Expanding Thurston Maps and Fractal Geometry

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In nature, there are many phenomena that exhibit fractal features, such lightning bolts, growth patterns of plants and crystals, snowflakes, or coastlines and river networks. In mathematics, fractal objects often appear in the study of dynamical systems. This project will explore the geometry of certain fractal spaces. The principal investigator will develop better analytic and geometric tools for an improved understanding of fractals that arise from very specific dynamical systems, namely so-called expanding Thurston maps. The involvement of early-career researchers in this activity will contribute to increasing the expertise in the field, and will help to maintain a scientific community that provides the necessary mathematical knowledge for progress in science and engineering. The project includes the training of graduate students. Expanding Thurston maps provide a surprisingly rich landscape with ties to fractals, Teichmüller theory, geometric group theory, and hyperbolic geometry. While there are many interesting questions in this field, in this project specific problems will be singled out whose resolution will lead to advanced insights into the subject. These problems are related to Thurston obstructions, the induced pull-back map on Teichmüller space, and the geometry of the visual sphere associated with an expanding Thurston map. An important numerical invariant of these spheres will be studied, namely their Ahlfors regular conformal dimension. A fundamental technical tool for this investigation is the notion of combinatorial modulus of path families. It seems that obstructions for Thurston maps are tied to path families of degenerating modulus. One of the goals of this project is to get a better grasp of this connection, which is only poorly understood at present.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.