Computational and Combinatorial Techniques in Conformal Dynamics

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This project will use combinatorial and computational techniques to further our understanding of conformal dynamics, in particular studying the dynamics of rational maps from a topological and algorithmic point of view. This is a continuation of previous work of the PI on characterizing rational maps among self maps of the sphere. The project will also conduct outreach to the broader community and mathematical training at the undergraduate and graduate levels.The project will utilize the PI's graphical techniques for understanding and analyzing the topology of these maps, and especially the energies introduced for maps between graphs. In the past, this led to a positive characterization theorem, giving a concrete combinatorial object that certifies when a given topological map from the sphere to itself is equivalent to a rational map. This project will improve the positive characterization theorem to cover more cases like: extensions to other related questions, including dynamics of transcendental maps and Fuchsian groups; better understanding and estimates of the Ahlfors regular conformal dimension, a measure of the "thickness" of sets, for Julia sets and other dynamically interesting subsets of the plane; a polynomial time algorithm for finding rational maps; and a better understanding of convexity in the context of geodesic currents and measured laminations.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.