Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an irreducible and aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Sigma_{Delta}$ be the subshift of allowable paths in the graph of $Sigma_{A}^{+}$ which
only passes through the vertices of $Delta$. For a random point $x$ chosen with respect to an equilibrium state $mu$ of a Holder potential $phi$ on $Sigma_{A}^{+}$, let $tau_{n}$ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $S^k x$ belong to $Delta$. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of $phi$ to $Sigma_{Delta}$ and the parameters of the limit law are explicitly computed.
Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Delta_{n}$ be the union of cylinders in $Sigma_{A}^{+}$ corresponding to the points $x$ for which the fi
rst $n$-symbols of $x$ belong to $Delta$ and let $mu$ be an equilibrium state of a Holder potential $phi$ on $Sigma_{A}^{+}$. We know that $mu(Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $mu(Delta_{n})$ and compare it with the pressure of the restriction of $phi$ to $Sigma_{Delta}$. The present paper extends some results in cite{CCC} to the case when $Sigma_{Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.
