The applicant and the host intend to investigate dynamics and entropy of homeomorfhism groups (resp.,pseudogroups) of fractals. They are going to apply those results for construction of a foliated space of co-dimension 2 and positive entropy, which minimal set is locally homeomorphic to a given fractal (for example the Sierpinski gasket).
Moreover, they would like to find what types of minimal sets and entropies can be realised by foliated spaces with restricted geometry. The applicant and the host are going to study relations between the geometry of fractals and dynamics of foliated spaces of higher co-dimension.
Results in foliation theory provide information about solutions of partial equations without referring to the explicit form of their solutions. Therefore, one can find applications of foliation theory in many areas of partial differential equations.
The knowledge and experience gained during the Fellowship will cause not only individual development of the applicant but also create the possibility to implement new results in science and practical applications of partial differential equations in Europe. Fractals, which can be realised as transversals of foliated spaces, and its dynamics can be applied not only in physics, chemistry or medicine (to describe the growth of cancer) but also for Internet traffic.
The joint research with the Department of Mathematics at the University of Illinois at Chicago can increase the competitiveness of Europe in new area of intersection of foliated spaces of higher co-dimension with fractal geometry and dynamical systems. The possibility of implementations of the results of research in industry makes this area of investigations very attractive.
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