3/26/2010

Mario Bonk
Michigan

Thurston maps

A continuous map $f\colon S2\rightarrow S2$ on a $2$-sphere $S2$ is called a branched cover if near each point it can be written as $z\mapsto z^n$ for some $n\in {\bf N}$ in suitable local coordinates in domain and image. A critical point of such a map $f$ is a point in $S2$ where $f$ is not a local homeomorphism, that is, where $n\ge 2$. Thurston considered branched covers of $S2$ for which the forward orbit of each critical point under iteration is finite. These maps are now called Thurston maps. Basic examples include maps on the Riemann sphere such as $z\mapsto z2+i$ or $z\mapsto 1-2/z2$.

The study of these maps provides links to areas such as dynamical systems, classical conformal analysis, hyperbolic geometry, geometric group theory, and analysis on metric spaces. In my talk I will give a survey on this subject and discuss some recent joint work with Daniel Meyer.