Uniformization and Rigidity of Sierpinski Carpets and Schottky Sets

本项目旨在研究拟对称映射下Sierpinski地毯及其相关集的几何性质。标准的Sierpinski地毯是用简单的迭代程序从平面上的一个正方形得到的。第一步包括将正方形细分为更小的正方形，并去除一个或多个不相互接触或原始外部正方形的小正方形的内部。在剩下的每一个小方块上重复这个过程，这些步骤无限重复。自20世纪50年代以来，特别是在Whyburn给出这种集合的拓扑特征之后，Sierpinski地毯的拓扑特性已经得到了很好的理解。例如，所有标准的Sierpinski地毯都是同态的。然而，在准对称映射下，即与准共形映射密切相关的度量空间之间的映射，Sierpinski地毯表现出更多的刚性。例如，有几对标准的Sierpinski地毯不是准对称等效的。Sierpinski地毯在项目中解决的两个最重要的问题是统一和刚性的问题。在均匀化方面，研究给定空间是否准对称等效于模型空间；在刚性方面，研究两个给定空间是否准对称等效。该项目解决了度量空间（包含距离概念的集合）分析领域的问题。用于解决这些问题的技术起源于复杂分析。反过来，复杂分析植根于物理和工程，特别是流体力学和电气工程，并在历史上为解决这些领域出现的问题提供了工具和方法。在数学中，这个项目研究的Sierpinski地毯出现在分析中，作为分形维数的集合，在动力学中作为Julia集，在Kleinian群的理论中作为极限集，在几何中作为Gromov双曲群的无穷边界，举几个例子。如果成功实施，该项目将对几何群论领域中研究的Gromov双曲群理论产生影响。特别是，首席研究员希望该项目能够为kapoovich - kleiner猜想提供线索，这是Gromov双曲群的分类陈述，其边界是标准Sierpinski地毯的连续变形。分形集的应用，如Sierpinski地毯，已经在物理学、工程学，以及最近的大气科学和地球科学中被发现。例如，分形形状最近被用于创建分形天线，不仅具有前所未有的频率覆盖和多功能性，而且非常紧凑。首席研究员希望，理解分形空间的几何特性将导致更好地理解分形物理对象或以分形空间为模型的对象，如分形天线，而这反过来将导致其他应用。

The aim of the project is to investigate the geometric properties of Sierpinski carpets and related sets under quasisymmetric maps. Standard Sierpinski carpets are obtained from a square in the plane using simple iterative procedures. The first step involves a subdivision of the square into smaller squares and the removal of the interior of one or more of these smaller squares that do not touch each other or the original outer square. The procedure is repeated on each of the smaller squares that remain, and the steps are repeated infinitely. The topological properties of Sierpinski carpets have been well understood since the 1950s, especially after Whyburn gave a topological characterization of such sets. For example, all standard Sierpinski carpets as just described are homeomorphic to each other. However, under quasisymmetric maps, which are maps between metric spaces closely related to quasiconformal maps, Sierpinski carpets exhibit much more rigidity. For example, there are pairs of standard Sierpinski carpets that are not quasisymmetrically equivalent. The two most important questions for Sierpinski carpets addressed in the project are the questions of uniformization and rigidity. Regarding uniformization, the project studies whether a given space is quasisymmetrically equivalent to a model space, and as to rigidity, it investigates whether two given spaces are quasisymmetrically equivalent.The project addresses questions in the area of analysis on metric spaces (sets in which there is a notion of distance). The techniques used to attack these questions originate in complex analysis. Complex analysis, in turn, has roots in physics and engineering, in particular in fluid mechanics and electrical engineering, and historically has provided tools and methods for attacking problems that arise in those areas. Within mathematics the Sierpinski carpets under investigation in the project arise in analysis as sets of fractal dimension, in dynamics as Julia sets, in the theory of Kleinian groups as limit sets, and in geometry as boundaries at infinity of Gromov hyperbolic groups, to mention a few examples. If carried out successfully, the project would have implications for the theory of Gromov hyperbolic groups that are studied in the area of mathematics known as geometric group theory. In particular, the principal investigator hopes that the project would provide clues to the Kapovich-Kleiner conjecture, which is a classification statement for Gromov hyperbolic groups whose boundaries are continuous deformations of standard Sierpinski carpets. Applications of fractal sets, such as Sierpinski carpets, have been found in physics, engineering, and more recently in atmospheric science and geoscience. For example, fractal shapes have recently been used to create fractal antennas that not only have unprecedented frequency coverage and versatility but also are very compact. The principal investigator hopes that understanding geometric properties of fractal spaces will lead to a better understanding of fractal physical objects or objects modeled on fractal spaces, such as fractal antennas, and that this in turn will lead to other applications.