Holomorphic Dynamical Systems

In holomorphic dynamics one studies iteration of holomorphic maps, in particular polynomials. The dynamical space (typically the complex plane or the Riemann sphere) is divided into the Fatou set, where the dynamics is stable, and the Julia set where the dynamics is chaotic. The goal is not only to understand the topology and geometry of Julia sets of individual holomorphic maps, but also to understand how the Julia set and the dynamics vary with the map, in particular to understand bifurcation sets of maps where the dynamics change qualitatively. The scope of the project is broad. It concentrates on describing special types of results and techniques, namely those for which a transfer of results is possible from dynamical spaces to parameter spaces, parametrizing families of holomorphic maps. For instance, results obtained using puzzles in dynamical spaces and para-puzzles in parameter spaces as a tool.