Hyperbolic Dynamics and Noncommutative Geometry

20世纪80年代，已经因在C*-代数上的工作而获得菲尔兹奖的阿兰·康尼斯发展了一个新的数学分支，称为非对易几何。该理论的部分灵感来自于量子力学对亚原子现象的描述，该理论旨在用代数术语描述各种各样的几何对象，其中“点”被“算符”取代。康纳斯能够将经典微分几何的大多数工具扩展到这种设置，但该理论足够灵活，可以描述不太规则的对象：行为糟糕的商空间、叶空间，特别是与此应用相关的分形集。事实上，“分形非对易几何”本身已经成为一个非常活跃的领域。最新的发展是将分形非对易几何与双曲动力系统理论相结合。后者是混沌动力系统的典型例子，其特征是局部分解成指数扩张和收缩方向。它们具有丰富的轨道结构和许多重要的特征，例如不变度量，可以通过对轨道族进行平均来恢复。这样的轨道族也可以用来构造动力学系统几何的非对易描述所需的对象，这是我们打算利用的一个方面。我们的主要目标是描述双曲动力系统的不变集，以及一类重要的不变度量，称为Gibbs测度，用非对易几何或更严格地说，用称为谱三元组的对象来描述。这包括一个称为狄拉克算符的运算符，它提供了类似的微分。我们的目的也是为了将这一理论推广到Klein群的极限集上，它可以表现为二维球面上的复杂的分形图案。我们进一步的目标是发展一种非对易的或谱的动力学理论和Kleian群作用。为此，我们将研究与来自最简单的Klein群，即肖特基群的代数对象有关的谱度量空间。在这些示例中，极限集是Cantor集，这是最常见的分形集之一。Bellisard和Pearson最近也研究了与Cantor集相关的谱三元组，并在此背景下定义了Laplace-Beltrami算子。我们的目标是将这项工作扩展到更广泛的背景下。最后，我们的目标是在非对易几何的基础上发展多重分形分析--研究动力系统的精细分形结构。

In the 1980s, Alain Connes, who had already won the Fields Medal for his work on C*-algebras, developed a new branch of mathematics called noncommutative geometry. Partly inspired by the description of subatomic phenomena given by quantum mechanics, the theory aimed to describe a wide variety of geometric objects in algebraic terms, where "points" are replaced by "operators". Connes was able to extend most of the tools of classical differential geometry to this setting but the theory is sufficiently flexible to allow a description of less regular objects: badly behaved quotient spaces, spaces of foliations and, particularly relevant to this application, fractal sets. Indeed, "fractal noncommutative geometry" has become a very active field in its own right.A very recent development has been the combination of fractal noncommutative geometry with the theory of hyperbolic dynamical systems. The latter are the prototypical examples of chaotic dynamical systems and are characterized by a local decomposition into exponentially expanding and contracting directions. They have a rich orbit structure and many important characteristics, for example invariant measures, can be recovered from averaging over families of orbits. Such families of orbits can also be used to construct to objects required for a noncommutative description of the geometry of the dynamical system and this is an aspect we intend to exploit.Our principle objective is to describe the invariant set of a hyperbolic dynamical system, together with an important class of invariant measures, called Gibbs measures, in terms of noncommutative geometry or, more technically, in terms of an object called a spectral triple. This includes an operator, called a Dirac operator, which provides the analogue of differentiation. We also aim to develop this theory for the limit sets of Kleinian groups, which can appear as intricate fractal patterns on the the two dimensional sphere.We further aim to develop a noncommutative, or spectral, theory of dynamics and Kleinian group actions. To this end, we will study spectral metric spaces associated to algebraic objects coming from the simplest Kleinian groups, namely Schottky groups. In these examples, the limit set is a Cantor set, one of the most familiar examples of fractal set. Spectral triples assiciated to Cantor sets have also been studied recently by Bellisard and Pearson and they were led to define a Laplace-Beltrami operator in this setting. We aim to extend this work to a wider setting.Finally, we aim to develop a mutifractal analysis -- the study of the fine fractal structure of dynamical systems -- in terms of noncommutative geometry.