On the asymptotic measure of periodic subsystems of finite type in symbolic dynamics

Let $\mathrm{\Delta }⊊\text{V}$ be a proper subset of the vertices $\text{V}$ of the defining graph of an aperiodic shift of finite type $({\mathrm{\Sigma }}_A^{+},\mathrm{\S })$. Let ${\mathrm{\Delta }}_n$ be the union of cylinders in ${\mathrm{\Sigma }}_A^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $\mathrm{\Delta }$ and let $\mu$ be an equilibrium state of a H"older potential $\varphi$ on ${\mathrm{\Sigma }}_A^{+}$. We know that $\mu ({\mathrm{\Delta }}_n)$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $\mu ({\mathrm{\Delta }}_n)$ and compare it with the pressure of the restriction of $\varphi$ to ${\mathrm{\Sigma }}_{\mathrm{\Delta }}$. The present paper extends some results in \cite{CCC} to the case when ${\mathrm{\Sigma }}_{\mathrm{\Delta }}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.