Dynamical properties of critical exponent functions

The paper rigorously proves Proposition 3.5, namely that for every finite binary word w one can achieve any target critical exponent α ≥ |w| by a suitable infinite extension, i.e., [|w|, +∞] ⊂ E(P(w)) . It also documents the failure of the stronger (incorrect) statement from [1] and replaces it with a conjecture (Conjecture 3.2) . The candidate solution gestures at a mirrored Currie–Rampersad construction but leaves key steps unproven and contains a false claim (that w is α+-free whenever α ≥ |w|). Hence the model’s argument is incomplete/incorrect, while the paper’s argument for the stated weaker result is correct.