RUI: Generalized Gauss Maps and Divergent Orbits

该项目的主要重点是开发预测某些数学定义的动力系统行为的技术，这些动力系统是用于描述真实的世界日常物理过程的广泛数学模型的柏拉图理想化。 除了帮助我们理解随机波动中的自然现象外，这项技术对于理解人类为满足现代社会需求而构建的复杂系统也变得越来越必要。 首席研究员将参加世界各地的各种研讨会和会议，以传播他的研究计划的最新发现，并跟上该领域的最新发展。 这些旅行也为PI提供了与合作者合作的机会。 PI还将邀请他的一些国际合作者到美国与他合作开展他们的联合项目。 这个项目解决了一个长期存在的问题，推广了著名的组合描述测地线的模块表面的动力学行为是密切相关的理论的无理数的有理数的近似。 主要研究者提出了几个推广的无缝集成数论和动力学更高的维度使用几个精心挑选的庞加莱节的均匀流，以取代全能的高斯连分数算法，并引入Farey tilings采取连分数的作用，并提供了一个组合描述的特殊线性群下的全对角子群。 这些概括萌芽在主要研究人员以前的工作确定的Hausdorff维数的一组奇异向量和承诺，导致新的见解的根本意义，同时逼近理论。

The main focus of this project is on the development of technology that predicts the behavior of certain mathematically defined dynamical systems that are platonic idealizations of a broad class of mathematical models used to describe everyday physical processes of the real world. Apart from helping us understand natural phenomena amid random fluctuations, this technology is also becoming increasingly necessary for understanding complex systems built by humans to meet the demands of modern society. The principal investigator will participate in various workshops and conferences around the world to disseminate the most recent discoveries of his research program and to keep up with latest developments in the field. These trips also provide the opportunity for PI to work with his collaborators. PI will also invite some of his international collaborators to the U.S. to work with him on their joint projects. This project addresses a long-standing problem to generalize the well-known combinatorial description of geodesics on the modular surface whose dynamical behavior is intimately tied to the theory of approximation of irrational numbers by rationals. The principal investigator proposes several generalizations of the seamless integration of number theory and dynamics to higher dimensions using several well-chosen Poincare sections of homogeneous flows to replace the all-powerful Gauss continued fraction algorithm, and introducing Farey tilings to take the role of continued fractions and provide a combinatorial description of the special linear group under its full diagonal subgroup. These generalizations germinated in the principal investigators previous work on the determination of the Hausdorff dimension of the set of singular vectors and promise to lead to new insights of fundamental significance in simultaneous approximation theory.