The Stability of Pointwise Hyperbolic Systems

The paper’s main stability theorem is largely coherent (shadowing-based semiconjugacy, continuity, extension to M\N, injectivity under an expansivity scale), but Step 3 of the proof invokes “Because N is compact …” to extract a uniform lower bound for a positive continuous function ψ on N, even though N was only assumed to be a connected open subset of a compact manifold. This uniformity is not justified and is used to conclude surjectivity; hence the paper’s argument has a gap that needs repair (e.g., by a different surjectivity argument or an added hypothesis) . The model’s solution matches the overall structure (graph-transform/unique shadowing, semiconjugacy, identity off N, and injectivity from expansivity) and correctly cites preservation of pointwise hyperbolicity under K via a ξ(·) scale (Theorem 4.8) , but it makes two unjustified leaps: (i) surjectivity is proved by shadowing with respect to g without establishing the required control windows for g (or comparability of g’s Q, δ scales to those of f), and (ii) injectivity is concluded by assuming one can make Q(g^n x) ≤ (1/2) r^g_n(x) for all n by shrinking ε, which is stronger than the expansivity premise and not implied by the hypotheses; the paper’s contradiction argument avoids this quantifier issue . The model also appeals to graph-transform uniqueness at times when g^n(x)∉N, where the hyperbolic structure is not defined, instead of using the paper’s clean extension h|M\N = id .