Zeta functions in fractal geometry and analysis

分形几何和分析中的 Zeta 函数

• 批准号: RGPIN-2019-05237
• 负责人: Mendivil, Franklin
• 金额: $ 1.09万
• 依托单位: Acadia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715430
• 关键词: Zeta; functions; fractal; geometry; analysis

Fractals are mathematical objects which possess detailed structure at all levels of resolution (the middle-1/3 Cantor set is a classical example). They often arise as invariant sets in a dynamical system, but they also have been the subject of extensive study in their own right. A geometric analysis of fractal sets usually proceeds by examining the asymptotics of some measure of their structure as the size scale shrinks to zero. The various notions of dimension (for metric spaces) are all examples of this process. Zeta functions have been used in the analysis of asymptotic properties ever since Riemann first introduced them into the study of the prime number counting function. Zeta functions are powerful tools in this type of analysis and now they are used in many different areas of mathematics outside of number theory (including the study of fractals). The goal of this project is to significantly extend the use of zeta functions in the study of the geometry of fractal sets and measures. The existing geometric fractal zeta functions encode the Minkowski (box-counting) dimension and Minkowski content of the set via the poles of a suitable meromorphic extension. In some cases, explicit formulae are available which give precise quantitative information about oscillations of the geometry. The proposed project will extend the geometric reach of fractal zeta functions by defining new zeta functions which are associated with the Hausdorff, Packing and Assouad dimensions and also by defining zeta functions which give "local" information. In addition, the project will explore how these zeta functions are transformed under mappings of the underlying space and define new zeta functions with good mapping properties. The goal here is to create new tools for analyzing projections and slices (intersection with a line) of fractal sets as well as deciding when two fractals are bi-Lipschitz equivalent. The results of the project will introduce substantial tools for the examination of many geometric measure-theoretic-properties of sets and measures. The fractal zeta functions previously introduced by M. Lapidus and his collaborators have already influenced research in dynamical systems, non-commutative geometry, and theoretical physics (to name just three areas). It seems likely that a broadened class of fractal zeta functions will also find significance in these (and other) areas.

分形是数学对象，在各个级别的分辨率上都具有详细的结构（中间1/3 Cantor集是一个经典的示例）。它们通常是动态系统中不变的设置，但它们本身就是广泛研究的主题。分形集的几何分析通常是通过检查其结构的某种度量的渐近学来进行的，因为尺寸尺度缩小到零。 维度的各种概念（对于度量空间）都是此过程的示例。 自从Riemann首次将其引入质量数计数函数的研究以来，Zeta函数已用于分析渐近性特性。 Zeta功能是这种类型分析的强大工具，现在它们已用于数字理论之外的许多不同数学领域（包括分形的研究）。 该项目的目的是显着扩展ZETA功能在分形集和度量的几何形状中的使用。现有的几何分形Zeta函数编码了Minkowski（盒子计数）尺寸和集合的Minkowski内容，该尺寸通过合适的Meromorthic Extension的极点。在某些情况下，可用明确的公式提供有关几何振荡的精确定量信息。 所提出的项目将通过定义与Hausdorff，包装和Assouad尺寸相关的新Zeta函数以及定义提供“局部”信息的Zeta函数，扩展分形Zeta函数的几何影响。 此外，该项目将探讨如何在基础空间的映射下转换这些ZETA函数，并定义具有良好映射属性的新Zeta函数。 这里的目的是创建用于分析分形集的预测和切片（与一条线的交集）的新工具，并决定何时两个分形相当于Bi-Lipschitz。 该项目的结果将引入实质性工具，以检查许多几何测量方法和措施。 M. Lapidus及其合作者先前引入的分形Zeta功能已经影响了动态系统，非交通性几何形状和理论物理学的研究（仅举三个领域）。在这些（和其他）区域中，宽阔的分形Zeta功能似乎也可能会发现重要性。