A short note on the orbit growth of sofic shifts

The paper proves that for an irreducible sofic shift with minimal right-resolving presentation of period p and entropy ln λ, the Artin–Mazur zeta function has the local form ζσ(z) = α(z)/(1 − λ^p z^p) with α analytic and nonzero beyond |z| = λ−1, by using the signed-subset determinant formula (equation (2)) and showing ρ(Aj) < λ for j ≥ 2 via a synchronizing-word argument; applying a general Tauberian theorem (their Theorem 1) then yields the prime-orbit and Mertens-type asymptotics stated in Theorem 3 . The candidate solution follows the same chain: factorization of ζσ, Perron–Frobenius simple poles at p-th roots, strict spectral radius separation for the other signed-subset matrices using a proper-subshift/entropy drop argument, and application of the same Tauberian scheme. One minor overstatement in the model—claiming periodic points occur only at lengths divisible by p—conflicts with the paper’s remark that the shift’s period need not equal the presentation’s period; nonetheless this does not affect the asymptotics or the proof scheme (cf. paper’s Remark 4.1) .