Zeta functions in fractal geometry and analysis

分形图是数学对象，它在所有分辨率级别上都有详细的结构(中间1/3的康托集是一个经典的例子)。它们通常作为动力系统中的不变集出现，但它们本身也是广泛研究的主题。对分形集的几何分析通常是在规模缩小到零时，通过检查其结构的某些度量的渐近性来进行的。维度的各种概念(对于度量空间)都是这个过程的例子。自从Riemann首次将Zeta函数引入素数计数函数的研究以来，Zeta函数一直被用于渐近性质的分析。Zeta函数是这类分析的强大工具，现在它们被用于数论以外的许多不同的数学领域(包括研究分形学)。 该项目的目标是显著扩展Zeta函数在研究分形集和测度的几何学中的使用。现有的几何分形Zeta函数通过适当的亚纯扩张的极点来编码集合的Minkowski(计盒)维度和Minkowski内容。在某些情况下，可以得到关于几何振荡的精确定量信息的显式公式。 该项目将通过定义与Hausdorff、Packing和Asourad维度相关的新的Zeta函数，以及通过定义给出“局部”信息的Zeta函数，来扩展分形Zeta函数的几何范围。此外，该项目将探索这些Zeta函数如何在底层空间的映射下进行转换，并定义具有良好映射属性的新Zeta函数。这里的目标是创建新的工具来分析分形集的投影和切片(与直线相交)，以及确定两个分形图何时是双Lipschitz等价的。 该项目的结果将为检验集合和测度的许多几何测度论性质提供重要的工具。拉皮德斯和他的合作者之前引入的分形Zeta函数已经影响了动力系统、非对易几何和理论物理(仅举三个领域)的研究。一类扩大的分形Zeta函数似乎也将在这些(和其他)领域找到意义。

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.