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拟对称图下谢尔宾斯基地毯和相关集的几何特性。标准谢尔宾斯基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细分为较小的正方形，并去除一个或多个这些较小正方形的内部，这些较小正方形彼此不接触，也不与原始外部正方形接触。对剩下的每个较小的方块重复该过程，并且无限地重复这些步骤。自 20 世纪 50 年代以来，谢尔宾斯基地毯的拓扑特性已得到很好的理解，特别是在 Whyburn 给出了此类集合的拓扑特征之后。例如，刚才描述的所有标准谢尔宾斯基地毯都是彼此同构的。然而，在拟对称映射（与拟共形映射密切相关的度量空间之间的映射）下，谢尔宾斯基地毯表现出更大的刚性。例如，有一些标准谢尔宾斯基地毯不是准对称等效的。该项目中解决的谢尔宾斯基地毯的两个最重要的问题是均匀性和刚性问题。关于均匀化，该项目研究给定空间是否与模型空间拟对称等效，而关于刚性，它研究两个给定空间是否拟对称等效。该项目解决了度量空间（其中存在距离概念的集合）分析领域的问题。用于解决这些问题的技术源于复杂的分析。反过来，复杂分析植根于物理学和工程学，特别是流体力学和电气工程，并且历史上为解决这些领域中出现的问题提供了工具和方法。在数学中，该项目中所研究的谢尔宾斯基地毯在分析中出现为分形维数集，在动力学中出现为朱莉娅集，在克莱因群理论中出现为极限集，在几何中出现为格罗莫夫双曲群无穷大边界，仅举几个例子。如果成功实施，该项目将对数学领域（称为几何群论）研究的格罗莫夫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卡波维奇-克莱纳猜想提供线索，该猜想是格罗莫夫双曲群的分类陈述，其边界是标准谢尔宾斯基地毯的连续变形。分形集（例如谢尔宾斯基地毯）的应用已在物理学、工程学以及最近的大气科学和地球科学中得到应用。例如，分形形状最近被用来创建分形天线，它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。首席研究员希望了解分形空间的几何特性将有助于更好地理解分形物理对象或在分形空间上建模的对象（例如分形天线），而这反过来又会带来其他应用。