The goal of the project is the formation of a new research group around a new research direction. It is meant as an interplay of areas of the team members: we plan to relate the qualitative theory of dynamical systems with the theory of complex dimensions of fractal sets, through fractal analysis of orbits. By fractal properties of orbits, we mean their box dimension and generalizations. We are motivated by the ‘inverse question’: can we extract information on the behavior of a system by fractal analysis of a single trajectory? The complex dimensions of fractal sets generalize the notion of their box dimension and reveal the geometrical structure more precisely. Analysing complex dimensions of orbits seems to be well-adaptedto measuring the complexity of the system. The theory of complex dimensionsis based on the analysis of singularities of complex functions (fractal zeta functions introduced by Lapidus). It is a very interesting field in itself, related to spectral theory, mathematical physics and number theory (the well-known Riemann hypothesis).We plan to apply this theory to important questions in discrete dynamical systems, such as their classifications. In short, key points of the research are: classifications (formal and analytic) of 1-dim discrete systems through complex dimension analysis of attached orbits, and understanding their bifurcations by the set of complex dimensions of attached orbits. Our aim is also to promote the field of dynamical systems in Croatia by organizing a workshop and mini-courses. We plan to expand our small group by engaging a new PhD student and a postdoc. In Croatia, there is only a small group of researchers in dynamical systems, scattered in different areas, which we intend to bring closer together. In the course of the project, we will put a particular effort in scientific growth of the new PhD student. We plan to provide her/him with the opportunity of international collaborations from the beginning of her/his career.