Ergodic theory of infinite interval exchange transformations

无限区间交换变换的遍历理论

• 批准号: 1101233
• 负责人: William Hooper
• 金额: $ 14.3万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101233&HistoricalAwards=false
• 关键词: Ergodic; theory; infinite; interval; exchange

This project involves the extension of ergodic theoretic results about interval exchange transformations (IETs) to cases of interval exchanges involving infinitely many intervals. The primary question of interest is to classify the locally finite ergodic invariant measures of an infinite IET. Interest in the infinite case is motivated by connections between infinite IETs and more classical topics. Progress on these classes of examples will make significant contributions to the field of infinite ergodic theory, and to the subject of dynamical systems. Recently, the work of the principal investigator and others in the field have shown the existence of parallels between the theory of infinite IETs, and the much more well developed theory of horocyclic flows on non-compact hyperbolic surfaces. Motivated and directed by these connections, known results about infinite IETs will be extended and new results will be produced.Dynamical systems is the study of any system which updates according to a fixed rule. For example, a finite number of particles behaving according to the rules of Newtonian mechanics is a dynamical system. This project concerns systems which are said to have low complexity, meaning that similar states for the system take a very long time to become dissimilar. These systems are of interest because they arise naturally from geometric, algebraic and physical questions. Despite the relevance of these systems, questions involving low complexity systems often can not be directly addressed by approaches prevalent in the theory of dynamical systems, because, for instance, these systems are not hyperbolic. This is particularly true for questions about the long term statistical behavior of low complexity systems. This project will extend the collection of tools which do apply to low complexity systems, and further extend the classes of systems to which these tools apply. Progress on these questions is of importance to the development and applicability of the theory of dynamical systems, which is one of the primary mathematical approaches available to understand the physical world. The PI will be working with undergraduates on related research during the summer.

本课题将区间交换变换的遍历理论结果推广到无限多个区间的区间交换情形。我们感兴趣的主要问题是对无限IET的局部有限遍历不变测度进行分类。对无限情况的兴趣是由无限的IET和更经典的主题之间的联系所激发的。这类例子的进展将对无限遍历理论领域和动力系统学科做出重大贡献。最近，该领域的主要研究人员和其他人的工作表明，在无限IETs理论和发展得更好的非紧双曲曲面上的环流理论之间存在着相似之处。在这些联系的激励和指导下，将会推广已有的关于无限IETs的结果，并产生新的结果。动力系统是对任何按照固定规则更新的系统的研究。例如，按照牛顿力学的规则运行的有限数量的粒子就是一个动力系统。这个项目涉及的系统据说具有低复杂性，这意味着系统的相似状态需要很长时间才能变得不同。这些系统之所以令人感兴趣，是因为它们自然地源于几何、代数和物理问题。尽管这些系统是相关的，但涉及低复杂性系统的问题通常不能用动力系统理论中流行的方法直接解决，因为例如，这些系统不是双曲的。对于有关低复杂性系统的长期统计行为的问题，这一点尤其正确。该项目将扩展适用于低复杂性系统的工具集合，并进一步扩展这些工具适用的系统类别。在这些问题上的进展对动力系统理论的发展和适用性具有重要意义，动力系统理论是理解物理世界的主要数学方法之一。暑假期间，PI将与本科生一起进行相关研究。