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。自从远古时代以来，人们一直对这种对称现象着迷，并且可以轻松地在视觉艺术，尤其是建筑，诗歌和音乐中找到这些例子。通常，这些示例中的对称性是镜像，即，当对象在平坦的表面上加倍，或者当在相同大小的空间或时间间隔中重复图案时转换。自然也有对称性丰富。最广泛认可的例子是闪电，蕨类植物和花椰菜或西兰花。这里的对称性是一种不同的类型，即它是自相似性，并且具有这种对称性的对象称为分形。分形在不同的尺度上大致相同，即分形的部分看起来像整个分形的较小拷贝。该项目研究具有上述对称性的几何空间，例如准自我相似性。更具体地说，该项目研究了支持动态系统的分形空间的动力学特性和曲率分布。这里的动力学通常是指给定分形上地图或地图系统的迭代。更准确地说，PI计划根据其支持组，它们支持的准对象动力学或与此类分形相关的包装的曲率分布函数的渐近分布式分类。该项目主要涉及分形，这些分形是朱莉娅（Julia）的朱莉娅（Julia）批判性有限的理性地图或各种自相似结构（例如阿波罗尼亚垫片）的残差集。要采用的方法来自组和理性图的动力学，复杂的分析以及其最新的对应物。例如，最近的事态发展表明，可以使用组合工具和公制空间分析技术有效地研究许多分形空间。此外，动态分形的几何形状会影响与此类分形相关的包装的某些分析特性。例如，各种动力包装的曲率分布函数的渐近分数与相应残差集的分形维度有关。 PI的希望是，该项目将添加新的方法和想法，尤其会给这些领域的关系提供更多的启示，即几何，动态和相关包装的曲率分布。复杂的分析通常提供了解决来自自然科学，工程和其他数学领域的问题的工具和方法。与物理学有关的最新示例包括研究统计物理学和量子重力应用中二维晶格模型的连续性限制的共形不变性。 PI预计他的调查将揭示出尤其是在自然科学中出现的分形的其他几何和其他特性。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子的评估来获得支持的，并具有更广泛的影响。