Workshop - Thermodynamic Formalism: Ergodic Theory and Geometry

研讨会 - 热力学形式主义：遍历理论和几何

• 批准号: EP/S020969/1
• 负责人: Richard Sharp
• 金额: $ 3.25万
• 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: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FS020969%2F1
• 关键词: Workshop; Thermodynamic; Formalism; Ergodic; Theory

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.

在过去的十年里，在基于所谓的热力学形式主义的遍历理论及其应用方面取得了惊人的、持续的进展。这种植根于统计物理学的方法有两个方面。一方面，动力系统的有趣的不变度量--称为平衡度量，包括自然几何度量--是由涉及熵的变分原理产生的，或者，包括位势加权，其加权版本的拓扑压。另一方面，拓扑力及其相关度量被编码为一族有界线性算子的特征数据，称为转移算子，这使得可以使用丰富的算子理论工具。在最有利的情况下，系统和势是解析的，这些算符将是核的(在Grothendieck意义上)，因此它们的谱由本征值组成，并且它们有明确的迹。特别地，这允许将相关Zeta函数定义为复平面上的亚纯函数，并精确计算许多数值特征。更广泛地说，人们在寻找新的函数空间以促进对一系列系统的分析方面付出了相当大的努力。这一理论在双曲线背景下得到了最好的发展，基本框架由Bowen，Ruelle和Sinai在20世纪70年代建立。Dolgopyat在20世纪90年代的工作给出了一个重大的进步，它允许随势的变化对转移算子进行估计，他用这些估计证明了一大类双曲型流动的指数混合率。我们现在看到了这一理论的新一波发展，在双曲背景下取得了新的进展，并将其扩展到一致双曲性之外的情况，如具有奇点的系统(如Lorenz吸引子)和非一致双曲系统，以及开放系统。这反过来又打开了新的几何应用，例如，对非正曲线流形和表现出粗负曲率的空间进行排名。该提案的目的是举办一次密集的研究研讨会，以促进热力学形式论作为遍历理论和应用中的一种方法的使用。这样的研讨会是及时的，因为这一主题在过去几年中发展迅速，既有对其他领域的新应用，也有来自其他分析分支的新想法。前者的例子包括利用热力学思想严格计算分形集的维度和其他数字特征，以及将洛伦兹系统关联衰变工作中的思想应用到几何中的Weil-Petersson测地线流的特征混合问题。后者的例子有：利用作为分数数学分析理论一部分而发展起来的所谓“分形测不准原理”来研究开放动力系统的基本光谱间隙，以及利用某些系统中存在的固有负曲几何来使用热力学方法，而不需要建立符号模型，而这种方法只在较有限的环境中可用。其目标是解决一些具体问题，以解决方案和部分解决方案衡量进展，并引入新的想法。