符号矩阵的拓扑共轭及其相关关系

Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided e…

On symbolic factors of $\mathcal{S}$-adic subshifts of finite topological rank

This paper studies several aspects of symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $π^{-1}(y)$ of factor maps $π\colon(X,T)\to(Y,T)$ between minimal $\mathcal{S}$-adic subshifts of finite alp…

Symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank

This paper studies several aspects of symbolic ({\em i.e.}\ subshift) factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $π^{-1}(y)$ of factor maps $π\colon(X,T)\to(Y,S)$ between minimal $\mathcal{S}$-adic …

符号动力学

Symbolic dynamics

本章介绍了自动机理论和符号动力学之间的一些联系。重点是两点。第一个是一些特定类别的自动机之间的相互作用，如局部自动机和有限型移位嵌入的结果。第二部分是句法半群与直到共轭的sofic移位分类之间的联系。

用符号动力学分析连续统上的拓扑混合与混沌

Analyzing Topological Mixing and Chaos on Continua with Symbolic Dynamics

本文提出一种分析机制，通过与传统符号动力学范畴外的定性动力系统共轭，证明了连续统上离散动力系统的拓扑混合与混沌的存在性，并在经典模型上验证了该方法。

On Graph Induced Symbolic Systems

\begin{abstract} In this paper, we investigate a shift arising from graph $G$. We prove that any $k$-dimensional shift of finite type can be generated through a $k$-dimensional graph. We investigate the structure of the shift space using the generating matrices for the shift space. We prove that a two dimensional shift space has a horizontally (vertically) periodic point if and only if it possesses a $(m,n)$-periodic point (for some $m,n\in \mathbb{Z}\setminus \{0\}$). We prove that a shift space is finite if and only if it can be generated by permutation matrices. We study the non-emptiness p…

符号动力学与矩阵的稳定代数

Symbolic dynamics and the stable algebra of matrices

本文介绍与符号动力学相关的矩阵稳定代数，探讨一般环上矩阵的移位等价与强移位等价，联系代数K理论与非负矩阵逆谱问题，并回顾有限型移位自同构群特征及Williams猜想的反例。

Cuntz-Krieger代数的状态分裂、强移位等价和稳定同构

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

证明了若两个非负矩阵是强移位等价的，则相应的具有广义规范作用的稳定Cuntz-Krieger代数是共轭的。利用纯泛函分析的方法，通过强移位等价矩阵从二部有向图构造不可分双模，从而阐明了相应的稳定Cuntz-Krieger代数之间的稳定共轭的K-理论性质.我们还研究了我们的机器的状态分裂图得到的矩阵，使拓扑共轭的拓扑马尔可夫位移描述的一些等价关系的Cuntz-Krieger代数与规范Masas和规范行动没有稳定。

双曲3空间中台球问题的编码

Coding of billiards in hyperbolic 3-space

本文将符号动力学扩展至3维双曲空间中的理想多面体，构建了台球动力学的新型编码系统，建立了带基点轨迹空间与编码移位空间的共轭关系，并发现其闭包为有限型子移位。

拓扑Markov移位的拓扑共轭性与Cuntz-Krieger代数

Topological conjugacy of topological Markov shifts and Cuntz-Krieger algebras

对于不可约的非置换矩阵$A$，Cuntz-Krieger代数${\mathcal{O}_A}$的三元组$({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$，它的标准极大交换$C^*$ -子代数${\mathcal{D}_A}$和它的规范作用量$ρ^A$被称为Cuntz-Krieger三元组。在Cuntz-Krieger三元组中引入了强Morita等价的概念，证明了两个Cuntz-Krieger三元组$({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$和$({\mathcal{O}_B},{\mathcal{D}_B},ρ^B)$是强Morita等价的当且仅当$A$和$B$是强移位等价的.我们还证明了，如果基础矩阵是强移位等价的，则稳定的Cuntz-Krieger代数上的广义规范作用是上循环共轭的。通过阐明上循环共轭的K-理论行为，研究了稳定Cuntz-Krieger代数上规范作用量的上循环共轭性与拓扑Markov位移的拓扑共轭性之间的关系.

双曲5-空间等距的代数刻划

Algebraic characterization of the isometries of the hyperbolic 5-space

利用四元数的除代数$\H$上的2 × 2可逆矩阵的等距表示，给出了双曲5-空间的保向等距的动力学类型的代数刻画.我们还确定了$GL(2, \H)$中的共轭类和中心化子或z-类的共轭类。

单位移位等价

Unital shift equivalence

本文提出有限非负整数方阵的单位移位等价概念，证明其不同于单边最终共轭，并推出相应有限型单边移位连续轨道等价，从而得到拓扑全群和Leavitt路径代数的同构，关联Hazrat分次分类猜想。