Dr. Lukas Geyer (Dept. of Mathematical Sciences, MSU)

10/2/2023  4:10pm

Abstract:  A major goal in the theory of complex dynamics is the classification of dynamical systems given by the iterates of rational maps, preferably through combinatorial and/or topological models. One of the most successful tools towards this goal are Thurston maps, introduced in the 1980’s by William Thurston as flexible topological models of “postcritically finite” rational maps. In particular, it enabled a complete classification of such maps in the polynomial case.

 In the first part of this talk, I will explain the definition and some basic properties of Thurston maps and associated Teichmüller spaces, as well as their connection to rational maps and complex dynamics. In the second part, I will introduce anti-Thurston and anti-rational maps as orientation-reversing versions of these maps as well as recently obtained classifications of critically fixed Thurston and anti-Thurston maps through simple combinatorial models. If time permits, I will sketch connections to the theory of Kleinian groups, as well as to gravitational lensing.