Topological complexity and dynamical invariants
Final Report Abstract
The general goal of this project was to obtain a better understanding of systems in the zero entropy regime. Thereby, we put a particular focus on systems which have close connections to the low-complexity notions of mean equicontinuity and amorphic complexity. In the course of the project, we gained a lot of insights in how precisely these two notions intertwine and we were able to set up their general theory in the realm of locally compact σ-compact amenable group actions. By doing this, we also showed why these notions play out particular well for systems having discrete spectrum with continuous eigenfunctions. In fact, this turns out to be crucial because these kind of systems take up a key position in the theory of mathematical quasicrystals. Moreover, we observed an intimate relationship between amorphic complexity and fractal geometry which in turn allowed us to deploy geometric methods from the general theory of fractal dimensions and iterated function systems to compute amorphic complexity for zero entropy symbolic dynamics, in particular, substitutive subshifts. These geometric methods also allowed us to obtain general upper bounds for amorphic complexity of Delone dynamical systems which naturally relate to mathematical quasicrystals, too. Finally, let us stress, that a substantial part of this project was also devoted to obtain many new examples of mean equicontinuous group actions. In doing so, we established intriguing relations to topological full groups and to self-similar groups, like the Basilica and Grigorchuk group, from geometric group theory.
Publications
- The structure of mean equicontinuous group actions, 2018
G. Fuhrmann, M. Gröger, and D. Lenz
- Constant length substitutions, iterated function systems and amorphic complexity. Mathematische Zeitschrift, 295:1385–1404, 2020
G. Fuhrmann and M. Gröger
(See online at https://doi.org/10.1007/s00209-019-02426-2) - Measures and stabilizers of group Cantor actions. Discrete & Continuous Dynamical Systems - A, 2020
M. Gröger and O. Lukina
(See online at https://doi.org/10.3934/dcds.2020350)
