Poisson processes for subsystems of finite type in symbolic dynamics
Let $Δ\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an irreducible and aperiodic shift of finite type $(Σ_{A}^{+},§)$. 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 $Σ_{A}^{+}$, 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 $§^k x$ 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.