Thermodynamic formalism and flows on moduli space

从最广泛的意义上说，遍历理论是上个世纪发展最迅速的分析学的分支，特别是在过去的几十年里，它取得了许多引人注目的成就。特别是在数论的应用方面，这是可以理解的。值得注意的重要亮点是沃尔夫奖赢家弗斯滕伯格的证明Szemerdi定理的算术级数;领域'奖章获得者马古利斯'证明的奥本海姆猜想和Einsideler-Katok-Lindenstrauss（另一个领域'奖章获得者）的贡献经典Littlewood猜想。许多这些证明使用了一个特殊的几何观点。将遍历理论应用于几何的一般原理现在既已确立，又是基本的。这是由基本的和经典的Mostow刚性定理的例子来证明的（其中，当然，表明在更高的维度模空间是平凡的，并强调了兴趣的表面）和开创性的工作马古利斯格点和封闭轨道计数的负弯曲流形，和超刚性李群。历史上，遍历理论有其根源的理论物理学，特别是，统计力学，一般关注确定性动力系统的长期随机行为。此外，我们的分析的关键方法之一，热力学形式主义，是一个特别富有成果的遍历理论的分支，与统计mechanics.The潜在的主题在拟议的研究计划是研究遍历理论和热力学形式主义的应用，以获得更好地了解度量黎曼曲面及其几何。在我们的建议中，遍历理论和几何之间的联系来自于研究测地线流动力学的经典观点。然而，考虑模空间上的流动，而不是经典的黎曼流形，导致更具挑战性的技术问题。首先，研究了Weil-Petersson测地线流的动力学性质。这是过去几年来取得重大进展的一个领域，我们为此作出了特别贡献。特别是，威尔-彼得森度规是一个具有负曲率的度规，因此可以用遍历理论中的许多经典技术来处理，比如散射台球理论（尽管存在一些相当大的技术问题）。此外，动力学和几何学之间的微妙相互作用使我们对这两个方面有了更深入的了解。第二个领域是Teichmuller测地线流的研究。这是一个受到数学界主要专家相当关注的话题（例如，菲尔兹的奖牌获得者麦克马伦和孔采维奇）。然而，这种流的统计性质可以用热力学形式主义的技术来研究，因为这种流可以方便地实现为可数分支扩展映射上的悬浮流。第三个研究领域涉及拉普拉斯算子的行列式，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，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Sarnak问题。最后一个研究领域是表面本身的水平。我们希望给出一个新的解释的典型不变量发现的Forni Flaminio在特殊情况下的曲面的常曲率和理论扩展到更一般的表面。基本方法使用我们最近的工作动态zeta函数。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.