In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen collections of disks. They generalize the classical notion of polynomial-like mapping, as we allow for infinitely many disk components in the domain and finitely many components in the range (while the restriction of the map to each domain component is still a branched covering). In this way, one can talk about generalized renormalization of a given map every time this map restricts to a non-trivial box mapping (even if the map is non-renormalizable in the classical Douady-Hubbard sense). This concept turned out to be extremely useful for tackling diverse problems in natural families of maps.
In our talk, we will discuss rigidity properties of complex box mappings, e.g. quasiconformal rigidity, shrinking of puzzle pieces, etc. This will give us a toolbox of useful results. We will then show how to apply the toolbox almost as a “black box” to conclude similar results in those families of rational maps which are renormalizable in this generalized sense. We will illustrate the success of our strategy with several examples, including polynomials with Siegel disks of bounded type rotation number (work in progress) and Newton maps of polynomials (in both cases, polynomials are of arbitrary degree). The talk is based on joint work with several people, including the participants of the MSRI semester D. Schleicher, S. van Strien, and J. Yang.