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，遍历理论是上个世纪发展最快的分析分支，特别是在过去的几十年里取得了许多引人注目的成就。这在数论的应用方面尤其值得注意。值得注意的重要亮点是沃尔夫奖获得者弗斯滕伯格对算术级数 Szemerdi 定理的证明；菲尔兹奖获得者马古利斯对奥本海姆猜想的证明以及艾因塞德勒-卡托克-林登斯特劳斯（另一位菲尔兹奖获得者）对经典利特尔伍德猜想的贡献。许多这样的证明都使用了特殊的几何观点。将遍历理论应用于几何的一般原理现已得到充分确立和基础。基本和经典的莫斯托刚性定理的例子（当然，该定理表明，在更高维度中，模空间是微不足道的，并且强调了表面的兴趣）以及马古利斯在格点和闭合轨道上的开创性工作证明了这一点计算负曲流形，以及李群的超刚性。历史上，遍历理论根源于理论物理学，特别是统计力学，并且通常关注长期确定性动力系统的随机行为。此外，我们分析的关键方法之一，热力学形式主义，是遍历理论的一个特别富有成果的分支，与统计力学有着密切的联系。拟议研究计划的基本主题是研究遍历理论和热力学形式主义的应用为了更好地了解黎曼曲面及其几何形状的度量。我们的建议中遍历理论与几何之间的联系来自研究测地流动力学的经典观点。然而，考虑模空间上的流动，而不是经典黎曼流形，会导致更具挑战性的技术问题。所提出的研究计划分为四个关键领域。首先，研究Weil-Petersson测地流的动力学。这是一个在过去几年中取得了相当大进展的领域，我们为此做出了特别的贡献。特别是，Weil-Petersson 度量具有负曲率，因此适用于遍历理论中的许多经典技术，类似于散射台球理论（尽管存在一些相当大的技术问题）。此外，动力学和几何之间的微妙相互作用可以让我们更深入地了解这两个方面。第二个领域是泰希米勒测地流的研究。这个话题受到了顶尖数学专家（例如菲尔兹奖得主麦克马伦和康采维奇）的极大关注。然而，这种流动的统计特性可以使用热力学形式主义的技术来研究，因为这些流动可以方便地实现为可数分支扩展图上的悬浮流。第三个研究领域涉及拉普拉斯行列式，其起源与数学有关物理。这是一个定义在函数空间上的函数，其行为特别神秘。使用我们多年来开发的技术，我们将确定与该函数相关的有趣值和点。特别是，我们希望解决萨尔纳克在这一领域长期存在的问题。最后的研究领域是表面本身的水平。我们希望对 Forni-Flaminio 在恒定曲率曲面的特殊情况下发现的正则不变量给出新的解释，并将该理论扩展到更一般的曲面。基本方法使用我们最近关于动态 zeta 函数的工作。这提供了开辟一个全新研究领域的可能性。