Expanding Thurston Maps and Fractal Geometry

在自然界中，有许多现象表现出分形特征，如闪电、植物和晶体的生长模式、雪花、海岸线和河网。在数学中，分形对象经常出现在动力系统的研究中。这个项目将探索某些分形空间的几何结构。首席研究员将开发更好的分析和几何工具，以提高对来自非常具体的动力系统的分形的理解，即所谓的扩展瑟斯顿图。早期职业研究人员参与这项活动将有助于增加该领域的专门知识，并将有助于维持一个科学共同体，为科学和工程的进步提供必要的数学知识。该项目包括培养研究生。扩展Thurston地图提供了令人惊讶的丰富景观，与分形，teichmller理论，几何群论和双曲几何有关。虽然在这个领域有许多有趣的问题，但在这个项目中，具体的问题将被挑选出来，这些问题的解决将导致对该主题的深入了解。这些问题与Thurston障碍物、teichmller空间上的诱导回拉图以及与扩展Thurston图相关的视球几何有关。本文将研究这些球的一个重要的数值不变量，即它们的阿尔福斯正则共形维数。这项研究的一个基本技术工具是路径族的组合模的概念。瑟斯顿映射的障碍物似乎与退化模量的路径族有关。这个项目的目标之一是更好地掌握这种联系，目前人们对这种联系知之甚少。该奖项反映了美国国家科学基金会的法定使命，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。

In nature, there are many phenomena that exhibit fractal features, such lightning bolts, growth patterns of plants and crystals, snowflakes, or coastlines and river networks. In mathematics, fractal objects often appear in the study of dynamical systems. This project will explore the geometry of certain fractal spaces. The principal investigator will develop better analytic and geometric tools for an improved understanding of fractals that arise from very specific dynamical systems, namely so-called expanding Thurston maps. The involvement of early-career researchers in this activity will contribute to increasing the expertise in the field, and will help to maintain a scientific community that provides the necessary mathematical knowledge for progress in science and engineering. The project includes the training of graduate students. Expanding Thurston maps provide a surprisingly rich landscape with ties to fractals, Teichmüller theory, geometric group theory, and hyperbolic geometry. While there are many interesting questions in this field, in this project specific problems will be singled out whose resolution will lead to advanced insights into the subject. These problems are related to Thurston obstructions, the induced pull-back map on Teichmüller space, and the geometry of the visual sphere associated with an expanding Thurston map. An important numerical invariant of these spheres will be studied, namely their Ahlfors regular conformal dimension. A fundamental technical tool for this investigation is the notion of combinatorial modulus of path families. It seems that obstructions for Thurston maps are tied to path families of degenerating modulus. One of the goals of this project is to get a better grasp of this connection, which is only poorly understood at present.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.