论文标题
有限拓扑等级最小康托尔系统，$ \ Mathcal S $ - adic subshifts及其复杂性之间的相互作用
Interplay between finite topological rank minimal Cantor systems, $\mathcal S$-adic subshifts and their complexity
论文作者
论文摘要
已知有限拓扑等级的最小康托尔系统（可以用均匀限制的每个级别的顶点数量的曲折Vershik图表示），已知具有动态刚度的性能。我们确定，当这种系统膨胀时，它们定义了相同类别的系统，直到拓扑结合，为原始且可识别的$ {\ Mathcal S} $ - ADIC subsifts。这是为了确定最小的子移位的必要条件，以使其具有有限的拓扑等级。作为应用程序，我们表明具有非典型复杂性（如所有经典的零熵示例）具有有限拓扑等级的最低次要缩影。相反，我们分析了$ {\ Mathcal s} $ - ADIC子缩短的复杂性，并为有限的拓扑等级子移动提供了足够的条件，以具有非典型的复杂性。这包括由Bratteli-vershik表示的最小插槽系统，其塔楼的高度和所谓的左至右$ {\ Mathcal s} $ - ADIC subshifts。我们还表明，有限的拓扑等级并不意味着非典型的复杂性。在拓扑等级2子缩影的特定情况下，我们证明它们的复杂性始终是沿子序列的亚次级，它们的自动形态群体是微不足道的。
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.
