Dynamical properties of minimal Ferenczi subshifts

The paper proves the equivalence between minimal Ferenczi subshifts and S-adic subshifts generated by the specific directive sequence τW, via a careful two-step construction: first showing XW = X_{\tilde τW} using the right-permutative morphisms \tilde τn and the exact identity \tilde τ[0,n+1)(a)=w_n1^a (Lemma 3.1 and Proposition 3.3), and then proving X_{τW}=X_{\tilde τW} by exploiting rotational conjugacy (equations (17)–(18)) and a nontrivial combinatorial lemma (Lemma 3.4), culminating in Theorem 3.7 . By contrast, the model’s proof hinges on a claimed ‘shift’ identity asserting that τ[0,N)(a) is a fixed shift of w_{N−1}1^a and then concludes that every factor of τ[0,N)(a) is a factor of w_{N−1}1^a. This step is false if the “shift” is a rotation (rotating a finite word does not preserve its set of linear factors), and it is inconsistent with the known equality of lengths |τ[0,N)(a)|=|w_{N−1}|+a (Lemma 3.8) if interpreted as deleting a prefix . Hence the model’s argument is incorrect, whereas the paper’s argument is correct and complete.