Julia Münch

Extending rational expanding Thurston maps

In this talk I will present an extension result. There are two main motivations, one comes from the theory of quasi-conformal mappings and one comes from generalising complex dynamics.

Quasi-conformal mappings occur naturally in many areas of analysis, however the notion is not preserved under products and it is not easy to extend a given quasi-conformal map f:Rⁿ → Rⁿ to a quasi-conformal map F: Rⁿ⁺¹ → Rⁿ⁺¹. Our result can be seen in that context, but we start with a map that is not assumed to be injective.
The second motivation is to generalise holomorphic dynamics to higher dimensions. Quasi-regular mappings on Rⁿ are a natural generalisation of holomorphic maps in C. The dynamics is particularly nice if the same eccentricity bound on ellipses holds for all iterates of the map, i.e., if we restrict to uniformly quasi-regular mappings, but it is difficult to find interesting examples of such maps.

I will talk about extending a certain class of holomorphic maps on the sphere f: S² → S² to a uniformly quasi-regular map F: R³ → R³. I will also present an application of the extension to illustrate a correspondence between the theory of Kleinian groups and holomorphic dynamics in the framework of Sullivan's dictionary.
This is joint work with Daniel Meyer, partially about work in progress.