Hyperbolic Dynamics and Noncommutative Geometry

在20世纪80年代，阿兰·康纳斯（Alain Connes）已经因为他在C*-代数方面的工作获得了菲尔兹奖，他发展了一个新的数学分支，称为非交换几何。部分受到量子力学对亚原子现象描述的启发，该理论旨在用代数术语描述各种几何对象，其中“点”被“算子”取代。康纳斯是能够扩大大多数工具的经典微分几何这一设置，但该理论是足够灵活的，以允许描述不太经常的对象：表现不佳的商空间，空间的foliations和，特别是有关本申请，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。最近的发展是分形非对易几何与双曲动力系统理论的结合。后者是混沌动力系统的典型例子，其特征在于局部分解成指数膨胀和收缩方向。它们有丰富的轨道结构和许多重要的特征，例如不变测度，可以通过对轨道族进行平均来恢复。这类轨道族也可以用来构造动力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测度，称为吉布斯测度，在非对易几何方面，或者更技术上，用一个叫做光谱三元组的物体来表示。这包括一个称为狄拉克算子的算子，它提供了微分的模拟。我们的目标也是发展这一理论的极限集的Kleinian群，这可能会出现在复杂的分形图案的二维sphere。我们进一步的目标是发展一个非交换的，或频谱，理论的动力学和Kleinian群行动。为此，我们将研究与来自最简单的Kleinian群，即肖特基群的代数对象相关的谱度量空间。在这些例子中，极限集是康托集，分形集最常见的例子之一。谱三元组关联到康托集也已研究最近由Bellisard和皮尔逊，他们导致定义一个Laplace-Beltrami算子在这种设置。我们的目标是将这项工作扩展到更广泛的setting.Finally，我们的目标是开发一个多重分形分析-动力系统的精细分形结构的研究-在非交换几何。

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.