Uniformization and Rigidity of Sierpinski Carpets and Schottky Sets

谢尔宾斯基地毯和肖特基集的均匀化和刚性

• 批准号: 0653439
• 负责人: Sergiy Merenkov
• 金额: $ 11万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-05-01 至 2011-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653439&HistoricalAwards=false
• 关键词: Uniformization; Rigidity; Sierpinski; Carpets; Schottky

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.

该项目的目的是研究sierpinski地毯的几何特性和在准对象图下的相关集。标准的Sierpinski地毯是使用简单的迭代程序从平面中的正方形获得的。第一步涉及将正方形的细分分为较小的正方形，并去除一个或多个较小的正方形的内部，这些正方形不会相互接触或原始的外部正方形。在保留的每个较小正方形上重复该过程，并无限重复步骤。自1950年代以来，Sierpinski地毯的拓扑特性就已经得到充分了解，尤其是在Whyburn给出了此类集合的拓扑表征之后。例如，所有标准的Sierpinski地毯刚刚描述的地毯彼此同构。但是，在准对称地图下，这是与准形式图密切相关的度量空间之间的地图，Sierpinski地毯表现出更大的刚性。例如，有许多标准的sierpinski地毯不是准对称等效的。项目中解决的Sierpinski地毯的两个最重要问题是统一和僵化的问题。关于统一化，该项目研究给定的空间是否在准对称上等同于模型空间，并且关于刚度，它研究了两个给定的空间是否是准对称的。是距离的概念）。用于攻击这些问题的技术源于复杂分析。反过来，复杂的分析源于物理和工程，尤其是流体力学和电气工程，历史上为攻击这些领域出现的问题提供了工具和方法。在数学中，该项目中所调查的Sierpinski地毯作为分形维度的集合，动态为朱莉娅集合，在克莱恩人群体的理论中，作为极限集，以及在gromov双层群的无限范围内的几何形状，要提及一个几个例子。如果成功执行，该项目将对在被称为几何群体理论的数学领域进行研究的Gromov双曲线群体具有影响。特别是，主要研究人员希望该项目将为Kapovich-Kleiner猜想提供线索，这是Gromov双曲线群体的分类声明，其边界是标准Sierpinski地毯的连续变形。在物理，工程学以及大气科学和地球科学中发现了分形集的应用，例如Sierpinski地毯。例如，分形状最近被用来创建分形天线，不仅具有前所未有的频率覆盖范围和多功能性，而且非常紧凑。主要研究者希望理解分形空间的几何特性将使​​人们更好地了解在分形空间（例如分形天线）上建模的分形物体对象或对象，而这又会导致其他应用。