The zero entropy locus for the Lozi maps

The paper’s Theorem 1.1 defines the region R via the alternating itinerary of the point Z and proves h_top(La,b)=0 by: (i) showing there are two saddle fixed points X,Y and a unique 2-cycle {P′,P} (with explicit formulas); (ii) taking the convex hulls of the odd/even iterates of Z, on which La,b is globally linear and whose union is attracted to {P′,P}; (iii) enlarging this with specific branches of stable/unstable manifolds to form a non-separating invariant set M; and (iv) applying the Brouwer plane translation theorem to L^2 on U=R^2\M to deduce that all points of U are wandering, so the nonwandering set is finite and entropy is zero. These elements appear explicitly in the uploaded paper (definition of R and Theorem 1.1; formulas for Z, P′, P; convex-hull argument; and the BPTT step) . The candidate solution follows the same blueprint: it defines K± as convex hulls of even/odd iterates of Z, notes that L^2 is affine on each, and verifies contraction by computing the constant matrix JL·JR with det=b^2 and applying a Jury stability check; it then augments K with appropriate invariant branches and applies BPTT to conclude zero entropy. The model’s extra use of the Jury test strengthens the contraction claim but is fully consistent with the paper’s statement that P′,P are attracting in 1−b<a<1+b. No substantive disagreements with the paper’s logic were found; minor presentation differences (e.g., naming of the invariant set as a “plane tree,” and not explicitly adding ∞) do not affect correctness. Overall, both are correct and essentially the same proof, with the model giving a slightly more explicit linear-algebra check.