Metric vs topological receptive entropy of semigroup actions

The paper proves, for continuous actions on compact metric spaces, that (i) ĥ(T,Γ) defined via open covers equals the receptive topological entropy h̃(T,Γ) (Theorem 5.3), (ii) the Bowen–type quantity b(T,Γ) equals h̃(T,Γ) when G is a finitely generated commutative semigroup and Γ is standard (Theorem 5.5), and (iii) the Pesin–Carathéodory quantity c(T,Γ) equals b(T,Γ) for any semigroup and regular system (Theorem 5.8). Consequently, in the stated setting, h̃(T,Γ)=b(T,Γ)=c(T,Γ) (Corollary 5.9). These results and their dependencies are clearly stated in the paper and are correct . The candidate’s solution reaches the same conclusion under the same hypotheses but follows a different route: (a) it argues directly that c(T,Γ)=h̃(T,Γ) using dynamic balls and separated sets (valid under the standard Γ assumption, which guarantees N_N⊆N_n), and (b) it sketches an identification b_A(T,X,Γ)=limsup_n (1/n)log N(∨_{g∈N_n}g^{-1}A|X), then takes sup over A to get b(T,Γ)=h̃(T,Γ). This per-cover identification is not proved in the paper and in the sketch needs extra care (notably, the reduction to covers by “tuple-sets” E_g and control of the overhead factor), but under commutativity and standard Γ the claimed growth-rate comparisons are plausible because |N_n| grows only polynomially in n. Net: the paper’s result is correct; the model’s proof of the same equality is essentially correct in spirit but compresses some technical steps.