Julia Münch: Extending rational expanding Thurston maps

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

A continuously differentiable homeomorphism is quasi-conformal when an infinitesimal field of circles is pulled back to an infinitesimal field of ellipses with a uniform bound on eccentricity. It is a generalisation of conformality and 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:R^n → R^n to a quasi-conformal map F: R^(n+1) → R^(n+1). Our result can be seen in that context, however we start with a map that is not assumed to be injective.

The second motivation is to generalise holomorphic dynamics to higher dimensions. If we consider maps f: R^n → R^n that have the same control on the derivative as quasi-conformal maps but are not necessarily injective, we consider quasi-regular maps. The dynamics is particularly nice if the same eccentricity bound on ellipses holds for all iterates of the map, 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 (not injective) f: S^2 → S^2 to a uniformly quasi-regular map F: Ω → R^3 defined on a subset Ω of R^3. If the time permits, I will outline properties of this map with respect to the hyperbolic metric in the Poincare ball. 

This is joint work with Daniel Meyer.