(S,w)-GAP SHIFTS AND THEIR ENTROPY

The paper proves that h(X_w(S)) = log λ where λ > 0 uniquely solves 1 = ∑_{n∈S} ϕ_w(n) λ^{-(n+1)}, using synchronizing word 0, extender sets, and an MME with µ[0] > 0 (Theorem 1.1 and its proof) . The candidate solution derives the same formula via a renewal/coded-shift generating function argument (codewords u0 with |u|∈S, C(z)=∑ ϕ_w(n) z^{n+1}, and A(z)=1/(1−C(z))). Both approaches are coherent and lead to the same value of entropy; the paper’s proof is measure-theoretic, while the model’s is combinatorial/analytic. A minor fixable gap in the paper is the use of |B_n(w)| ≥ e^{n h(O)} “for all n” inside Lemma 5.3; one should choose the finitely many n_i along a subsequence where (1/n)log|B_n(w)| is sufficiently close to h(O). This does not affect correctness of the main result .