Uniformization and Rigidity of Sierpinski Carpets and Schottky Sets

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The aim of the project is to investigate the geometric properties of Sierpinski carpets and related sets under quasisymmetric maps. Standard Sierpinski carpets are obtained from a square in the plane using simple iterative procedures. The first step involves a subdivision of the square into smaller squares and the removal of the interior of one or more of these smaller squares that do not touch each other or the original outer square. The procedure is repeated on each of the smaller squares that remain, and the steps are repeated infinitely. The topological properties of Sierpinski carpets have been well understood since the 1950s, especially after Whyburn gave a topological characterization of such sets. For example, all standard Sierpinski carpets as just described are homeomorphic to each other. However, under quasisymmetric maps, which are maps between metric spaces closely related to quasiconformal maps, Sierpinski carpets exhibit much more rigidity. For example, there are pairs of standard Sierpinski carpets that are not quasisymmetrically equivalent. The two most important questions for Sierpinski carpets addressed in the project are the questions of uniformization and rigidity. Regarding uniformization, the project studies whether a given space is quasisymmetrically equivalent to a model space, and as to rigidity, it investigates whether two given spaces are quasisymmetrically equivalent.The project addresses questions in the area of analysis on metric spaces (sets in which there is a notion of distance). The techniques used to attack these questions originate in complex analysis. Complex analysis, in turn, has roots in physics and engineering, in particular in fluid mechanics and electrical engineering, and historically has provided tools and methods for attacking problems that arise in those areas. Within mathematics the Sierpinski carpets under investigation in the project arise in analysis as sets of fractal dimension, in dynamics as Julia sets, in the theory of Kleinian groups as limit sets, and in geometry as boundaries at infinity of Gromov hyperbolic groups, to mention a few examples. If carried out successfully, the project would have implications for the theory of Gromov hyperbolic groups that are studied in the area of mathematics known as geometric group theory. In particular, the principal investigator hopes that the project would provide clues to the Kapovich-Kleiner conjecture, which is a classification statement for Gromov hyperbolic groups whose boundaries are continuous deformations of standard Sierpinski carpets. Applications of fractal sets, such as Sierpinski carpets, have been found in physics, engineering, and more recently in atmospheric science and geoscience. For example, fractal shapes have recently been used to create fractal antennas that not only have unprecedented frequency coverage and versatility but also are very compact. The principal investigator hopes that understanding geometric properties of fractal spaces will lead to a better understanding of fractal physical objects or objects modeled on fractal spaces, such as fractal antennas, and that this in turn will lead to other applications.