Topological conjugacy of topological Markov shifts and Ruelle algebras

One-sided topological conjugacy of normal subshifts and gauge actions on the associated $C^*$-algebras

The class of normal subshifts includes irreducible infinite topological Markov shifts, irreducible infinite sofic shifts, synchronized systems, Dyck shifts, $β$-shifts, substitution minimal shifts, and so on. We will characterize one-sided topological conjugacy classes of normal subshifts in terms of the associated $C^*$-algebras and its gauge actions with potentials.

${\mathbb Z}^k$与高秩图相关的作用

Actions of ${\mathbb Z}^k$ associated to higher rnak graphs

${\mathbb Z}^k$的动作与满足温和假设的较高秩图$Λ$相关联。这推广了由非负整数矩阵产生的拓扑马尔可夫移位的构造。证明了$Λ$的稳定Ruelle代数与$C^*(Λ)$是强Morita等价的.因此，如果$Λ$满足非周期性条件，则稳定Ruelle代数是简单、稳定和纯无限的。

Coded equivalence of one-sided topological Markov shifts

We introduce a notion of coded equivalence in one-sided topological Markov shifts. The notion is inspired by coding theory. One-sided topological conjugacy implies coded equivalence. We will show that coded equivalence implies continuous orbit equivalence of one-sided topological Markov shifts.

扭共轭的非交换Riesz定理和弱Burnside型定理

Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy

本文件由两部分组成。在第一部分中，我们证明了Riesz（-Markov-Kakutani）定理的一个非交换的类似定理，该定理是关于连续函数代数上的泛函在基空间上的正则测度表示。在第二部分中，我们利用这个结果证明了任意离散群的扭共轭的Burnside型定理的一个弱形式。

State splitting, strong shift equivalence and stable isomorphism of Cuntz-Krieger algebras

We prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz-Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behavior of the stable conjugacy between the associated stable Cuntz-Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Marko…

$C^*$-algebras associated with two-sided subshifts

This paper is a continuation of the paper entitled "Subshifts, $λ$-graph bisystems and $C^*$-algebras", arXiv:1904.06464. A $λ$-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility condition on their edge labeling. For any two-sided subshift $Λ$, there exists a $λ$-graph bisystem satisfying a special property called FPCC. We will construct an AF-algebra ${\mathcal{F}}_{\frak L}$ with shift automorphism $ρ_{\frak L}$ from a $λ$-graph bisystem $({\frak L}^-,{\frak L}^+)$, and define a $C^*$-algebra ${\mathcal R}_{\frak L}$ by the crossed product ${\…

On a family of $C^*$-subalgebras of Cuntz-Krieger algebras

In this paper, we study a family of $C^*$-subalgebras defined by fixed points of generalized gauge actions of a Cuntz-Krieger algebra, by introducing a family of étale groupoids whose associated $C^*$-algebras are these $C^*$-subalgebras. We know that topological conjugacy classes of one-sided topological Markov shifts are characterized in terms of the isomorphism classes of these étale groupoids.

Strongly continuous orbit equivalence of one-sided topological Markov shifts

We will introduce a notion of strongly continuous orbit equivalence in one-sided topological Markov shifts. Strongly continuous orbit equivalence yields a topological conjugacy between their two-sided topological Markov shifts $(\bar{X}_A, \barσ_A)$ and $(\bar{X}_B, \barσ_B)$. We prove that one-sided topological Markov shifts $(X_A, σ_A)$ and $(X_B, σ_B)$ are strongly continuous orbit equivalent if and only if there exists an isomorphism bewteen the Cuntz-Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ preserving their maximal commutative $C^*$-subalgebras $C(X_A)$ and $C(X_B)$ and gi…

单计数器码的子移位和C*-代数

Subshifts and C*-algebras from one-counter codes

我们引入了一类子移位，称为“标准单计数器移位”。标准的单计数器移位是属于单计数器语言家族的某些马尔可夫码的马尔可夫编码系统。研究了标准单计数器移位的拓扑共轭性和流等价性。子移位通过其$λ$ -图系统与C*-代数相关联。我们描述了一类标准的一计数器移位，其性质是与它们相关联的C*-代数是简单的，而与它们的逆相关联的C*-代数不是。这给出了子移位的例子，这些子移位不是流等价于它们的逆。对于一个高度结构化的标准单计数器移位族，我们计算K-群。

Thermodynamic Formalism for Generalized Countable Markov Shifts

Countable Markov shifts, denoted by $Σ_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Σ_A$ in the locally compact case. The set $Σ_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal…

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As …

Ahlfors regularity and fractal dimension of Smale spaces

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen's construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry's measure which is Ahlfors regular. By extending Bowen's construction we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff,…