Recurrence rates for shifts of finite type

2022/09/05 by Demi Allen, Allen, Demi, Simon Baker +3 · 3 citations
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2209.01919

openalex publication_date 2022/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Hölder 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 R_ψ=\\texttt i∈ Σ_A:i_n+1… i_n+ψ(n)+1=i₁… i_ψ(n)\textrm for infinitely many n∈ℕ\. We provide sufficient conditions for μ(R_ψ)=1 and sufficient conditions for μ(R_ψ)=0. As a corollary of these results, we discover a new critical threshold where the measure of R_ψ 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.

Cited by

• Quantitative recurrence properties for piecewise expanding maps on [0,1]^d
• Shrinking Targets versus Recurrence: a brief survey
• Dynamical covering sets in self-similar sets

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