Iterated Monodromy Groups

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Iterated monodromy groups were introduced in 2001 as groups naturally associated with (partial) self-coverings of topological spaces, for instance arising from the action of complex rational functions on a punctured Riemann sphere. These groups are used as algebraic encoding of combinatorial information about the dynamical system. If the covering is expanding, then all the essential information (for instance the Julia set) can be recovered from the iterated monodromy groups. The project is devoted to the study of iterated monodromy groups and their application to topology, dynamics and group theory. Iterated monodromy groups are naturally defined for more general structures than partial self-covering. This generalized definition can be used to construct simplicial approximations of Julia sets of multi-dimensional dynamical systems. There are very few examples of rational functions of several variables for which there is a satisfactory understanding of the topology of their Julia set. The project will provide a general method of constructing approximations of such Julia sets and will lead to new homological invariants of dynamical systems. The aim of the project is to study new connections between algebra and geometric group theory on one side and dynamical systems and the associated fractal Julia sets on the other. The iterated monodromy groups provide a bridge between these two branches of mathematics. Geometric group theory studies large-scale properties of groups of symmetries. Dynamical systems study chaotic dynamics of iterations of maps, which are models of chaotic systems in Science. Iterated monodromy groups encode in a computationally effective way combinatorial information about the dynamical systems. In particular, they give a method to construct approximations by polyhedra of complicated fractals associated with the dynamical systems. This approach can be used in the study of multi-dimensional dynamical systems, where usual computer visualization can not be applied. Iterated monodromy groups are also very interesting and exotic examples of groups. This way dynamical systems can be used to reach better understanding of group theory. This project is jointly funded by the Topology Program and the Analysis Program.