Poisson processes for subsystems of finite type in symbolic dynamics

Let $\Delta\subsetneq\V$ 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 H\"older 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.