Dynamical Systems Seminar
A notion of puzzles provides a powerful combinatorial tool for studying local connectivity and rigidity questions for complex mappings. Pioneered by B. Branner and J. Hubbard in the study of cubic polynomials, it was further substantially developed by J.-C. Yoccoz in his famous result on finitely renormalizable quadratic polynomials, and is now standard in the polynomial dynamics. However, for non-polynomial rational maps, puzzles were defined and successfully exploited only for some low-dimensional families of maps. In the talk we will present a puzzle construction for multi-dimensional family of rational maps arising from the Newton root-finding method for polynomials of degree $d$, $d \ge 3$ (so-called Newton maps), and outline further applications of the result. The talk is based on work in progress with Dierk Schleicher.