On the Asymptotic Measure of Periodic Subsystems of Finite Type in Symbolic Dynamics

Let ∆ ( 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 ∆n be the union of cylinders in Σ + A corresponding to the points x for which the first n-symbols of x belong to ∆ and let μ be an equilibrium state of a Hölder potential φ on Σ+A. We know that μ(∆n) converges to zero as n diverges. We study the asymptotic behaviour of μ(∆n) and compare it with the pressure of the restriction of φ to Σ∆. The present paper extends some results in [2] to the case when Σ∆ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.