Ergodic theory of infinite interval exchange transformations

这个项目涉及到的遍历理论结果的推广区间交换变换（IECHO）的情况下，区间交换涉及无限多个区间。主要的问题是分类的局部有限遍历不变测度的无限IET。对无限情况的兴趣是由无限Icons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，主要研究者和其他领域的工作表明，存在的相似之处之间的理论无限的Irons，和更发达的理论horocyclic流动的非紧双曲曲面。在这些联系的推动和指导下，关于无穷大的已知结果将得到推广，并产生新的结果。动力系统是研究任何按固定规则更新的系统。例如，有限数量的粒子按照牛顿力学的规则运动就是一个动力系统。该项目涉及据说复杂度较低的系统，这意味着系统的相似状态需要很长时间才能变得不相似。这些系统是感兴趣的，因为它们自然产生于几何，代数和物理问题。尽管这些系统的相关性，涉及低复杂性系统的问题往往不能直接解决的方法流行的动力系统理论，因为，例如，这些系统不是双曲。对于低复杂度系统的长期统计行为问题，尤其如此。这个项目将扩展适用于低复杂度系统的工具集，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是重要的发展和适用性的动力系统理论，这是一个主要的数学方法可用于了解物理世界。PI将在夏季与本科生一起进行相关研究。

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.