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Non-Generic Teichmueller Dynamics

非通用 Teichmueller 动力学

基本信息

批准号：
EP/I019030/1
负责人：
Corinna Ulcigrai
金额：
$ 8.67万
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI019030%2F1
关键词：
Non Generic Teichmueller Dynamics

项目摘要

Imagine playing billiard in a table whose shape is not rectangular as usual, but polygonal, for example in an octagonal table. Imagine that the ball is a point that moves without friction, so that it keeps moving forever if it does not hit any pockets. This is a mathematical billiard. These idealized billiards are studied by mathematicians because they arise naturally as models of various systems in the physical sciences. Moreover, they represent one of simplest models of chaos. Billiard trajectories that never hit a pocket become quickly complicated and chaotic. One of the main goals in dynamical systems and ergodic theory is to describe and quantify this chaotic properties and to be able to predict thei asymptotic behaviour.One of the objectives of this research is to prove chaotic properties of a class of billiards in polygons. The goal of understanding billiards in polygons has led to the development of very deep mathematical theories, in particular to the study of an abstract space (the moduli space) of deformations of surfaces. The study of moduli spaces has independent interest and connections and applications to different areas of fundamental research in mathematics (as algebraic geometry and number theory) and in theoretical physics. The beautiful philosophy that has guided the research in the field is that properties of a single billiard are reflected by the properties of the dynamics in the moduli space. Unfortunately, some of the deep results obtained for these abstract spaces do not apply directly to billiards, which form a small set in this abstract space.In our proposal we plan to develop methods and tools to solve some of the open questions for polygonal billiards and other concrete systems which cannot currently be attacked by the standard tools. We have recently developed a renormalization technique which allows to attack these problems. We plan to get very precise information which is not usually accessible for generic systems. Other related objectives concern the properties of maps of the interval, called interval exchange transformations, which arise naturally in connection with billiards and have been studied as a basic model of dynamical system. We plan to investigate long-standing open problems about their spectral properties and a new generalization of classical Diophantine properties.

想象一下，在一张形状不是通常的矩形而是多边形的桌子上打台球，例如在一张八角形的桌子上。想象一下，球是一个没有摩擦力的点，所以如果它没有击中任何口袋，它就会永远移动。这是一个数学台球。这些理想化的台球被数学家研究，因为它们自然地作为物理科学中各种系统的模型出现。此外，它们代表了最简单的混沌模型之一。从未击中球袋的台球轨迹很快就会变得复杂和混乱。动力系统和遍历理论的主要目标之一是描述和量化这种混沌性质，并能够预测其渐近行为.本研究的目标之一是证明一类多边形台球的混沌性质.理解多边形中台球的目标导致了非常深入的数学理论的发展，特别是对曲面变形的抽象空间（模空间）的研究。模空间的研究对数学（如代数几何和数论）和理论物理的基础研究的不同领域具有独立的兴趣和联系和应用。指导该领域研究的美丽哲学是，单个台球的属性由模空间中的动力学属性反映。不幸的是，这些抽象的空间得到的一些深刻的结果不直接适用于台球，形成一个小的集合在这个抽象的spaces.In我们的建议，我们计划开发的方法和工具来解决一些开放的问题多边形台球和其他具体的系统，目前不能攻击的标准工具。我们最近开发了一种重正化技术，它允许攻击这些问题。我们计划获得非常精确的信息，这是一般系统通常无法获得的。其他相关的目标涉及的属性映射的时间间隔，所谓的时间间隔交换变换，这自然出现在与台球和已被研究作为一个基本模型的动力系统。我们计划调查长期存在的公开问题，他们的光谱性质和一个新的推广经典丢番图性质。