Geometric Properties of Fractals That Arise in Various Dynamical Settings

这个项目的主要目标是研究各种形式的对称。自古以来，人们就对对称现象着迷，在视觉艺术，特别是建筑，诗歌和音乐中可以很容易地找到对称的例子。通常，这些示例中的对称性是镜像的，即，当图案在相同大小的空间或时间间隔中重复时，对象在平坦表面上加倍或平移。自然界也充满了对称性。最广为人知的例子是闪电，蕨类植物，花椰菜或花椰菜。这里的对称性是一种不同的类型，即它是自相似的，拥有这种对称性的物体被称为分形。分形在不同的尺度上大致相同，即，分形的一部分看起来像是整个分形的较小副本。本计画研究具有上述对称性的几何空间以及这种对称性的推广，例如，准自相似性更具体地说，本项目研究支持动力系统的分形空间的动力学性质和曲率分布。在这里，动态通常指的是在给定分形上的映射或映射系统的迭代。更确切地说，PI计划根据它们的拟对称群、它们支持的拟正则动力学或与这些分形相关的填充的曲率分布函数的渐近性，将分形空间从不同的家族中分类。该项目主要关注分形，即后临界有限有理映射的Julia集或各种自相似结构的剩余集，如Apollonian垫片。所采用的方法来自群体和理性映射的动力学，以及复分析及其最近的同行。例如，最近的发展表明，许多分形空间可以有效地研究使用组合工具和技术的分析度量空间。此外，动态分形的几何形状影响与这种分形相关的填充的某些分析性质。例如，在一个示例中，各种动力学填充的曲率分布函数的渐近性与相应剩余集的分形维数有关。PI希望该项目将增加新的方法和想法，特别是将更多地揭示这些领域的关系，即几何，动力学和相关包装的曲率分布。复变函数分析经常为解决来自自然科学、工程和其他数学领域的问题提供工具和方法。最近与物理学有关的例子包括统计物理学中二维晶格模型连续极限的共形不变性的研究和量子引力的应用。PI希望他的研究能够揭示分形的其他几何和其他性质，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知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.