Geometric Properties of Fractals That Arise in Various Dynamical Settings

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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.