Skew product dynamics and multifractal analysis
Final Report Abstract
The aim of the scientific network “Skew product dynamics and multifractal analysis” was to bring together experts on dynamical systems theory and fractal geometry in order to work on the interface of the two topics and to advance the state of the field, with a particular focus on the fractal strucutes of invariant curves and graphs-like repellers in skew product systems. A key element for the interaction within the network was a workshop program that has been implemented as a part of the project, with four major workshops organised on an annual basis in the years 2012–2015. Due to the partial school-like character of these events, they specifically fostered the development of several young scientists participating in the network and allowed them to acquire profound background knowlegde on various issues of high topicality. At the same time, the meetings allowed for a stimulating exchange on the ongoing research of the participants, and numerous publications were initiated at or benefited from these events. During the project, substantial progress was made on several important aspects of the topic. For instance, a 25 year old conjecture on the fractal structure of so-called strange nonchaotic attractors has been confirmed, and methods have been established to determine the dimensions of such attractors in broad classes of systems. Further, members of the network have taken an active role in recent advances in the study of the graph of the classical Weierstrass function. The latter had been introduced by Weierstrass already around the beginning of the 20th century as an example of a continuous, but nowhere differentiable curve. However, a precise description of its fractal structure and dimensions has only been completed very recently. Further topics on which several advances were made include the study of bifurcation pattern and blowout bifurcations in forced systems and the fractal structure of so-called hyperbolic graphs, which occur in skew product systems with chaotic base dynamics and contracting fibres. Finally, surprising connections have been discovered between skew product dynamics and the mathematical theory of quasicrystals, and a substantial and fruitful transfer of ideas and methods between the two fields has been initiated.
