Combinatorial Structures in Holomorphic Dynamical Systems

Indiana University, Bloomington IN

Investigators

Abstract

This project is concerned with several problems at the interface between holomorphic dynamics, geometric group theory, and combinatorics. Its goal is to obtain a deeper understanding than currently exists of combinatorial structures that arise in dynamical problems and to use this information in the enhancement of known analytical results. The project involves research on the following topics: the structure of iterated monodromy groups (in particular, their growth and spectral properties); the development of a puzzle technique for Julia sets in the Devaney family of rational maps, with a view towards establishing the local connectivity of its connectivity locus; and recursive properties of the linearizing map of quadratic Siegel disks. The objects of study in dynamical systems have very complicated (fractal) structures that require detailed combinatorial descriptions. Not only are these descriptions a prerequisite for tackling hard dynamical problems, but they also serve the purpose of exposing connections between dynamics and other areas of mathematics, as well as between dynamics and other sciences. The proposed research will focus on three such structures, linking holomorphic dynamical systems to geometric group theory and combinatorics.