Topological complexity and dynamical invariants

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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.