Complexity in Dynamical systems, Algebra, and Geometry

The overall goal of the project is to study relations between different measures of complexity of dynamical systems, fractal sets, and groups.

The main objects of our interest are iterated monodromy groups (IMGs), which are self-similar groups naturally associated to certain dynamical systems, such as the iteration of a rational map on the Riemann sphere.

IMGs provide a prominent bridge between dynamical systems and geometric group theory, and their study has been a vibrant topic in the last 20-years. In the project, we will focus on three aspects of this modern research.

Subproject A: Decomposition theory of maps and groupsIn a recent work with collaborators, I established a novel decomposition theorem for rational maps based on the structure of their Julia sets.

I aim to extend this result to the case of contracting self-similar groups, which will provide a new entry to the renowned Sullivan dictionary.

I will also explore computational aspects of IMGs and implement the decomposition in the computer algebra system GAP.Subproject B: Algebraic properties of IMGsQuite unexpectedly, the IMGs of even very simple maps provide examples of groups with interesting properties that are exotic from the point of view of classical group theory.

However, we still lack general theory that will unify these nice examples.

The main objective in this research direction is to relate dynamical properties of maps to algebraic properties, such as growth and amenability, of the respective IMGs.Subproject C: Spectral properties of Schreier graphs of IMGThe study of the Laplacian spectrum and spectral measures occupies a significant place in the geometric group theory.

Computations of spectra for the Schreier graphs of self-similar groups have recently been an active filed of research. Surprisingly, it connects to multidimensional dynamics and Schroedinger operators associated to aperiodic order. The goal of this subproject is to explore such connections in the case of IMGs.