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Description
Dynamical systems are models of all kind of phenomena evolving in time.
Thereby, the best possible situation we could imagine is to completely classify all possible outcomes
of processes with arbitrary accuracy. Indeed, in a sense, this is the quintessentially goal of the theory
of dynamical systems, namely, to classify all possible dynamical systems via their long-term behavior
in a meaningful and absolute fashion. A very common scheme and successful approach to obtain useful classification of dynamical systems is the following: first, we have to define when we want to consider two dynamical systems to be the same. Here, two very prominent concepts are topological conjugacy and measure-theoretic isomorphism. Second, we have to come up with convenient dynamical notions which do not change for systems which are isomorphic. Such notions are called dynamical invariants and they usually reflect diffrent kinds of dynamical behaviour and complexity of a system. In the best case possible, these invariants can be expressed as a single number.
Now, one of the most prominent dynamical invariants is the notion of entropy. It measures how
much disorder is present in a system, by quantifying the exponential growth rate for the number of
initial states that can be separated within a certain accuracy while time passes and accuracy increases.
As it turns out, entropy is especially useful for classifying systems that exhibited a lot of complexity
(sometimes referred to as chaotic systems). However, this project is devoted to explore and study
dynamical invariants for systems showing only low-complexity behavior (zero entropy). For these
systems one needs to develop new invariants. One aim of our project will be to demonstrate that these dynamical invariants induced by pseudometrics are suitable to study low-complexity systems and to take a step towards providing fundamental classiffications-schemes for these systems.
Summary of project results
The completed project was directed towards advancing our understanding of dynamical systems theory, a field of research lying at the interface between physics and mathematics, specifically studying systems exhibiting low complexity behavior. This class of systems is particularly interesting due to its inclusion of systems akin to quasicrystals, which, from the standpoint of physics, lack translational symmetries yet maintain a type of long-range order. This long range order results in pure point diffraction patterns, typically seen in periodic structures like crystals. Additionally, we also took a look at substitutive systems, which are characterized by a finite set of local rules reminiscent of Conway''s Game of Life, which can also exhibit very interesting properties while having low complexity. Our primary goal was to better understand these systems and, if possible, to classify them in an appropriate sense. In doing so, we have built on recent findings and insights while introducing new concepts and tools to provide a foundation that we hope will prove useful for future research on these types of systems. An important question we were interested in was what kind of general strategies can be applied to identify useful dynamical invariants for low complexity system. In particular, our approach involved the investigation of specific pseudometrics that measure how different the future behavior of nearby initial states of a dynamical system will be on average. A surprising result of this approach was that these pseudometrics can potentially be used to distinguish certain families of topological amenable groups, and this will be investigated in more detail as a subject for future research.
The project focused on the following three main research tasks: Mean equicontinuity beyond amenable groups, periodically iterated morphisms and amorphic complexity as well as characterizing topological models of ergodic actions. The project team, in collaboration with external partners, worked on these problems and related topics leading to various research outputs including peer-reviewed articles, preprints as well as the completion of a PhD and master''s theses. Moreover, the obtained results were shared through presentations at conferences, workshops, and research seminars worldwide. The project''s sponsors were always mentioned during these presentations, thus raising awareness of the Norway Grants and the POLS program. To further strengthen this awareness and draw the attention of various experts to the outcomes of the project, the PI of the project has co-organized two local events at the Jagiellonian University in 2022 and 2023. The project funding also made it possible to invite several project collaborators to Kraków to work together with the PI on joint research projects and to give research talks at the local Dynamical Systems Seminar at the Faculty of Mathematics and Computer Science. Lastly, a project homepage was set up to inform about project activities, which has been accessed over 400 times to date.
The project allowed the PI and the co-investigator to focus on their research topics efficiently, resulting in two already published peer-reviewed articles, one preprint currently undergoing review and one preprint almost ready for submission. Additionally, within the project framework, the co-investigator successfully completed and defended their Phd thesis under the co-supervision of the PI. Furthermore, during the project, the PI closely supervised a master''s student in writing a thesis closely aligned with one of the project''s main research tasks, which the student successfully defended in 2023. It is important to emphasize that both the doctoral and master''s students gained valuable experience in working in a rigorous and methodical mathematical approach, while the PI was able to acquire and enhance valuable insights into thesis supervision. Furthermore, project funding enabled all team members to participate in scientific meetings, facilitating numerous research talks to disseminate project results and gain additional experience in presenting research. It also provided opportunities for the team members for networking with other mathematicians and field experts. By acknowledging the project''s sponsors in all talks (both local and non-local) and by co-organizing two local workshops in Kraków, including a revival of the so-called Wandering Seminar series that had been interrupted due to the pandemic, awareness of the Norway Grants and the POLS program was heightened. Furthermore, these talks and co-organized workshops also strengthened the visibility of the Dynamical Systems research group in Kraków. Project funding facilitated the invitation of several project collaborators to Kraków, enhancing the local Dynamical Systems Seminar with their research talks, too. And last but not least, the project raised several new and intriguing open problems, which will be pursued in future research efforts.