Quantitative recurrence properties in the historic set for symbolic systems

摘要

Let(∑_A,σ) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols. Let Φ:∑_A→ R be a continuous function. For a positive function ψ:N→ R~+ and a continuous positive function f: ∑_A→R~+, define the sets of recurrence points R(ψ,Φ) :={x ∈ ∑_A: d(σ~nx,x) ＜ψ(n) for infinitely many n and lim_(n→∞)1/n∑_(i=0)~(n-1)Φ(σ~ix)dose not exist} and R(f,Φ) :={x ∈ ∑A: d(σ~nx,x) ＜ e~(-S_nf(x)) for infinitely many n and lim_(n→∞)1/n∑_(i=0)~(n-1)Φ(σ~ix)dose not exist}.In this paper, we give quantitative estimates of the sets of recurrence points, that is,dim_HR(ψ,Φ) can be expressed by ψ and dim_HR(f,Φ) is the solution of the Bowen equation of topological pressure.