Multifractal analysis for g-measures and singular potentials

• Applicant: Dr. Philipp Gohlke
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 509427705

The aim of this project is to enhance the understanding and applicability of a mathematical technique, known as multifractal analysis. Such a multifractal analysis is an important tool to obtain refined quantitative information about objects that exhibit an irregular and fractal-like structure. It has been used successfully for a wide range of models, data sets, images and physical examples in many different disciplines such as Geophysics, Medicine and Biology. In Mathematics, multifractal analysis has been central to understand the behavior of chaotic dynamical system. Here, it is often not feasible to describe the trajectories of each individual point and one resorts to a study of ``typical" behavior. What is deemed to be ``typical" often depends on an underlying topological or probabilistic structure. A multifractal analysis provides a more complete picture in the sense that it informs on the abundance of behavior that is neither typical in the probabilistic nor in the topological sense.Despite the fact that multifractal analysis is typically applied to highly irregular objects, many such objects of interest can be obtained from a rather smooth function (called a potential) via an iterative procedure. Indeed, if the potential is sufficiently regular, there is a good understanding of how to obtain the relevant multifractal data and what kind of data can be expected. To explore what happens in less regular situations is an active area of research. A particularly drastic way to break the standard regularity assumptions is to consider a singular potential, that is, a potential that diverges to infinity at some point. This is relevant because there are natural objects (for example arising in the theory of aperiodic order) that are indeed related to singular potentials. A first study of multifractal analysis in the presence of singular potentials has been initiated for specific examples. With this project, we have three principal objectives. First, we want to obtain a more complete understanding of the new multifractal phenomena that arise in the presence of singularities. Second, we aim at a systematic classification of the effects that certain types of singularities have. Finally, this will enable us to perform a multifractal analysis on new classes of examples that are relevant in the context of number theory and aperiodic order.

• DFG Programme: WBP Fellowship
• International Connection: Sweden
• Host: Professor Dr. Jörg Schmeling