EMERGENCE FOR DIFFEOMORPHISMS WITH NONZERO LYAPUNOV EXPONENTS

The paper’s Theorem A states exactly the lower bound dim_H(E_f) ≥ dim(E^s) + (1 − (1 − d) χ_l/χ_1) dim(E^u) under Conditions A1–A2, and proves it via a pressure-based horseshoe construction and a reduction to a subshift-of-finite-type result on high emergence, see the statement and assumptions in Section 2 and Theorem A, and the horseshoe Theorem B . The proof controls the unstable Hausdorff dimension using the potential −d log|det Df|_{E^u}| and a careful product/covering argument along intermediate unstable foliations (Section 5), culminating in a bound of the required form (cf. the layered-cover construction and the final lower bound on dim_H of the unstable projection) . By contrast, the model’s Step 4 asserts a lower bound dim_H(X ∩ W^u_loc) ≥ h_μ(f)/χ_1 by counting global n-cylinders via topological entropy and bounding unstable diameters by e^{-(χ_1−ε)n}. This mixes global topological entropy with leaf-wise coverings and ignores the nonconformal structure; for a single unstable leaf, the relevant complexity is governed by u-entropy/pressure, not h_top over the whole basic set. The claimed h_μ/χ_1 lower bound can easily exceed dim(E^u), so the covering argument is not valid in general. The paper avoids this pitfall by working with pressure for −d log|det Df|_{E^u}| on the horseshoe and by a refined multi-scale unstable covering (Sections 3–5) . Therefore, while the model’s final inequality matches the paper’s statement, its justification is flawed at a key step; the paper’s argument is correct.