Statistical Properties of Hyperbolic Groups and Dynamics

该项目是遍历理论的一个分支，它是数学分析的一个分支，从概率的角度研究动力系统和更一般的群体行为，以及研究模空间的几何领域，即参数上升几何结构的空间。后者的一个例子是所谓的Teichmueller空间，它将固定亏格的紧致曲面上的双曲度量参数化为矩阵群，或者等价地，将基本群的表示化为矩阵群。该项目的目的是加深我们对离散群的统计性质的理解，这些离散群作为模空间中的点和更一般的类似类型的群-这是一个在过去几年中发展起来的领域，尽管建立在与几何相关的混沌动力系统的良好遍历理论的基础上，测地线流动。例如，Calegari和Fujiwara证明了某些自然类的可观测对象满足中心极限定理，即它们的行为就像它们是从简单的随机系统(如扔硬币)派生出来的一样。对于紧致曲面的特殊情况，其中模空间是Teichmueller空间，Pollicott和Sharp得到了更精确的概率结果，例如处理布朗运动逼近随机变量的几乎必然不变原理，以及一个局部版本的中心极限定理。这项研究的一个目标是将这些更精确的结果推广到更广泛的群体中。该方法将基于对所谓转移算符家族的光谱性质的分析，这些转移算符是与编码关于群行动的信息的符号动力系统相关的热力学形式的一部分。到目前为止(2018年9月)，学生已经(A)完成了关于双曲(混沌)流周期轨道的中心极限定理的工作，并且(B)成功地将Calegari-Fujiwara的结果推广到一般非初等双曲群的更广泛的可观测类别。(B)中的工作已接近完成，学生已获得中心极限结果和大偏差结果。学生还可以进一步研究更精细的结果，如局部极限定理或统计不变性原理。潜在的受益者是遍历理论、几何群论、双曲几何和混沌动力系统的研究人员。这项研究是在几何和拓扑学和数学分析的研究领域，完全在数学科学主题范围内。。

The project is at the interface of ergodic theory, a branch of mathematical analysis that studies dynamical systems and, more generally, group actions, from a probabilistic point of view, and the area of geometry that studies moduli spaces, i.e. spaces that parametrise geometric structures. An example of the latter is the so-called Teichmueller space that parametrises the hyperbolic metrics on a compact surface of fixed genus or, equivalently, representations of the fundamental group into a matrix group. The aim of the project is to further our understanding of the statistical properties of the discrete groups that arise as points in moduli spaces and more general groups of similar type -- a field which has grown up over the last few years, though building on the well developed ergodic theory of the chaotic dynamical systems, geodesic flows, which are associated to the geometry. For example, Calegari and Fujiwara showed that certain natural classes of observable satisfy a Central Limit Theorem, i.e. that they behave as if they were derived from a simple random system such as coin tossing. For the special case of compact surfaces, where the moduli space is the Teichmueller space, Pollicott and Sharp obtained more precise probabilistic results such as the Almost Sure Invariance Principle, which deals with the approximation of random variables by Brownian motion, and a local version of the Central Limit Theorem. An objective of the research is to extend these more precise results to a wider class of groups. The methodology would be based on an analysis of the spectral properties of families of so-called transfer operators that are part of the thermodynamic formalism associated to symbolic dynamical systems that encode information about the group actions. To date (September 2018), the student has (a) completed work on a central limit theorem for periodic orbits of hyperbolic (chaotic) flows and (b) succeeded in extending the results of Calegari-Fujiwara to a wider class of observables for general non-elementary hyperbolic groups . The work in (b) is close to being completed and the students has obtained both central limit results and large deviations results. There is scope for the student to pursue further investigation of more refined results such as local limit theorems or statistical invariance principles. The potential beneficiaries are researchers in ergodic theory, geometric group theory, hyperbolic geometry and chaotic dynamical systems. The research is in the research areas of Geometry and Topology and Mathematical Analysis, and is wholly within the Mathematical Sciences theme. .