To Symbolic Dynamics Through The Thue–Morse Sequence

The paper’s Method A outlines the same construction as the model (code Thue–Morse via a 2-block map to a ternary sequence, relabel, then map d↦1^d0) and asserts minimality and uncountability of the orbit closure, and that distinct length-triples yield disjoint examples. However, key steps are only asserted: (i) preservation of uniform recurrence through the variable-length coding is not justified; (ii) disjointness of orbit closures for different triples is claimed without proof; and (iii) the section introduction says [M] is uncountable but the argument only establishes that [M] is infinite. The model supplies complete, correct proofs for these points, including explicit lemmas establishing uniform recurrence under the sliding-block code and under bounded-length prefix morphisms, a clean aperiodicity argument, and a simple invariant proving disjointness. The paper also contains a problematic conclusion in Theorem 4.1’s proof (it momentarily concludes M = {0,1}^N), inconsistent with its own theorem statement that M is dense. Overall, the model’s solution is correct and fills the paper’s gaps.