Tangential Homoclinic Points for Lozi Maps

The paper’s Theorem 1.1 states that, under the border/tangency assumption, all homoclinic points for X are exactly the iterates of Z and/or V, i.e., the homoclinic set is orbit(Z), orbit(V), or their union, with a proof built on (i) polygonal geometry of W^u_X and W^s_X, (ii) reduction to the fundamental stable segment [V,V^1]^s, and (iii) a triangular trapping region with vertices X, Z^1, V that confines all relevant post-critical intersections. The candidate’s solution proves the same classification using the same structural steps (polygonal manifolds, vertices = post-critical/V-points, reduction to [V,V^1]^s, and the XVZ^1 trap), differing mainly in presentation and level of detail. Hence, both are correct and substantially the same proof. See Theorem 1.1 and the surrounding arguments, including Lemmas 3.5, 3.6, 3.8, and 3.9 for the paper’s structure and key steps .