Low-Dimensional Dynamics

Indiana University, Bloomington IN

Investigators

Abstract

The Principal Investigator will perform research in the theory of dynamical systems, mainly low-dimensional (both smooth and continuous). He will apply analytical, topological and combinatorial tools in order to investigate real and complex systems. A substantial part of the project is aimed at providing bridges between real and complex, one- and two-dimensional dynamics. Investigation of a class of branched coverings of the plane will be continued. Semigroups of affine maps will be investigated from the point of view of ergodic theory. For other systems attractors, periodic orbits, rotation sets and topological entropy will be in the center of attention. For almost every phenomenon from Physics, Chemistry, Biology, Medicine, Economy and other sciences one can make a mathematical model that can be regarded as a dynamical system. Sometimes the study of this system is extremely difficult, but quite often it can be reduced to the study of a much simpler, low dimensional one. This project is devoted mainly to the investigation of the properties of these types of systems from the mathematical point of view. The results that can be obtained are usually much stronger than in the general case and often lead to much deeper understanding of the original systems.