A PERSISTENTLY SINGULAR MAP OF Tn THAT IS C1 ROBUSTLY TRANSITIVE.

The paper proves the existence, for every n ≥ 2, of a C^1-robustly transitive endomorphism F: T^n → T^n with a persistent critical set. It first builds a robustly transitive local diffeomorphism f using a strongly robustly minimal IFS on the (n−1)-torus and a protoblender located over two overlapping base intervals in S^1; this yields dense stable and unstable manifolds and hence robust transitivity (Theorem 3.1) . Then it perturbs f inside a small ball disjoint from the protoblender region to obtain F as in (4.1), computes DxF as in (4.2), exhibits points where det DxF takes opposite signs, and uses continuity to conclude persistent singularities (Lemma 4.1) . Cone invariance/expansion are checked inside the ball (Lemmas 4.2–4.3), giving robust transitivity for F (Lemma 4.4, Theorem 4.1) . The model follows the same overall two-step program (robustly transitive G + local surgery S away from the blender; set F = S ∘ G; persistence of singularities via det DG > 0 and det DS taking both signs). However, its explicit surgery map s(u,t) = (u, β(|(u,t)|) φ(t)) is incorrect: it is not identity on the boundary of W (hence S is not C^1 when patched to the identity), and the Jacobian is not lower block-triangular as claimed, so det Ds = β φ′ is wrong. This breaks the key regularity and sign-change assertions in Step 8. By contrast, the paper’s perturbation (4.1) and Jacobian computation (4.2) are consistent with the required estimates and cone invariance checks . Therefore: the paper’s argument is correct; the model’s is flawed at the crucial surgery step even though the high-level strategy mirrors the paper.