Workshop - Thermodynamic Formalism: Ergodic Theory and Geometry

在过去的十年里，遍历理论及其应用的方法在所谓的热力学形式主义的基础上取得了惊人的和持续的进展。这种方法植根于统计物理学，有两个方面。一方面，动力系统的有趣的不变测度--称为平衡测度，包括自然几何测度--是由于涉及熵的变分原理而产生的，或者，包括用势加权，它的加权版本拓扑压力。另一方面，拓扑压力和相关的措施被编码为一个家庭的有界线性算子，称为转移运营商，这使得丰富的运营商理论的工具被采用的特征数据。在最有利的情况下，系统和势是解析的，这些算子将是核的（在格罗滕迪克意义下），因此它们的谱由本征值组成，并且它们有明确的迹。这，特别是，允许定义相关的zeta函数作为一个亚纯函数的复平面和精确计算的许多数值特征。更一般地说，相当大的努力已经进入寻找新的功能空间，以方便分析的范围systems.This理论是最好的发展，在双曲设置和基本框架是到位的鲍恩，Ruelle和西奈由20世纪70年代。Dolgopyat在20世纪90年代的工作取得了重大进展，允许估计转移算子的潜力变化，他用来证明指数混合率为一类广泛的双曲流。我们现在看到了这个理论的新一波发展，在双曲设定和扩展到超越一致双曲性的情况，如奇异系统（如洛伦兹吸引子）和非一致双曲系统，以及开放系统的新进展。这反过来又开辟了新的几何应用，例如秩1非正弯曲的流形和空间表现出粗糙的负曲率。该提案的目的是举行一个密集的研究研讨会，以推进使用热力学形式主义作为遍历理论和应用的方法。这样一个研讨会是及时的，因为这个主题在过去几年中发展迅速，既有新的应用到其他领域，也有来自其他分析分支的新想法。前者的例子是使用热力学的想法进行严格的计算维数的分形集和其他数值特征，并应用的想法，从工作的衰减相关性的洛伦兹系统的问题，表征混合的韦尔-彼得森测地线流的几何。后者的例子是使用所谓的“分形不确定性原理”，作为分形数学分析理论的一部分，来研究开放动力系统的基本谱隙，以及使用某些系统中存在的固有负弯曲几何来采用热力学方法，而不需要构建符号模型，这只适用于更受限制的设置。目标是解决一些具体问题，并以解决和部分解决以及提出新的想法来衡量进展情况。

The last decade has seen spectacular and continuing advances in an approach to ergodic theory and its applications based on the so-called Thermodynamic Formalism. This approach, which is rooted in statistical physics, has two aspects. On one hand, interesting invariant measures for dynamical systems - known as equilibrium measures and including natural geometric measures - arise as a result of variational principles involving entropy or, to include weighting by a potential, its weighted version topological pressure. On the other, the topological pressure and the associated measures are encoded as eigendata of a family of bounded linear operators, called transfer operators, which enables a wealth of operator-theoretic tools to be employed. In the most favourable circumstances, where the system and potential are analytic, these operators will be nuclear (in the sense of Grothendieck) and so their spectrum consists of eigenvalues and they have a well-defined trace. This, in particular, allows the definition of the associated zeta function as a meromorphic function on the complex plane and the accurate computation of many numerical characteristics. More generally, considerable effort has gone into finding new function spaces that facilitate the analysis of a range of systems.This theory is best developed in the hyperbolic setting and the basic framework was put in place by Bowen, Ruelle and Sinai by the 1970s. A major advance was given by Dolgopyat's work in the 1990s, which allowed estimates on transfer operators as the potential varies, which he used to prove exponential mixing rates for a wide class of hyperbolic flows. We are now seeing a new wave of development of this theory, with new advances in the hyperbolic setting and extensions to situations beyond uniform hyperbolicity, such as systems with singularities (such as the Lorenz attractor) and non-uniformly hyperbolic systems, and to open systems. This, in turn, has opened up new geometric applications, for example to rank 1 non-positively curved manifolds and to spaces exhibiting coarse negative curvature.The aim of the proposal is to hold an intensive research workshop to progress the use of Thermodynamic Formalism as a method in ergodic theory and applications. Such a workshop is timely as the subject has seen rapid development in the last few years, with both new applications to other areas and fresh injections of ideas coming from other branches of analysis. Examples of the former are the use of thermodynamic ideas to carry out rigorous computations of dimensions of fractal sets and other numerical characteristics, and the application of ideas from work on the decay of correlations for Lorenz systems to the problem of characterising mixing for the Weil-Petersson geodesic flow in geometry. Examples of the latter are the use of the so-called `fractal uncertainty principle', developed as part of the theory of mathematical analysis on fractals, to study essential spectral gaps for open dynamical systems, and the use of the intrinsic negatively curved geometry present in some systems to employ thermodynamic methods without the requirement of constructing a symbolic model, which is only available in more restricted settings. The objectives are to attack a number of specific problems with progress being measured in terms of both solutions and partial solutions, and the introduction of new ideas.