Research will be conducted on continuous dynamical systems on manifolds of dimension three and higher, with emphasis on compact invariant subsets on which the flow is minimal. Continuing the work related to the real analytic solution to the Seifert conjecture on aperiodic flows acting on the three-dimensional sphere, the PI plans to classify the minimal sets in such flows with respect to their homological and shape theory properties. Minimal sets shape-equivalent (in the sense of Borsuk) to a polyhedron often appear in flows as attractors, and are of special interest. Minimal sets of non-polyhedral shape are usually approximated by circular orbits or by other movable sets with the so called UV-property. They have very complicated dynamics around them, which can be investigated through the structure of the family of the approximating invariant sets. Circular orbits are extremely important in celestial mechanics as they appear as paths of movement. Smaller particles, dust, can align along a braided orbit, and in the continuous case, in a mathematical model, this "orbit" could be an invariant set, aperiodic but "circle-like", or "graph-like" such as the well-known Denjoy continua. A volume-preserving dynamical system, associated with incompressible fluids, imposes restrictions on the flow outside the special invariant collection and is an important area of study as well.
The theory of dynamical systems was developed in an effort to provide a mathematically rigorous description of real physical phenomena. It is therefore closely connected to many domains of science: mechanics, various areas of physics, biology, and economics. Dynamical systems connect several branches of mathematics such as analysis, topology, geometry, algebra, and combinatorics.
