GRAPH MAPS WITH ZERO TOPOLOGICAL ENTROPY AND SEQUENCE ENTROPY PAIRS

The paper proves two main claims for graph maps f with h_top(f)=0: (i) f is Li–Yorke chaotic iff there exists an NS-pair (Theorem 1.1), and (ii) for any distinct x,y, the following are equivalent: 〈x,y〉 is an NS-pair, an IN-pair, or an IT-pair (Theorem 1.3). Both statements and their proofs are clearly stated and internally consistent. In particular, the paper shows that any non-diagonal IN-pair is asymptotic and lies in a solenoidal ω-limit set (Lemma 5.3), which is then used to prove IN ⇒ NS and ultimately the full equivalence NS ⇔ IN ⇔ IT, and it establishes Li–Yorke chaos from an NS-pair via a reduction to an interval subsystem and a known interval theorem (Smítal) . By contrast, the candidate solution asserts a key step that is incompatible with the paper: it claims that from an NS-pair one can “produce a scrambled pair” by aligning iterates inside nested periodic subgraphs. This contradicts the paper’s Lemma 5.3 together with Theorem 1.3, which imply that (for zero entropy graph maps) every non-diagonal IN-pair—and hence every NS-pair—is asymptotic, so it cannot be scrambled . The candidate also overstates the structure of ω-limit sets under zero entropy (“any infinite ω-limit set is solenoidal”), overlooking the circumferential (rotation-like) alternative that the paper keeps (Theorem 3.12) . While the candidate cites correct literature and gets the high-level equivalences right, the incorrect NS ⇒ scrambled step undermines its proof of Li–Yorke chaos from an NS-pair (their route relies on first producing a scrambled pair and then invoking Ruette–Snoha; the paper uses a different, correct route). The paper’s results and proofs are sound; the model’s argument contains substantive errors.