论文标题
有限类型偏移的复发率
Recurrence rates for shifts of finite type
论文作者
论文摘要
令$σ_{a} $为有限类型的拓扑混合转移，让$σ：σ_{a} \ todTo_ {a} $成为通常的左移，然后让$μ$成为gibbs的gibbs度量，用于Hölder连续潜能，这是不稳定的。在本文中，我们研究了动态系统的复发率$（σ_{a}，σ）$，肯定会肯定。特别是，给定一个函数$ψ：\ mathbb {n} \ to \ mathbb {n} $我们对以下设置$$ r_in = \ {{\ texttt i} \ inσ_{a} {a} i_ {ψ（n）} \ textrm {对于\ mathbb {n} \}的无限多} n \ 我们为$μ（r_in）= 1 $提供足够的条件，并提供$μ（r_in）= 0 $的足够条件。作为这些结果的推论，我们发现了一个新的关键阈值，其中$r_ψ$从零到一个的度量。即使在整个偏移上定义的不均匀的伯努利度量的特殊情况下，该阈值以前是未知的。我们结果的证据结合了概率理论和热力学形式主义的思想。在最后一部分中，我们将结果应用于自相似集合的动态研究。
Let $Σ_{A}$ be a topologically mixing shift of finite type, let $σ:Σ_{A}\toΣ_{A}$ be the usual left-shift, and let $μ$ be the Gibbs measure for a Hölder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system $(Σ_{A},σ)$ that hold $μ$-almost surely. In particular, given a function $ψ:\mathbb{N}\to \mathbb{N}$ we are interested in the following set $$R_ψ=\{{\texttt i}\in Σ_{A}:i_{n+1}\ldots i_{n+ψ(n)+1}=i_1\ldots i_{ψ(n)}\textrm{ for infinitely many }n\in\mathbb{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.
