Derived-from-expanding endomorphism on T2

The paper’s main theorem (Theorem 1.1) assumes f is a C^r special absolutely partially hyperbolic endomorphism on T^2 homotopic to an expanding linear A with irrational real eigenvalues, adds Ec of class C^1, and imposes the pointwise lower bound inf_x |det Df(x)|·sin α > 1 (α is the minimum angle between Ec and Eu). Under these hypotheses, the semi-conjugacy h is a C^{1+α} conjugacy; for r≥5, h∈C^{r−3+α}; for r=∞, h∈C^∞ . The alignment of f’s invariant foliations by h with A’s eigendirections is established (Propositions 2.5 and 2.6), using bounded-distance and quasi-isometry arguments . The C^2 regularity of Eu follows from the product inequality via a section theorem (Theorem 2.3), which, together with Denjoy’s theorem on a C^2 Poincaré return map with irrational rotation number, yields injectivity of h and thus topological conjugacy (Proposition 3.2) . The paper then proves periodic-data rigidity along the center direction (Propositions 3.3–3.4), ultimately recovering the precise center multiplier of A and deriving higher regularity through a cohomological equation on a circle rotation and Journé’s lemma . By contrast, the candidate solution asserts an equivalence “h is a conjugacy iff f is area expanding.” The ‘⇒’ direction in their outline is incorrect as stated: from a merely topological conjugacy (h a homeomorphism), one cannot ‘pull back’ a Riemannian metric to deduce uniform expansion, hence cannot conclude area expansion in the differential sense used later. The paper does not claim nor need this equivalence. The candidate also claims periodic-data equalities along Eu and appeals to a general Livšic theorem in both leaf directions, whereas the paper only establishes and uses the center-direction periodic data and a circle-rotation cohomological equation; an unstable-direction Livšic step is neither stated nor needed in the paper. Finally, the model’s analytic (C^ω) upgrade is not part of the paper’s results. Overall, the paper’s argument is coherent and suffices for its stated theorem under its stated hypotheses, while the model introduces statements not justified by the paper’s assumptions or proofs.