Poisson processes for subsystems of finite type in symbolic dynamics

Let Δ ⊂ V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type (Σ_A⁺, T). Let Σ_Δ be the subshift of allowable paths in the graph of Σ_A⁺ which only passes through the vertices of Δ. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on Σ_Δ, let τ_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 T^kx belong to Δ. 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 φ to Σ_Δ and the parameters of the limit law are explicitly computed.