In the last decades, the field of complex dynamical systems has established itself not merely as a stand-alone subject, but as a source of deeply influential examples and paradigms for many other areas of mathematics, including topology, conformal geometry, and geometric group theory. Even more, the tools and methods developed within this field have provided new approaches to long-standing problems in other fields, such as statistical physics.
One of the fundamental objects of study in all these areas is that of fractal spaces, i.e. spaces exhibiting an infinitely complex self-similar structure (a property also observed in natural phenomena such as crystals, coastlines, or proteins). In the mathematical settings, such fractal spaces are often encoded or approximated by combinatorial data (such as finite graphs). One of the central research questions is then how the global and local structure of these spaces depends on this combinatorial data, and conversely, how this data influences the properties of the associated dynamical systems, groups, and related objects. In this vein, the overall goal of the project was to study the intricate relations between different measures of complexity of dynamical systems, fractal sets, groups, and graphs. The project was organized around three complementary research directions, each strengthening a conceptual bridge between dynamical systems and another mathematical domain.
A. Decomposition and approximation theory
Two natural ways to study an object with a complicated structure are decomposition and approximation. These strategies have proven very fruitful in complex dynamics and geometric group theory. In this subproject, we bridge these two subjects by developing a parallel decomposition theory and a unified approximation framework, thereby providing a toolbox for subsequent studies of more intricate properties of the underlying dynamical systems and spaces.
B. Iterated monodromy groups (IMGs)
The theory of IMGs forms a prominent bridge between dynamical systems and geometric group theory. IMGs are groups with an intrinsic self-similar structure naturally associated with specific dynamical systems. Over the past two decades, IMGs have attracted significant attention, as they provide an algebraic, and computationally efficient tool for addressing dynamical problems, while also yielding examples of groups that exhibit “exotic” behavior from the perspective of classical group theory. In this subproject, we systematically studied the algebraic properties of the IMGs associated with particular classes of polynomial maps. We particularly focused on characterizing the structure of these groups in terms of the dynamical properties of the corresponding maps.
C. Partition functions and spectral graph theory
The study of partition functions in mathematics is inspired by their fundamental role in statistical physics, where they encode the equilibrium properties of physical systems. In this subproject, we focused specifically on the independence polynomial—a partition function modeling “hard-core” gases, in which each particle occupies an exclusive region of space. The location of the complex roots of this polynomial is intimately connected to phase transitions in the underlying system. In a similar spirit, the study of the Laplacian spectrum and spectral measures is related to diffusion processes in physics and plays a central role in geometry and probability theory. In this subproject, we employed methods from (multidimensional) complex dynamics to investigate the structure of the zero-loci of independence and Laplacian polynomials for recursive graph sequences converging to fractal spaces.
