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映射下Sierpinski地毯及其相关集合的几何性质。标准的谢尔宾斯基地毯是从一个正方形的平面上使用简单的迭代程序。第一步是将正方形细分为更小的正方形，并移除一个或多个彼此不接触或与原始外部正方形不接触的较小正方形的内部。在剩下的每一个较小的方块上重复这个过程，这些步骤无限重复。Sierpinski地毯的拓扑性质自20世纪50年代以来就得到了很好的理解，特别是Whyburn给出了这种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胚的。然而，在准对称映射下，这是度量空间之间的映射，与准共形映射密切相关，谢尔宾斯基地毯表现出更大的刚性。例如，有几对标准谢尔宾斯基地毯不是准对称等价的。Sierpinski地毯在该项目中解决的两个最重要的问题是均匀化和刚性问题。关于一致化，研究给定空间与模型空间是否准对称等价;关于刚性，研究两个给定空间是否准对称等价;解决度量空间（有距离概念的集合）的分析问题。用于解决这些问题的技术起源于复分析。复变分析，反过来，有根源的物理学和工程学，特别是在流体力学和电气工程，并在历史上提供了工具和方法来解决出现在这些领域的问题。在数学的谢尔宾斯基地毯下调查的项目出现在分析集的分形维数，在动力学作为朱莉娅集，在理论的克莱因集团作为极限集，并在几何边界在无穷大的格罗莫夫双曲群，举几个例子。如果成功进行，该项目将对Gromov双曲群理论产生影响，该理论在数学领域被称为几何群论。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声明，其边界是标准谢尔宾斯基地毯的连续变形。分形集的应用，如谢尔宾斯基地毯，已被发现在物理学，工程学，以及最近在大气科学和地球科学。例如，分形形状最近已被用于创建分形天线，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。首席研究员希望理解分形空间的几何性质将导致更好地理解分形物理对象或分形空间上建模的对象，如分形天线，这反过来又会导致其他应用。