Metric mean dimension of irregular sets for maps with shadowing

The paper’s Theorem 1.2 is proved via a careful pseudo-orbit construction adapted from Dong–Oprocha–Tian and a pressure distribution principle that yields the desired lower bounds for topological entropy and both metric mean dimension variants on I_Φ(f) ∩ B(Y, ε) ∩ CR(f). Crucially, the authors modify the construction to guarantee the shadowing points are nonwandering (hence chain recurrent, since Ω(f)=CR(f) for shadowing maps) by passing to minimal points that trace periodic pseudo-orbits, and then derive the scale-entropy bounds needed for metric mean dimension (see the statement of Theorem 1.2 and the proof using Proposition 3.1, together with the step ensuring A ⊂ Ω(f)=CR(f) and the final passage to (a)–(c) in the paper). These elements appear explicitly in the paper’s statement and argument . The candidate solution mirrors the overall strategy (gluing orbit segments inside a chain-recurrent class and forcing historicity from two ergodic measures), but makes two critical, unsubstantiated claims: (i) uniqueness of shadowing needed to build a topological conjugacy with a full shift, and (ii) that the shadowed set lies in CR(f) by density of periodic sequences and their ‘images’ being periodic—this would require a periodic shadowing property (or an extra expansivity-type assumption) that is not assumed. Since the target set is I_Φ(f) ∩ B(Y, ε) ∩ CR(f), the failure to justify CR-membership is a fatal gap. Hence, the paper is correct; the model’s proof is incomplete.