The research proposed lies at the interface between measurable and topological dynamics. In these and other areas of dynamics the problems concerning the relationship between the dynamical and cohomological properties of systems proved to be very important. One circle of the proposed problems can be informally called "inverse problems for coboundaries", and, roughly speaking, is aimed to understand how and to what extent the dynamics of a measurable group action can be recovered from the cohomological information, and in particular, from the collection of all measurable coboundaries. It turns out that, in the case of group actions, the algebraic properties of the acting group play an important role. Another line of the proposed research is an attempt to get a complete understanding of the possibility of constructing minimal topological models for arbitrary families of measurable dynamical systems. In the case when the family consists of one single ergodic automorphism, the answer is given by the famous Jewett-Krieger theorem.
The classical theory of dynamical systems originated with the study of qualitative behavior of the solutions of differential equations describing the motion of a mechanical system. The fundamental theorem of Liouville provided the bridge between the classical theory and modern measurable dynamics. The study of the connections between the different and seemingly unrelated properties of the measurable systems, as well as relations between the measurable and continuous structures in dynamics, strengthens our sense of connectedness of the entire building of mathematics.
