Kostiantyn Drach (IST Austria):
Title: Rigidity of rational maps
Abstract:
In one-dimensional complex dynamics, a branch of dynamical systems that studies iterations of holomorphic maps, the rigidity question has the classical form: Under which conditions can one promote topological conjugacy between a pair of maps to an analytic (conformal) conjugacy? We call this 'parameter rigidity', as it allows us to distinguish maps within parameter space starting with some 'soft' topological (or even combinatorial) data. On the other hand, for a given map, there is a parallel 'dynamical rigidity' question: Can one distinguish individual orbits in combinatorial terms (e.g. via symbolic dynamics)? For polynomials, and especially for quadratic polynomials, this circle of questions is well-studied in the works of Avila, Douady, Hubbard, Lyubich, McMullen, Sullivan, van Strien, Yoccoz, and many others. In this case, the progress was possible thanks to the fact that most polynomials have a naturally-defined Markov partition in their dynamical plane (via so-called Yoccoz puzzles). This is not true for general rational maps, and hence life becomes more complicated (or interesting?). In my talk, I will discuss what we know so far about dynamical and parameter rigidity of general rational maps. Our guiding example will be Newton maps, a family of maps naturally arising from Newton's root-finding method. I will also outline a 'toolbox' of techniques useful to attack these types of questions; the key 'tool' is a renormalization concept of complex box mapping. The main message of the talk hopefully will be that life is not that complicated after all (under certain conditions).