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.

中文翻译：

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有限拓扑秩最小康托系统、$\mathcal {S}$-adic subshifts 及其复杂性之间的相互作用

已知有限拓扑秩的最小康托系统（可以用 Bratteli-Vershik 图表示，每个级别的顶点数是统一有界的）具有动态刚度特性。我们确定这样的系统，当它们是膨胀的时，将同一类系统定义为原始和可识别的 ${\mathcal S}$-adic subshifts，直到拓扑共轭。这样做是为了建立一个具有有限拓扑等级的最小子位移的充分必要条件。作为一个应用，我们展示了具有非超线性复杂度的最小子位移（如所有经典的零熵示例）具有有限的拓扑秩。相反，我们分析了 ${\mathcal S}$-adic subshifts 的复杂性，并为有限拓扑秩次位移提供了足够的条件，使其具有非超线性复杂度。这包括由 Bratteli-Vershik 表示给出的最小康托系统，其塔层具有成比例的高度和所谓的从左到右 ${\mathcal S}$-adic subshifts。我们还展示了有限拓扑秩并不意味着非超线性复杂性。在拓扑秩为 2 的子移位的特殊情况下，我们证明它们的复杂性总是沿着子序列的二次二次，并且它们的自同构群是微不足道的。