Ergodic Theory of Foliated Spaces through Geometric Deformations

这个项目的目标是开发一套工具，用它来研究一个大类的动力系统。这门大课中的系统来自不同的领域，从多边形的弹跳球到奇异晶体的晶体学，等等。这些已经用不同的技术进行了研究。我们的目标是展示这门大课中的系统都有共同的属性，并提出一个适用于这门课中所有系统的观点。更具体地说，提出的项目旨在发展一个更统一的理论，翻译行为的遍历性质在某些叶状空间。这些系统中最著名的是平面上的平移流。用于研究这些系统的一个关键工具是在通过叶状空间的几何变形获得的模空间上使用重整化流。本项目旨在将这些工具和思想引入到其他（通常是更高级别）翻译行为的动态研究中。例如，准晶体，非周期平铺和Delone集的主题是一个特别好的主题，其中应用了在teichm<s:1> ller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，已经在平面的研究中被利用，也可以用来研究来自准晶体和非周期平铺的系统的性质。用于研究平铺的工具，如非交换几何，也可以用于研究平面上的平移流，包括无限型平面。田地的这种交叉施肥将是互利的。

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.