Topological complexity and dynamical invariants

在拓扑动力学中，拓扑熵的概念可以说是最基本的动力学不变量之一。它在获得对动力系统的结构和长期行为的基本见解方面起着至关重要的作用。然而，如果拓扑熵为零或无穷大，它并不能提供太多的信息。在这个研究项目中，我们的目标是更好地理解零熵制度下的系统，这是一个相当相关的问题，因为存在许多具有理论和实际意义的系统类别，由于结构原因，这些系统具有零熵。该项目将专注于与平均等度连续和无定形复杂性的低复杂性概念有密切联系的系统和概念，后者最近在作者的博士论文中被引入。具体地说，我们将通过将最近关于平均等连续性的结果与非晶态复杂性的行为联系起来来研究这两个概念的相互作用。此外，我们想要发展一种几何方法来计算符号系统中基于迭代函数系统理论的非晶态复杂性。此外，我们的目标是将非晶态复杂性的拓扑不变量扩展到平均等度连续系统之外。最后，我们渴望将非晶态复杂性推广到服从群作用，并将其应用于数学准晶理论。

In topological dynamics the notion of topological entropy is, arguably, one of the most fundamental dynamical invariants. It plays a crucial role in gaining essential insights into the structure and long-term behavior of dynamical systems. However, if topological entropy is zero or infinite, it does not provide very much information.Within this research project we aim at a better understanding of systems in the zero entropy regime, an issue of considerable relevance, since there exist many system classes of both of theoretical and practical importance which have zero entropy for structural reasons. The project will focus on systems and concepts that have close connections to the low-complexity notions of mean equicontinuity and amorphic complexity, where the latter was recently introduced in the PhD thesis of the author. Concretely, we will study the interplay of these two notions by relating recent results on mean equicontinuity to the behavior of amorphic complexity. Furthermore, we want to develop a geometric approach towards the computation of amorphic complexity in the context of symbolic systems based on the theory of iterated function systems. Additionally, we aim at an extension of amorphic complexity as a topological invariant beyond the realm of mean equicontinuous systems. Finally, we aspire to generalize amorphic complexity to amenable group actions and to pursue its application to the theory of mathematical quasicrystals.