Geometric Properties of Fractals That Arise in Various Dynamical Settings

各种动态环境中出现的分形的几何性质

基本信息

批准号：
1800180
负责人：
Sergiy Merenkov
金额：
$ 18万
依托单位：
CUNY City College
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-15 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800180&HistoricalAwards=false
关键词：
Geometric Properties Fractals Arise Various

项目摘要

The main goal of this project is to study various forms of symmetry. People have been fascinated with the symmetry phenomenon since ancient times and the examples can easily be found in visual arts, in particular architecture, poetry and music. Typically, the symmetry in these examples is either mirror, i.e., the objects are doubled across a flat surface, or translational when a pattern is repeated in space or time intervals of the same size. Nature is also abundant with symmetries. The most widely recognized examples are lightning, fern, and cauliflower or broccoli. The symmetry here is of a different type, namely it is self-similarity, and objects that possess such symmetries are called fractals. Fractals are roughly the same on different scales, i.e., parts of a fractal look like smaller copies of the whole fractal. This project studies geometric spaces that possess the above symmetries and generalizations of such symmetries, e.g., quasi-self-similarities. More specifically, this project investigates dynamical properties and curvature distribution of fractal spaces that support dynamical systems. Here dynamics typically refers to the iteration of a map or a system of maps on a given fractal. More precisely, the PI plans to classify fractal spaces from various families according to their quasisymmetry groups, quasiregular dynamics that they support, or the asymptotics of the curvature distribution function of the packings associated to such fractals. The project concerns mainly fractals that are the Julia sets of postcritically finite rational maps or residual sets of various self-similar constructions, such as Apollonian gaskets. The methods to be employed come from dynamics of groups and rational maps, and from complex analysis as well as its more recent counterparts. For example, recent developments have shown that many fractal spaces can be effectively studied using combinatorial tools and techniques of analysis on metric spaces. Moreover, geometry of dynamical fractals influences certain analytic properties of packings associated to such fractals. E.g., the asymptotics of the curvature distribution function of various dynamical packings is related to the fractal dimension of the corresponding residual sets. It is the hope of the PI that the project would add new methods and ideas that, in particular, would shed more light onto the relationship of these fields, namely geometry, dynamics, and curvature distribution of associated packings. Complex analysis has often provided tools and methods for solving problems that come from natural sciences, engineering, and other fields of mathematics. Recent examples related to physics include the investigation of the conformal invariance of continuum limits of two-dimensional lattice models in statistical physics and applications to quantum gravity. The PI expects that his investigations would reveal additional geometric and other properties of fractals that arise, in particular, in natural sciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的主要目标是研究各种形式的对称性。自古以来，人们就对对称现象着迷，在视觉艺术中很容易找到这样的例子，特别是建筑、诗歌和音乐。通常，这些示例中的对称性要么是镜像的，即对象在平面上加倍，要么是平移的（当图案在相同大小的空间或时间间隔中重复时）。自然界也充满了对称性。最广泛认可的例子是闪电、蕨类植物、花椰菜或西兰花。这里的对称性是另一种类型，即自相似性，具有这种对称性的物体称为分形。分形在不同尺度上大致相同，即分形的部分看起来像整个分形的较小副本。该项目研究具有上述对称性的几何空间以及此类对称性的推广，例如准自相似性。更具体地说，该项目研究支持动力系统的分形空间的动力学特性和曲率分布。这里的动力学通常指的是给定分形上的映射或映射系统的迭代。更准确地说，PI 计划根据分形空间的拟对称群、它们支持的拟正则动力学或与此类分形相关的堆积曲率分布函数的渐近性，对不同族的分形空间进行分类。该项目主要涉及分形，即后临界有限有理图的 Julia 集或各种自相似结构的残差集，例如阿波罗垫片。所采用的方法来自群体动力学和理性图，以及复杂分析及其最近的对应方法。例如，最近的发展表明，使用组合工具和度量空间分析技术可以有效地研究许多分形空间。此外，动态分形的几何形状影响与此类分形相关的填料的某些分析性质。例如，各种动态堆积的曲率分布函数的渐近性与相应残差集的分形维数有关。 PI 希望该项目能够增加新的方法和想法，特别是能够更多地阐明这些领域的关系，即相关填料的几何、动力学和曲率分布。复杂分析通常为解决自然科学、工程和其他数学领域的问题提供工具和方法。最近与物理学相关的例子包括统计物理学中二维晶格模型连续体极限的共形不变性的研究以及在量子引力中的应用。 PI 预计他的研究将揭示分形的额外几何和其他属性，特别是在自然科学中。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。