Thermodynamic formalism and flows on moduli space

热力学形式主义和模空间上的流动

基本信息

批准号：
EP/J013560/1
负责人：
Mark Pollicott
金额：
$ 33.88万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ013560%2F1
关键词：
Thermodynamic formalism flows moduli space

项目摘要

In the broadest sense, Ergodic theory is the branch of analysis which has developed most rapidly in the last century, and which has had many striking achievements, particularly in the past few decades. This is noticable, in particular, in terms of applications to number theory. Notable important highlights were Wolf prize winner Furstenberg's proof of Szemerdi's theorem on arithmetic progressions; Fields' medallist Margulis' proof of the Oppenheim conjecture and the Einsideler-Katok-Lindenstrauss (another Fields' medallist) contribution to the classical Littlewood conjecture. Many of these proofs use a particularly geometric viewpoint. The general principle of applying ergodic theory to geometry is now both well established and fundamental. This is bourne out by the examples of the fundamental and classical Mostow rigidity theorem (which, of course, show that in higher dimensions the Moduli space is trivial and emphasizes the interest in surfaces) and the seminal work of Margulis on lattice point and closed orbit counting for negatively curved manifolds, and super-rigidity for Lie groups.Historically, ergodic theory has its roots in theoretical physics and, in particular, statistical mechanics, and is generally concerned with the long term stochastic behaviour of deterministic dynamical systems. Moreover, one of the key methods of our analysis, thermodynamic formalism, is a particularly fruitful branch of ergodic theory, with strong connections to statistical mechanics.The underlying theme in the proposed programme of research is to study the application of ergodic theory and thermodynamic formalism in order to gain a better insight into metrics on Riemann surfaces and their geometry. The connection between ergodic theory and geometry in our proposal comes from the classical viewpoint of studying the dynamics of the geodesic flow. However, considering the flow on moduli spaces, instead of classical Riemannian manifolds, leads to more challenging technical problems.The programme of proposed research is divided into four key areas. Firstly, studying the dynamics of the Weil-Petersson geodesic flow. This is an area in which there has been considerable progress in the past couple of years, and we have made particular contributions to this. In particular, the Weil-Petersson metric is one which has negative curvature(s) and thus is amenable to many classical techniques in ergodic theory, by analogy with the theory of scattering billiards (notwithstanding some considerable technical problems). Moreover, the subtle interplay between the dynamics and the geometry gives a greater insight into both aspects. A second area is the study of the Teichmuller geodesic flow. This is a topic which has received considerable attention from leading experts in mathematics (e.g., Fields' medallists McMullen and Kontsevich). However, statistical properties of such flows can be studied using techniques from thermodynamic formalism since the flows can be conveniently realised as suspension flows over countable branch expanding maps.A third area of investigation relates to the determinant of the laplacian, whose origins are related to mathematical physics. This is a function defined on the space of function whose behaviour is particularly mysterious. Using techniques we have developed over several years we will determine interesting values and points associated to the function. In particular, we expect to resolve a long standing problem of Sarnak in this area.The final area of study is at the level of the surfaces themselves. We want to give a new interpretation for the canonical invariants discovered by Forni-Flaminio in the special case of surfaces of constant curvature and to extend the theory to more general surfaces. The basic approach uses recent work of ours on the dynamical zeta function. This offers the possibility of opening up a whole new field of research.

从最广泛的意义上讲，千古理论是上个世纪发展最快的分析分支，并且取得了许多惊人的成就，尤其是在过去的几十年中。就数字理论的应用而言，这尤其是值得注意的。值得注意的重要亮点是沃尔夫奖得主弗斯滕伯格（Furstenberg）证明了Szemerdi关于算术进程定理的证明；菲尔兹（Margulis）的奖牌获得者猜想的证据证明了奥普斯坦海姆（Oppenheim）和埃西德勒（Einsideler-Katok-Lindenstrauss）（另一个领域的奖章）对古典莱特伍德（Littlewood）猜想的贡献。这些证据中的许多都使用了特别的几何观点。现在，将千古理论应用于几何形状的一般原则既建立又基本。这是由基本和古典刚性定理的示例（当然表明，模态空间都是微不足道的，并且强调表面的兴趣）以及Margulis对Lattice Point的精确工作的兴趣和封闭的Orbit计算层次的折叠式和超级依赖的ORTODSISTISS，ERD的概念是ERD的，ER编的是，ERD的概念是，ER的概念是，ER的范围是，这是BOURNE的。特别是统计力学，通常与确定性动力学系统的长期随机行为有关。此外，我们分析的关键方法之一是热力学形式主义，是与统计力学的富有成果的分支。拟议的研究计划中的基本主题是研究麦芽胶质理论和热力学形式主义的应用，以便在riemann and riemann surface and is unemann cemement and is surface and is thermant andsight and Thermotynalismism中的应用。在我们的提案中，千古理论与几何形状之间的联系来自研究测量流动的动力学的经典观点。但是，考虑到模量空间上的流动，而不是经典的riemannian歧管，会导致更具挑战性的技术问题。拟议的研究计划分为四个关键领域。首先，研究Weil-Petersson测量流的动力学。在过去的几年中，这是一个取得很大进步的领域，我们为此做出了特别的贡献。特别是，Weil-Petersson指标是一种具有负曲率的指标，因此可以通过类似于散射台球理论（尽管存在一些相当大的技术问题）来适合厄戈德理论中的许多经典技术。此外，动力学和几何形状之间的微妙相互作用使得对这两个方面都有更深入的了解。第二个区域是对Teichmuller Geodesic流的研究。这是一个受到数学领先专家的关注的主题（例如，菲尔德的奖章麦克马伦和肯特塞维奇）。但是，可以使用热力学形式主义的技术研究此类流的统计特性，因为这些流可以方便地实现为悬架流在可数的分支扩展地图上的悬架流量。研究的第三个研究领域与拉普拉斯的决定因素有关，其起源与数学物理学有关。这是在功能空间上定义的函数，其行为特别神秘。使用几年来我们开发的技术，我们将确定与该功能相关的有趣值和点。特别是，我们期望在这一领域解决一个长期存在的Sarnak问题。研究的最终领域是表面本身的水平。我们希望为Forni-Flaminio在恒定曲率表面的特殊情况下发现的规范不变性提供新的解释，并将理论扩展到更一般的表面。基本方法使用我们的最新工作在动态Zeta功能上。这提供了开放一个全新研究领域的可能性。