标题： Toeplitz 拓扑子移位的有限秩 显示英文标题

标题： Toeplitz subshifts of finite rank

摘要： 本文研究了有限拓扑秩Toeplitz子移位的一些基本问题。我们通过结合从Bratteli-Vershik表示或$\mathcal{S}$-adic观点下关于Toeplitz性质和有限拓扑秩$K$的刻画，定义了强有限秩Toeplitz子移位的概念$K$。特征化问题询问对于每个$K\geq 2$，是否每个拓扑秩为$K$的Toeplitz子移位都是强有限秩为$K$的Toeplitz子移位。 我们通过对一个拓扑秩为$2$的Toeplit兹子移位进行构造，给出了一个否定回答来解决刻画问题，该Toeplit兹子移位不是强Toeplit兹子移位且其秩为$2$。 然而，我们证明了所有有限秩的强Toeplit兹子移位的集合在所有无限极小子移位的空间中是常见的。 在第二部分中，我们从描述集理论的角度考虑了拓扑秩为$2$的Toeplit兹子移位的几个分类问题。 通过证明它们作为等价关系是超有限且不光滑的，我们完全确定了共轭问题、翻转共轭问题和双因子问题的复杂性。 我们还考虑了所有Toeplit兹子移位的逆问题。 我们给出了一个准则，用于判断一个Toeplit兹子移位是否与其自身的逆共轭，并利用它证明了所有此类Toeplit兹子移位的集合在所有无限极小子移位的空间中是稀疏的。 最后，我们证明了任意有限秩Toeplitz子移位的自同构群都与某个有限循环群$C$的群$\mathbb{Z}\oplus C$同构，并且对于每个非平凡的有限循环群$C$， 存在一个强Toeplitz子移位，其自同构群的同构类为$\mathbb{Z}\oplus C$，并且该强Toeplitz子移位的秩大于$2$。 显示英文摘要

摘要： In this paper we study some basic problems about Toeplitz subshifts of finite topological rank. We define the notion of a strong Toeplitz subshift of finite rank $K$ by combining the characterizations of Toeplitz-ness and of finite topological rank $K$ from the point of view of the Bratteli--Vershik representation or from the $\mathcal{S}$-adic point of view. The characterization problem asks if for every $K\geq 2$, every Toeplitz subshift of topological rank $K$ is a strong Toeplitz subshift of rank $K$. We give a negative answer to the characterization problem by constructing a Toeplitz subshift of topological rank $2$ which fails to be a strong Toeplitz subshift of rank $2$. However, we show that the set of all strong Toeplitz subshifts of finite rank is generic in the space of all infinite minimal subshifts. In the second part we consider several classification problems for Toeplitz subshifts of topological rank $2$ from the point of view of descriptive set theory. We completely determine the complexity of the conjugacy problem, the flip conjugacy problem, and the bi-factor problem by showing that, as equivalence relations, they are hyperfinite and not smooth. We also consider the inverse problem for all Toeplitz subshifts. We give a criterion for when a Toeplitz subshift is conjugate to its own inverse, and use it to show that the set of all such Toeplitz subshifts is a meager set in the space of all infinite minimal subshifts. Finally, we show that the automorphism group of any Toeplitz subshift of finite rank is isomorphic to $\mathbb{Z}\oplus C$ for some finite cyclic group $C$, and for every nontrivial finite cyclic group $C$, $\mathbb{Z}\oplus C$ can be realized as the isomorphism type of an automorphism group of a strong Toeplitz subshift of finite rank greater than $2$.