The theory of dynamical systems and especially the study of chaotic systems has been one of the important breakthroughs in science in the 20th century. The field is still relatively young, but there is no question that it is becoming more and more important in a variety of disciplines. Scientists have been able to apply the geometric and qualitative techniques from dynamical systems to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics. Ergodic theory, topological, hyperbolic, symbolic dynamics, all has been developed as disciplines in their own right. Development of powerful computers and computer graphics has shown that the dynamics of low-dimensional systems can be at once beautiful and complicated.

The goal of the project is to study certain classes of these beautiful and complicated low-dimensional chaotic dynamical systems. More precisely, our objectives are to investigate the following problems:

(1) The Lozi and Lozi-like maps of the plane. There are a lot of open problems related to the dynamics of these families of maps. One of them, which we plan to study, is the transition between simple and chaotic dynamics, that is, the transition from zero to positive topological entropy. We also plan to further investigate the chaotic dynamics, especially the topological properties of the Lozi and Lozi-like attractors, and recently developed symbolic dynamics on these attractors.

(2) The area-preserving twist diffeomorphisms on a cylinder. There is a number of fundamental ergodic theoretical questions still unanswered for these classes of maps, in particular related to obtaining sharp bounds on topological and metric entropy and Lyapunov exponents. We plan to develop further recently developed variational techniques to address these problems.

(3) Ergodic theory on the Heisenberg group. We plan to derive estimates providing quantitative information on the norm convergence of double ergodic averages with respect to two flows/maps generating the Heisenberg group. This will be achieved by investigating properties of left-invariant flows and the corresponding analytical averages on the continuous Heisenberg group, which is an interesting structure on its own. As byproducts of this investigation we will also deduce boundedness and convergence of paraproducts with Heisenberg group dilations and characterize general pairs of flows l eading to bounded paraproducts.

The expected results of the project are to further develop some aspects of the low-dimensional dynamical systems and to contribute to deeper understanding of the theory.


The research group: