Ergodic Theory of Foliated Spaces through Geometric Deformations

这个项目的目标是开发一套工具来研究一大类动力系统。这一大类中的系统来自不同的地方，从多边形中的弹跳球到奇异晶体的结晶学，仅举几例。人们已经使用不同的技术对它们进行了研究。我们的目标是证明这一大类中的系统具有共同的性质，并发展一种适用于这类中所有系统的观点。更具体地说，该项目旨在发展关于某些叶状空间上的平移作用的遍历性质的更统一的理论。其中最著名的系统是平面上的平移流。用来研究这些系统的一个关键工具是利用模空间上的重整化流，这些重正化流是通过分叶空间的几何变形而获得的。这个项目试图将这些工具和想法引入到其他(通常是更高级别的)翻译行为的动态研究中。例如，准晶、非周期瓦片和Delone集的主题是一个特别好的主题，它应用了Teichmüler动力学中使用的思想。特别是，遍历理论、上同调和几何变形之间的关系，已经被用来研究平面的研究，也可以被用来研究来自准晶和非周期瓦片的系统的性质。用于研究平铺的工具，如非对易几何，也可以用来研究平面上的平移流，包括无限大类型的平移流。这种农田的交叉施肥将是互惠互利的。

The goal of this project is to develop a set of tools with which to study a large class of dynamical systems. The systems in this large class come from different places, from bouncing balls in polygons to the crystalography of exotic crystals, to name a few. These have been studied using different techniques. The goal here is to show that the systems in this large class have common properties and to develop a point of view which applies to all systems in this class.More specifically, the proposed project aims at developing a more unified theory of ergodic properties of translation actions on certain foliated spaces. The best known of these systems are translation flows on flat surfaces. A key tool used to study these systems is the use of renormalization flows on moduli spaces obtained through geometric deformations of the foliated spaces. This project seeks to introduce these tools and ideas to the study of the dynamics of other (usually higher rank) translation actions. For example, the subject of quasicrystals, aperiodic tilings and Delone sets is a particularly good one in which to apply the ideas used in Teichmüller dynamics. In particular, the relationship between ergodic theory, cohomology, and geometric deformation, which has been exploited in the study of flat surfaces, can also be exploited to study the properties of systems coming from quasicrystals and aperiodic tilings. Tools used in the study of tilings such as non-commutative geometry can also be used to study translation flows on flat surfaces including those of infinite type. This cross-fertilization of fields will be mutually beneficial.