Poisson processes for subsystems of finite type in symbolic dynamics

Details Poisson processes for subsystems of finite type in symbolic dynamics Poisson processes for subsystems of finite type in symbolic dynamics

2008-04-16

arxiv.org/abs/0804.2550v2

Mathematics

Dynamical Systems

21 pages, submitted

Scientific paper

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.

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