We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of the topological notion of amorphic complexity. For subshifts with discrete spectrum associated to constant length substitutions, this characterization allows us to derive bounds for the amorphic complexity by interpreting the subshift as the attractor of an iterated function system in a suitable quotient space. As a result, we obtain the general finiteness and positivity of amorphic complexity in this setting and provide a closed formula in case of a binary alphabet.

中文翻译：

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恒定长度替换、迭代函数系统和非晶复杂度

我们展示了如何使用分形维数和迭代函数系统的一般理论中的几何方法来研究零熵状态下的符号动力学。更准确地说，我们建立了非晶复杂性拓扑概念的维度特征。对于具有与恒定长度替换相关联的离散谱的子位移，这种表征允许我们通过将子位移解释为合适商空间中迭代函数系统的吸引子来推导出非晶复杂性的界限。因此，我们在此设置中获得了非晶复杂性的一般有限性和正性，并在二进制字母表的情况下提供了一个封闭的公式。