Recurrence rates for shifts of finite type

Let Σ_A be a topologically mixing shift of finite type, let σ : Σ_A → Σ_A be the usual left-shift, and let μ be the Gibbs measure for a H¨older continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system (Σ_A, σ) that hold μ-almost surely. In particular, given a function ψ : N → N we are interested in the following set

Rψ = {i ∈ ΣA : in+1 . . . in+ψ(n)+1 = i1 . . . iψ(n) for infinitely many n ∈ N}.

We provide sufficient conditions for μ(Rψ) = 1 and sufficient conditions for μ(Rψ) = 0.

As a corollary of these results, we discover a new critical threshold where the measure of Rψ transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets.