Topological Entropy for Countable Markov Shifts and Exel–Laca Algebras

The paper proves, with clear hypotheses and two complementary theorems, that ht(γA)=hG(σA) under (SH),(FS),(AF),(SI),(SD),(O) and the existence of a KMS state; it decomposes the argument into an upper bound (via cp-approximations using PCF/orthogonal divisions and KMS on OA) and a lower bound (via a noncommutative variational principle on the diagonal without any KMS assumption) . By contrast, the model solution makes several incorrect or unjustified claims: (i) it asserts “(PCF) (hence (UCF)),” but the paper separates these and never shows PCF⇒UCF; the definition and extension of γA require (UCF)+(FS) (Proposition 3.7) , while PCF is used later for cp-approximations on (OA)Γ and then OA; (ii) it replaces the paper’s upper bound in terms of w(m,n) (which carries extra factors log PA and the subexponential indices I and D) by a simpler bound using only |L(m,n)|, thereby missing the explicit correction terms established in Theorem 6.6/4.10 ; and (iii) it uses a KMS state to manufacture a measure of maximal entropy for the lower bound, whereas the paper proves the lower bound hG(σA)≤ht(γA) via the abelian diagonal and a variational inequality that needs (UCF) but not KMS (Theorem 4.11 and Appendix A.2) . Hence the paper’s result is correct and complete for the stated regime, while the model’s proof contains substantive gaps.