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Projekt Druckansicht

Dimension theory for skew products

Fachliche Zuordnung Mathematik
Förderung Förderung von 2011 bis 2014
Projektkennung Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 209502447
 
Erstellungsjahr 2015

Zusammenfassung der Projektergebnisse

This was a successful project investigating new classes of complex systems focusing on their dynamical long-term behaviour and their fractal-geometric aspects. The Julia set of a polynomial map acting on the Riemann sphere is a prominent example of a fractal object which reflects the chaoticity of the long-term behaviour of the iteration of the given polynomial. More precisely, the iteration of the polynomial exhibits a sensitive dependence on the initial point precisely for the points in the Julia set. In modern physics, biology and chemistry as well as in financial mathematics and medicine, fractals provide models of objects in applied sciences. One of the aims is to describe the geometry of Julia sets by means of the underlying dynamical system. In this project we focused on systems which have an infinite generator system. That is, instead of iterating a single polynomial on the Riemann sphere, we consider at each step a polynomial from an infinite collection of polynomials. Regarding real world phenomena it is very natural to allow for many different choices at each step since there are many random terms in nature which may cause different developments. Our main result in this context is a formula for the dimension (in the sense of Hausdorff) of the Julia set of a new class of infinitely generated systems. Regarding the random iteration of polynomial maps on the Riemann sphere, we have also investigated the phenomenon of noise-induced order. This phenomenon means that a random perturbation (or a random choice) stabilises the long-term behaviour of the system. That is, the Julia set disappears with respect to a certain random process. In this case, we study how the probability to experience a certain long-term behaviour depends on the initial condition. In fact, another main result of this project shows that, this probability function shows a very irregular behaviour which can again be investigated by means of fractal geometry. The irregularity can be seen as a gradation between chaos and order.

Projektbezogene Publikationen (Auswahl)

 
 

Zusatzinformationen

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