Lightly chaotic functional envelopes

The paper’s Theorem 4.2 asserts that for any metric space (X,d) the following are equivalent: (1) f is chaotic (Devaney), (2) the functional envelope F_f on S_k(X) is lightly chaotic, and (3) the functional envelope F_f on S_p(X) is lightly chaotic. The proof reduces (2) and (3) to “f is transitive and has dense periodic points,” then concludes “hence chaotic,” appealing to the claim that transitivity plus dense periodic points implies sensitivity (cf. Preliminaries). However, the paper states this implication for infinite spaces and cites Banks et al., but the correct hypothesis is that X has no isolated points. As stated, Theorem 4.2 is false on metric spaces with isolated points: for example, on the two-point discrete space with the flip map, F_f is lightly chaotic for both the compact-open and point-open canonical subbases (by the same constant-map construction used in the paper’s proof), yet f is not sensitive (hence not Devaney chaotic). This directly contradicts the claimed equivalence. The model identifies the missing hypothesis and supplies the corrected equivalence under the standard “no isolated points” assumption, in line with Banks–Brooks–Cairns–Davis–Stacey. See Theorem 4.2 statement and proof sketch in the paper, where the final step equating transitivity+dense periodic points to chaos is taken without excluding isolated points ; compare the Preliminaries’ discussion conflating the ‘infinite’ case with the sensitivity implication, which is not generally valid without excluding isolated points . Definitions of LS-transitive/LS-periodically dense/LS-chaotic and the functional-envelope setup appear earlier in the paper and align with the model’s reductions via constant maps .