Critical Exponents and Thermodynamic Formalism on Geometrically Infinite Spaces

群是一种描述几何对象对称性的方法，这些对称性通常可以用镶嵌的方式来看待；例如，艺术家M·C·埃舍尔创作的图片。这种类型的镶嵌描述了如何通过适当的对称从所谓的均匀覆盖空间中获得曲面。除了少数例外情况外，生成的曲面允许具有负曲率的几何图形，其中任何给定点周围的区域看起来像一个鞍点。有一个与这种几何有关的自然动力系统，称为测地线流，负曲率使这个系统变得混乱。此外，这种混沌行为类似于宇宙覆盖空间中的“无穷大”行为。同样的现象出现在更高的维度和几何结构是“粗糙”而不是“光滑”的情况下。这一理论中出现的群具有与它们相关的各种数字特征，特别是所谓的临界指数。这可以被描述为描述了在测地线流的动力学复杂性的群作用下万能覆盖的增长。它通常等于位于万能覆盖边界内的潜在复杂集合的分维。这个项目的主要目的是理解这个数量，因为一个人以特定的方式改变了群体。特别是，我们从一个固定的群开始，然后考虑各种子群。我们期望建立与这些子群的纯代数性质的关系。当子群产生几何无限的空间时，这个理论变得很有趣，因为许多标准理论不适用于这种情况。为了分析这些问题，我们将研究作为测地线流模型的符号动力系统。这种方法允许像临界指数这样的量由一套称为热力学形式主义的理论来描述。这一理论起源于统计力学，但已被成功地应用于理解混沌动力系统。我们的第二个目标是发展符号动力系统的无限群扩张理论。在分析测地线流和其他动力系统中，一个非常成功的工具就是所谓的系统的Zeta函数。这些是通过组合由系统的周期轨道给出的局部数据而获得的复变量的函数。它们是由无穷乘积在复数区域的收敛来定义的，但如果能扩展它们的解析域，就可以得到重要的信息，而获得这样的扩展与热力学形式密切相关。例如，可以为这些系统建立非常精确的渐近结果。然而，对于几何无限空间，即使从定义的角度来看，人们对这些函数的理解也很少。我们的第三个目标是开发和分析适用于这种情况下的这些函数的理论。

Groups are a way of describing symmetries of geometric objects and these symmetries may often be viewed in terms of tessellations; for example, the pictures produced by the artist M C Escher. Tessellations of this type describe how surfaces may be obtained from a so-called uniform covering space via appropriate symmetries. Apart from a small number of exceptions, the resulting surfaces admit geometries with negative curvature, in which the area around any given point looks like a saddle. There is a natural dynamical system associated to this geometry called the geodesic flow and the negative curvature makes this system chaotic. Furthermore, this chaotic behaviour parallels behaviour "at infinity" in the universal covering space. The same type of phenomena occur in higher dimensions and in situations where the geometric structure is "coarse" rather than "smooth".The groups that appear in this theory have various numerical characteristics associated to them, notably the so-called critical exponent. This can be characterised as describing the growth in the universal cover under the group action of the dynamical complexity of the geodesic flow. It is often equal to the fractal dimension of a potentially complicated set that sits inside the boundary of the universal cover. The principle aim of this project is to understand this quantity as one varies the group in specific ways. In particular, one starts with a fixed group and then considers various subgroups. We expect to establish relations with purely algebraic properties of these subgroups. The theory becomes interesting when the subgroups give rise to spaces which are geometrically infinite, since much of the stanard theory does not apply in this case.To analyse these problems, we shall investigate symbolic dynamical systems that serve as models for geodesic flows. This approach allows quantities such as the critical exponent to be described by a body of theory called thermodynamic formalism. This had its origins in statistical mechanics but has been applied with great success to to understand chaotic dynamical systems. Our second objective will be the development of this theory for infinite group extensions of symbolic dynamical systems.A very successful tool in the analysis of geodesic flows and other dynamical systems has been the so-called zeta functions of the systems. These are functions of a complex variable obtained by combining local data given by the periodic orbits of the system. They are defined by convergence of an infinite product in a suitable region of the complex place but important information can be obtained if one can extend their analytic domain, and obtaining such extensions is closely related to thermodynamic formalism. For example, it has been possible to establish very precise asymptotic results for these systems. However, these functions are poorly understood, even from the point of view of definition, for geometrically infinite spaces. Our third aim is to develop and analyse a suitable theory for these functions in this case.