Thermodynamic formalism and flows on moduli space

从最广泛的意义上讲，遍历理论是分析学的一个分支，它在上个世纪发展最为迅速，并取得了许多显著的成就，特别是在过去的几十年里。这是很明显的，特别是在数论的应用方面。值得注意的重要亮点是沃尔夫奖得主芙丝汀宝对等差数列的Szemerdi定理的证明；菲尔兹的奖牌获得者马古利斯对奥本海姆猜想的证明，以及埃内德勒-卡托克-林登施特劳斯（另一位菲尔兹的奖牌获得者）对经典利特尔伍德猜想的贡献。这些证明中有许多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在既确立又基本。这是由基本的和经典的Mostow刚性定理（当然，它表明在高维模空间是平凡的，并强调对曲面的兴趣）的例子和马古利斯关于负弯曲流形的格点和闭轨道计数以及李群的超刚性的开创性工作证明的。历史上，遍历理论起源于理论物理学，特别是统计力学，通常与确定性动力系统的长期随机行为有关。此外，我们分析的关键方法之一，热力学形式论，是遍历理论中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，以便更好地了解黎曼曲面及其几何上的度量。在我们的建议中，遍历理论与几何之间的联系来自于研究测地线流动动力学的经典观点。然而，考虑模空间上的流动，而不是经典的黎曼流形，会导致更具挑战性的技术问题。拟议的研究方案分为四个关键领域。首先，研究了Weil-Petersson测地线流的动力学。这是一个在过去几年中取得了相当大进展的领域，我们为此作出了特别的贡献。特别地，韦尔-彼得森度规是一个负曲率的度规，因此与散射台球理论类似，适用于遍历理论中的许多经典技术（尽管存在一些相当大的技术问题）。此外，动力学和几何之间微妙的相互作用使人们更深入地了解这两个方面。第二个领域是对Teichmuller测地线流的研究。这是一个受到数学领域顶尖专家（例如Fields的奖牌获得者McMullen和Kontsevich）相当关注的话题。然而，这种流动的统计性质可以使用热力学形式的技术来研究，因为流动可以方便地实现为可计数分支展开映射上的悬浮流动。第三个研究领域与拉普拉斯行列式有关，它的起源与数学物理有关。这是一个定义在函数空间上的函数，它的行为特别神秘。使用我们多年来开发的技术，我们将确定与函数相关的有趣值和点。特别是，我们希望解决这个地区长期存在的萨尔纳克问题。研究的最后一个领域是表面本身的水平。我们想对Forni-Flaminio在常曲率曲面的特殊情况下发现的正则不变量给出一个新的解释，并将该理论推广到更一般的曲面。基本方法使用了我们最近在动态zeta函数上的工作。这为开辟一个全新的研究领域提供了可能性。

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.