preceding Lemma 5.3 and its proof) . Hence, the model’s construction relies on an unstated and generally false identification of π-fibers with connected components of stable sets, and it bypasses the iterate-and-extend mechanism that the paper uses to overcome this obstruction. The paper’s proof is thus correct and the model’s is not.