Renormalization in piecewise isometric dynamical systems

其更改受与时间无关的更新规则控制的对象的集合称为动态系统。科学中的例子比比皆是，包括天体的运动、天气模式、流体通过管道的行为以及化学过程。这个主题中的一个基本问题是确定这样一个系统的行为。所谓的双曲动力系统是这一领域成功的一个主要例子。这些系统允许一种特殊的膨胀和收缩，这使得分析这些系统的行为的某些数学方法变得可行。这个项目的中心是理解处于相反极端的系统，分段等距系统，它不允许收缩或膨胀，因此迫使使用不同的方法来理解这些系统。这些系统通常是通过重整化来理解的。重整化是一种越来越密切地观察或放大重复行为的方法。重整化是这个项目的一个主要部分，PI将开发新的重整化方法并改进现有的方法来理解这些系统。更好地理解这些系统将对我们作为一门学科理解动力系统产生广泛的影响。除了项目的研究目标外，PI还将与学生合作，提供数学研究方面的指导和培训。PI将为使用重整化来理解分段等距动力系统及其相关系统中的动力学现象做出贡献。感兴趣的系统包括双曲曲面上的环环流、平坦流形上的测地线流和分段等距系统。这些系统中的许多都是通过与低维拓扑和几何的连接自然产生的。分段等距图是将公制空间分割成若干小块，并对每个小块应用等距图以重新组合整个空间的方法。例如区间交换变换(IETs)，其中空间是一个区间，该区间被分割成有限多个子区间。IETS的理论已经发展得相当成熟，证明了重整化方法的力量，但相比之下，许多相关的系统只被很少了解。一个主要的目标是将重整化方法的适用性扩展到更广泛的系统类，包括多边形交换变换和无限区间交换变换。新的动力学现象将被发现，并使用PI发展的重整化方法进行严格的研究。本提案中的研究成果将通过在研究期刊上发表和通过会议发言广泛传播。

A collection of objects whose changes are governed by a time-independent update rule is called a dynamical system. Examples are pervasive in the sciences, including the motion of celestial bodies, weather patterns, the behavior of fluids moving through a pipe, and chemical processes. A basic question in the subject is to determine the behavior of such a system. So called hyperbolic dynamical systems are a prime example of success in the field. These systems admit a special kind of expansion and contraction, which make available certain mathematical methods of analyzing the behavior of these systems. This project centers on understanding systems at the opposite extreme, piecewise isometric systems, which admit no contraction or expansion and thus force different approaches to be used to understand these systems. These systems are typically understood through renormalization. Renormalization is a method of looking more and more closely at, or zooming in on, repeated behavior. Renormalization is a major part of this project and the PI will develop new renormalization methods and improve existing methods to understand these systems. A better understanding of these systems will have broad consequences for our understanding of dynamical systems as a subject. In addition to the research aims of the project, the PI will work with students, providing mentoring and training in mathematical research.The PI will make contributions toward the use of renormalization to understand dynamical phenomena in piecewise isometric dynamical systems and related systems. Systems of interest include horocyclic flows on hyperbolic surfaces, geodesic flows on flat manifolds, and piecewise isometric systems. Many of these systems arise naturally through connections with low-dimensional topology and geometry. A piecewise isometry is formed by cutting a metric space into pieces and applying an isometry to each piece so as to reassemble the whole space. Examples include interval exchange transformations (IETs), where the space is an interval, which is cut into finitely many subintervals. The theory of IETs is quite well developed and stands as a testament to the power of renormalization methods, but in contrast, many related systems are only poorly understood. A primary goal is to extend the applicability of renormalization methods to wider classes of systems including polygon exchange transformations, and infinite interval exchange transformations. New dynamical phenomena will be discovered and rigorously studied using the renormalization methods developed by the PI. Results produced by research in this proposal will widely disseminated through publication in research journals and through conference presentations.