Strange attractors for the family of orientation preserving Lozi maps

The paper proves, for U− given by (C5), that the Lozi map with b<0 has a unique saddle fixed point in the lower half-plane and that the closure of its unstable manifold is a strange attractor on which the map is topologically mixing; it also establishes Hausdorff continuity of the attractor with respect to parameters. These claims, together with the parameter region, are explicitly stated and rigorously derived (see the definition of (C5), the fixed-point/eigenvalue computations, the construction of an invariant polygon H whose forward iterates shrink to Cl W^u(X), the mixing theorem, and the continuity lemma) . By contrast, the model asserts a uniform two-strip horseshoe and full two-shift symbolic dynamics on all of U−, but it neither specifies nor verifies the trapping parallelogram Π or the “two full-width vertical strips” inequalities. The paper’s proof does not proceed via such a Markov partition and does not claim a full two-shift; instead it uses a polygonal attracting region and mixing-by-crossing arguments. Given the non-differentiability along x=0 emphasized in the paper, the model’s global uniform-hyperbolicity/Markov-structure claim over U− is unsupported and likely false as stated. Its continuity argument also hinges on an unproved fixed-combinatorics Markov partition, whereas the paper’s proof uses area contraction and a general continuity lemma (Glendinning–Simpson) .