Dimension theory for skew products

• Antragsteller: Dr. Johannes Klaus Bernhard Jaerisch
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2011 bis 2014
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 209502447

Julia sets represent prominent examples of fractal objects. In modern physics, biology and chemistry as well as in financial mathematics and medicine, fractals provide models of objects in applied sciences. Typically, fractals are derived from dynamical processes. Notions of complexity of dynamical systems such as entropy, which is also important in the analysis of graphs and IT networks, are related to physical properties of fractals (for instance, roughness of surfaces). An important tool in the study of fractal geometry (for instance, dimensions) and measure theoretical dynamical systems is given by the thermodynamic formalism, which arose from theoretical physics and has nowadays become an extraordinary fruitful area in pure mathematics. The thermodynamic formalism for dynamical systems with an infinite symbolic representation has currently attracted a lot of interest and inspired modelling in mathematics significantly. However, due to the high complexity of these systems, they are mathematically not sufficiently well understood. This is the starting point of our project, in which we will concentrate on Julia sets of infinitely generated rational semigroups. The main idea is to investigate to which extent infinite systems can be determined by their finitely generated subsystems. For this we will exploit parallels to Kleinian groups in order to develop analogies. Concerning random dynamics, numerical experiments have surprisingly shown that noise can induce order into chaotic systems. First results indicate that this fascinating phenomenon can be explained with the help of random dynamical systems. We propose to study differentiability of devils staircases in order to measure chaos in these random dynamical systems.

• DFG-Verfahren: Forschungsstipendien
• Internationaler Bezug: Japan
• Gastgeber: Professor Dr. Hiroki Sumi