A Continuous Family of Non-Monotonic Toral Mixing Maps

The uploaded paper proves that the map H = G∘F (with F(x,y) = (x + f(y), y), f(y) = y/(1−η) for y ≤ 1−η and f(y) = (1−y)/η for y ≥ 1−η; G(x,y) = (x, y + x)) has the Bernoulli property on two explicit parameter windows: 0 < η < η1 and η2 ≤ η < η3, with η1 ≈ 0.324, η2 ≈ 0.415, η3 ≈ 0.491. This is stated as Theorem 1 and established by verifying the Katok–Strelcyn conditions and the Repeated Manifold Intersection Property (MR) via Propositions 1–3 (Cat-map side) and 4–6 (CG-map side) . The candidate solution asserts that the status was likely open as of the cutoff and that RMIP had not been proved for this exact H in those ranges, but the paper explicitly constructs invariant expanding cone fields, growth schemes for local (un)stable segments, and geometric quadrilaterals ensuring that v-segments and h-segments map to themselves, thereby proving (MR) and hence Bernoulli on the stated windows . Therefore, the model's status assessment is incorrect.