CAREER: Interval Exchange Transformations and Translation Surfaces

动力学系统可以被认为是对运动中的物体的研究，例如行星，其中运动根据一组固定的规则随着时间的推移而演变。进化可以是结构化的，也可以是相当复杂的，甚至是混乱的。这个研究项目研究了一系列高度结构化的动力系统，但其长期行为通常可以通过将动力学重新规范化的混沌系统来理解。这些问题的分析涉及动力学，几何和组合参数。该项目还有一个很大的组成部分，旨在平滑早期职业数学家的发展。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说明，同时为他们在真实的分析中的资格考试做准备，该领域主题的问题列表，以及遍历理论中的关键论点的集合。这个项目的目标是更好地理解区间交换变换的动力学，平移曲面上的流和Teichmueller测地线流。区间交换变换和平移曲面上的流是密切相关的对象。区间交换变换作为平移表面上的流的第一返回映射而出现，并且类似地，平移表面可以从具有附加悬浮数据的区间交换来构建。感兴趣的问题是广义的同构问题，非唯一遍历性及其变种，有界集的行为，以及行为的矩阵的行动空间上的翻译表面。这些问题将研究使用重整化动力学- Rauzy感应和Teichmueller测地流-以及组合结构，特别是在建设具有异国情调的性质的例子。

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.