On the asymptotic measure of periodic subsystems of finite type in symbolic dynamics

Abstract

Let $Δ\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an aperiodic shift of finite type $(Σ_{A}^{+},§)$. Let $Δ_{n}$ be the union of cylinders in $Σ_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $Δ$ and let $μ$ be an equilibrium state of a Hölder potential $ϕ$ on $Σ_{A}^{+}$. We know that $μ(Δ_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $μ(Δ_{n})$ and compare it with the pressure of the restriction of $ϕ$ to $Σ_Δ$. The present paper extends some results in \cite{CCC} to the case when $Σ_Δ$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.

Companion of the paper "Poisson processes for subsystems of finite type in symbolic dynamics"