Dynamics and Quasiconformal Geometry

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This research project explores the geometry of fractal spaces. Many natural phenomena exhibit fractal features, including lightning bolts, growth patterns of plants and crystals, snowflakes, coastlines, and river networks. In mathematics, fractal objects arise in the study of dynamical systems such as Julia sets of rational maps, limit sets of Kleinian groups, and attractors in iterated function systems, as well as in probabilistic models such as continuum random trees. The goal of this project is to develop better analytic and geometric tools for an improved understanding of fractals. The involvement of young researchers in this activity will contribute to increasing the expertise in the field and will help to maintain a scientific community with the mathematical knowledge necessary for progress. The project consists of three parts, concerning trees in dynamics and probability, expanding Thurston maps, and solenoids in dynamics. In each of these subprojects, geometric aspects of fractal spaces are studied that are related to their quasiconformal geometry. One is interested in geometric features that only depend on relative shape sizes and are scale invariant. In more technical terms, this means that one wants to study the geometry of the relevant spaces up to quasisymmetric equivalence. In the past, this conceptual framework has been quite successful for the study of geometric properties of self-similar fractals; the project focuses on some open questions that seem particularly amenable to further investigation.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.