Interplay between finite topological rank minimal Cantor systems, $\mathcal S$-adic subshifts and their complexity
Interplay between finite topological rank minimal Cantor systems, $\mathcal S$-adic subshifts and their complexity
复制标题
有限拓扑秩最小康托系统、$mathcal S$-adic 子移及其复杂性之间的相互作用
DOI：
10.1090/tran/8315
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
S. Petite
中科院分区：
文献类型：
--
作者：
S. Donoso;F. Durand;A. Maass;S. Petite
已知有限拓扑等级的最小康托系统（可以用每级顶点数量统一有限的 Bratteli-Vershik 图表示）具有动态刚性属性。我们建立这样的系统，当它们是可扩展的时，将同一类系统定义为拓扑共轭，作为原始且可识别的${\mathcal S}$-adic subshifts。这是为最小子移具有有限拓扑等级建立必要和充分的条件。作为一个应用，我们证明了具有非超线性复杂度的最小子移位（像所有经典的零熵示例）具有有限的拓扑等级。相反，我们分析了 ${\mathcal S}$-adic 子移的复杂度，并为有限拓扑秩子移具有非超线性复杂度提供了充分的条件。这包括由 Bratteli-Vershik 表示给出的最小康托系统，其塔层具有比例高度和所谓的从左到右 ${\mathcal S}$-adic subshifts。我们还表明，有限的拓扑秩并不意味着非超线性复杂性。在拓扑秩 2 子移的特殊情况下，我们证明它们的复杂度总是沿着子序列是次二次的，并且它们的自同构群是微不足道的。
Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic subshifts. This is done establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like all classical zero entropy examples) have finite topological rank. Conversely, we analyze the complexity of ${\mathcal S}$-adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so called left to right ${\mathcal S}$-adic subshifts. We also exhibit that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank 2 subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial.
影响因子：
2.6
作者：
Van Cyr;Bryna Kra
通讯作者：
Van Cyr;Bryna Kra
影响因子：
1
作者：
Cyr, Van;Kra, Bryna
通讯作者：
Kra, Bryna