Combinatorial Structures in Holomorphic Dynamical Systems

这个项目关注的是全纯动力学、几何群论和组合学之间的一些问题。它的目标是获得一个更深入的了解比目前存在的组合结构中出现的动力学问题，并使用这些信息在增强已知的分析结果。该项目涉及以下主题的研究：迭代单值群的结构（特别是，他们的增长和光谱特性）;发展的一个难题技术朱莉娅集在德瓦尼家庭的理性地图，以期建立当地的连通性，其连接轨迹;二次Siegel圆盘线性化映射的递归性质.动力系统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动力学问题的先决条件，而且还有助于揭示动力学与其他数学领域之间的联系，以及动力学与其他科学之间的联系。拟议的研究将集中在三个这样的结构，连接全纯动力系统的几何群论和组合。

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.