Extending rational expanding Thurston maps (Geometry Seminar)

Uniformly quasi-regular maps are a suitable class to study in order to extend the concepts and ideas of complex dynamics to non-holomorphic functions. The dynamics of quasi-regular mappings are particularly interesting in R^n for n equal to at least 3. However it is not easy to find non-trivial examples of uniformly quasi-regular maps.

In the talk I will present how one can extend any rational expanding Thurston map f defined on the Riemann sphere to a map F defined on an open neighbourhood Ω containing S^2. The map is uniformly quasi-regular as long as the iteration is defined. In a second part I will talk about properties of the extension with respect to the Poincare metric on the unit ball. I will show how to find a quasi-isometric embedding of the hyperbolic plane into B(0, 1) arising from the dynamics of f on the sphere. This is work in progress.