CAREER: Interval Exchange Transformations and Translation Surfaces

职业：区间交换变换和平移曲面

• 批准号: 1452762
• 负责人: Jonathan Chaika
• 金额: $ 44.7万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1452762&HistoricalAwards=false
• 关键词: CAREER; Interval; Exchange; Transformations; Translation

Dynamical systems can be thought of as the study of objects in motion, e.g. planets, where the motion evolves over time according to a fixed set of rules. The evolution can be structured or quite complex and even chaotic. This research project studies a family of dynamical systems that are highly structured but whose long term behavior can often be understood via a chaotic system that renormalizes the dynamics. The analysis of these problems involves dynamical, geometric, and combinatorial arguments. The project also has a large component geared towards smoothing the progression of early career mathematicians. This includes research mentoring for undergraduate and graduate students, professional mentoring for graduate students, notes to introduce first year graduate students to ergodic theory while preparing them for qualifying exams in real analysis, a problem list of topics in the area, and a collection of key arguments in ergodic theory.The goal of this project is to better understand the dynamics of interval exchange transformations, flows on translations surfaces, and Teichmueller geodesic flow. Interval exchange transformations and flows on translation surfaces are closely related objects. Interval exchange transformations arise as first return maps of flows on translation surfaces, and similarly a translation surface can be built from an interval exchange with additional suspension data. Questions of interest are generalizations of the isomorphism problem, non-unique ergodicity and its variants, the behavior of bounded sets, and the behavior of actions of matrices on the space of translation surfaces. These problems will be studied using renormalization dynamics -- Rauzy induction and Teichmueller geodesic flow -- as well as combinatorial constructions, especially in building examples with exotic properties.

动力学系统可以被认为是对运动中的物体(例如行星)的研究，其中运动根据一组固定的规则随时间演变。进化可能是结构化的，也可能非常复杂，甚至是混乱的。这项研究项目研究了一类高度结构化的动力系统，但其长期行为通常可以通过重新规格化动力学的混沌系统来理解。对这些问题的分析涉及动力学、几何和组合论证。该项目还有一个很大的组成部分，旨在为早期职业数学家的发展铺平道路。这包括本科生和研究生的研究指导，研究生的专业指导，向一年级研究生介绍遍历理论的笔记，同时为真实分析的合格考试做准备，该领域的问题清单，以及遍历理论的关键论点的集合。本项目的目标是更好地理解区间交换变换的动态，翻译表面上的流动，以及泰希穆勒测地线流动。平移曲面上的区间交换变换和流是密切相关的对象。区间交换变换作为平移面上的流动的第一返回映射而产生，类似地，可以从具有附加悬浮数据的区间交换建立平移面。感兴趣的问题是同构问题的推广，非唯一遍历性及其变体，有界集的行为，以及平移曲面空间上矩阵的作用的行为。这些问题将使用重整化动力学--劳兹诱导和泰希穆勒测地线流--以及组合结构来研究，特别是在构建具有奇异性质的例子时。