Straightening Maps for Higher Degree Polynomials

Little copies of the Mandelbrot set are ubiquous in complex one-dimensional families {fλ}Λ of holomorphic dynamical systems acting on (subsets of) the Riemann sphere. According to the work by Douady and Hubbard in the 80’s, the presence of these little copies is generally explained via renormalization and straightening. The purpose of this mini-course is to  explore generalizations of this phenomenon to natural families {fλ}Λ where Λ has complex dimension at least 2.

Our focus is on the simplest higher dimensional parameter space setting.
Indeed, we will discuss families that arise in higher degree polynomial dynamics. More precisely, given a hyperbolic polynomial f0 with connected Julia set, we will introduce the sets of f0-combinatorially renormalizable maps C(f0), f0-renormalizable maps R(f0) and the model space M0 for C(f0). Then we will introduce a straightening map χ from R(f0) to M0. We will continue exploring the basic properties of straightening maps. The idea is to discuss injectivity, surjectivity and (dis)continuity results, following the recent work by W. Shen and Y. Wang about surjectivity, and the work of H. Inou about (dis)continuity.