Mapping Class Semigroups and the Classification of Conformal Dynamical Systems

动力系统是描述系统随时间演变的数学对象。它们模拟了许多类型的行为，从动物神经元到流行病再到天气模型。在某些情况下，我们有最简单类型的分类，并且知道它们是如何排列的。例如，我们现在对实二次情况有了很好的理解。这一成就是来自不同领域的许多研究人员共同努力的结果。给出超出实际二次集的示例分类仍然是一个核心挑战。本项目使用来自几个不同数学领域的工具来应对这一挑战。它侧重于涉及复数的系统，以及更一般类型的共形系统。本项目研究的共形动力系统包括新发现的、更奇特的例子。虽然我们对这些更一般的系统的理解较差，但我们有许多新的工具，从代数到分析，可以用来研究它们。该项目将成熟的类群映射理论中的技术应用于分类问题。映射类群是二维物体的对称性，是数学家和物理学家研究的对象。这个项目推广了映射类群的概念，以一种包括这些二维复杂动力系统的方式。它还应用了最近发展起来的自相似群理论的新工具。这个研究领域有许多容易解决的问题。参与这些主题的学生将逐渐体会到数学的本质统一和研究的兴奋。这将有助于培养广泛训练的数学人才。为这项研究设计的特殊软件允许丰富的实验和技术技能的发展。复杂动力系统的组合基础是由A. Douady、J. Hubbard和W. Thurston奠定的。由于缺乏一个足够自然的代数框架，在基本问题上的进展被拖延了，直到2006年L. Bartholdi和V. Nekrashevych引入了新技术。这些自相似组技术现在是标准的。这个新领域的最新发展与映射类群理论的发展是平行的。分支点的前向轨道构成有限集的自然对象——球面的分支自覆盖，可以被有效地视为表示可数半群中的分支映射类。这个半群是纯映射类群上的一个二集，这使得组合结构非常丰富。就像映射类群一样，在半群方面，与Teichmueller理论、动力学、几何和算法问题也有类似的非常深刻的联系。更全面地探索这些联系是这个项目的重点。目前缺乏基本的动力有限性结果，以及相对双曲性的适当概念的发展。有趣的是，相对双曲性的概念似乎自然地导致了与伯科维奇空间上的粗糙几何和算术动力学的联系。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。

Dynamical systems are mathematical objects that describe the evolution of systems over time. They model many types of behavior, from animal neurons to epidemics to weather modeling. In some cases, we have a classification of the simplest types, and know how they are arranged. For example, we have now a good understanding in the real quadratic case. This achievement was the result of work by many researchers from different fields. It remains a central challenge to give a classification of examples beyond the real quadratic setting. This project takes up this challenge using tools from several different areas of mathematics. It focuses on systems that involve complex numbers, and on more general types called conformal. The conformal dynamical systems studied in this project include newly discovered, more exotic examples. Though our understanding of those more general systems is poorer, we have many new tools, from algebra to analysis, with which to study them. The project applies techniques from the well-developed theory of mapping class groups to the classification problem. Mapping class groups are symmetries of two-dimensional objects and are studied by both mathematicians and physicists. This project generalizes the notion of a mapping class group in a way that includes these two-dimensional complex dynamical systems. It also applies new tools from the recently developed theory of self-similar groups. This research area has an abundance of accessible problems. Students who engage with these topics will come to appreciate the essential unity of mathematics and the excitement of research. This will contribute to the development of a pool of mathematical talent that is broadly trained. Special software designed for this study allows for rich experimentation and the development of technical skills. The combinatorial foundations of complex dynamical systems were laid by A. Douady, J. Hubbard, and W. Thurston. The lack of a sufficiently natural algebraic framework delayed progress on fundamental problems until new techniques were introduced by L. Bartholdi and V. Nekrashevych in 2006. These selfsimilar group techniques are now standard. The recent developments in this new field are paralleling those in the theory of mapping class groups. The natural objects–branched self-covers of the sphere whose forward orbits of branch points form a finite set—may be fruitfully regarded as representing branched mapping classes in a countable semigroup. That this semigroup is in addition a biset over the pure mapping class group makes the combinatorial structure immensely rich. Just as with mapping class groups, on the semigroup side, there are similarly very deep connections to Teichmueller theory, dynamics, geometry, and algorithmic questions. The fuller exploration of these connections is the focus of this project. Basic dynamical finiteness results, and the development of an appropriate notion of relative hyperbolicity, are currently lacking. Intriguingly, the notions of relative hyperbolicity seem to lead naturally to connections with both coarse geometry and arithmetic dynamics on Berkovich spaces.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.