Combinatorial Structures in Holomorphic Dynamical Systems

这个项目涉及全纯动力学、几何群论和组合学之间的几个问题。它的目标是对动力学问题中出现的组合结构获得比目前存在的更深入的理解，并将这些信息用于增强已知的分析结果。该项目涉及以下主题的研究：迭代单元群的结构(特别是它们的增长性和谱性质)；Devaney有理映射族Julia集的拼图技术的发展，以期建立其连通性轨迹的局部连通性；以及二次Siegel圆盘的线性化映射的递归性质。动力系统中的研究对象具有非常复杂的(分形)结构，需要详细的组合描述。这些描述不仅是解决困难的动力学问题的先决条件，而且还用于揭示动力学和其他数学领域以及动力学和其他科学之间的联系。建议的研究将集中在三个这样的结构上，将全纯动力系统与几何群论和组合学联系起来。

This project is concerned with several problems at the interface between holomorphic dynamics, geometric group theory, and combinatorics. Its goal is to obtain a deeper understanding than currently exists of combinatorial structures that arise in dynamical problems and to use this information in the enhancement of known analytical results. The project involves research on the following topics: the structure of iterated monodromy groups (in particular, their growth and spectral properties); the development of a puzzle technique for Julia sets in the Devaney family of rational maps, with a view towards establishing the local connectivity of its connectivity locus; and recursive properties of the linearizing map of quadratic Siegel disks.The objects of study in dynamical systems have very complicated (fractal) structures that require detailed combinatorial descriptions. Not only are these descriptions a prerequisite for tackling hard dynamical problems, but they also serve the purpose of exposing connections between dynamics and other areas of mathematics, as well as between dynamics and other sciences. The proposed research will focus on three such structures, linking holomorphic dynamical systems to geometric group theory and combinatorics.