A TYPE OF SHADOWING AND DISTRIBUTIONAL CHAOS

The paper’s main theorem (Theorem 1.1) assumes: (1) chain transitivity, (2) shadowing along the ∼-classes D, (3) continuity of D(·) into K(X), and (4) positive topological entropy, and concludes that for every n ≥ 2 there exists δ_n > 0 such that for every class D, D^n ∩ DC1^δ_n_n(X,f) is residual in D^n . The proof strategy in the paper avoids specification and works even when the quotient X/∼ is infinite (e.g., an odometer), which the author emphasizes (Remark 1.6) . By contrast, the model’s Step 1 incorrectly asserts that the number of ∼-classes is finite and that f permutes them cyclically; this contradicts the paper’s allowance of an infinite set of classes and minimal odometer factors (used explicitly in Lemma 2.5’s proof) . The model further relies on a specification-type gluing for g = f^m|_D (Step 3) without justification under the paper’s hypotheses; the paper instead proves ordinary shadowing from (2)+(3) (Lemma 2.3) and uses an entropy-to-symbolic-factor construction to obtain distal n-tuples (Lemma 2.4), spreads them across all D via the minimal quotient (Lemma 2.5), and then applies a Baire argument (Lemma 2.6 and the R/S construction) to get residual DC1^δ_n in each D^n . The model also claims h_top(f^m|_D) = h_top(f^m) = m h_top(f) on each class (Step 4), which is unjustified; the paper makes no such claim and does not require it. Therefore, the paper’s argument is correct as written, while the model’s proof contains critical errors and missing assumptions.