Renormalization and Rescaling in Conformal Dynamics

The theory of dynamical systems describes how mathematical structures change over time according to prescribed constraints and laws. Dynamical systems model a host of complex phenomena, ranging from celestial mechanics to financial systems to human social behavior. An important but difficult question is to understand when small perturbations of the initial state of a system will qualitatively change the long-term behavior. The stability problem is frequently investigated through a rigorous study of the classifying space of all relevant dynamical systems, known as moduli space. This project focuses on a broad class of one-dimensional dynamical systems satisfying a geometric constraint known as conformality, along with the associated moduli spaces. Tools from complex analysis, hyperbolic geometry, and arithmetic geometry will be combined to address longstanding conjectures and to open new directions for investigation. The project will generate research opportunities for undergraduate and graduate students and will facilitate collaboration among early-career researchers via the organization of seminars and workshops. The research will also result in visually compelling representations, including intricate fractal images and videos, which will be shared with the broader public. Three distinct but interrelated directions lie at the core of this research project. First, the investigator will use recently developed techniques for the study of degenerations of rational maps and a priori renormalization bounds to study boundedness questions in conformal dynamics. These methods suggest promising approaches to tackle longstanding conjectures about the boundaries of hyperbolic components for rational maps. Next, the investigator will extend the correspondence between rational maps and Kleinian groups. This extension yields novel hybrid dynamical systems combining rational maps and Kleinian groups, where renormalization and rescaling methods can be used to understand rigidity and the deformation spaces. Finally, the investigator will pursue a recently developed renormalization theory for infinite circle packings. These new techniques hold promise in solving various conjectures regarding the quasiconformal geometry of circle packings, thereby addressing some open questions about uniformization and offering insights into conjectures from geometric group theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.