Computational and Combinatorial Techniques in Conformal Dynamics

这个项目将使用组合和计算技术来加深我们对保角动力学的理解，特别是从拓扑和算法的角度研究有理映射的动力学。这是PI之前关于刻画球体自映射中的有理映射的工作的继续。该项目还将扩展到更广泛的社区，并在本科生和研究生层面进行数学培训。该项目将利用PI的图形技术来理解和分析这些地图的拓扑，特别是为图形之间的地图引入的能量。在过去，这导致了一个正特征刻画定理，给出了一个具体的组合对象，证明了一个给定的从球面到它自己的拓扑映射何时等价于一个有理映射。这个项目将改进正特征定理，以涵盖更多的情况：对其他相关问题的扩展，包括超越映射和Fuchsian群的动力学；更好地理解和估计Julia集和其他动态有趣的平面子集的Ahlfors正则共形维度(集合“厚度”的度量)；寻找有理映射的多项式时间算法；以及更好地理解测地线洋流和测量的分层背景下的凸性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.