Critical Exponents and Thermodynamic Formalism on Geometrically Infinite Spaces

群是描述几何物体对称性的一种方式，这些对称性通常可以从镶嵌的角度来看待；例如，艺术家M·C·埃舍尔的作品。这种类型的镶嵌描述了如何通过适当的对称性从所谓的均匀覆盖空间中获得表面。除了少数例外情况外，最终的曲面允许几何形状具有负曲率，其中任何给定点周围的区域看起来都像马鞍。有一个自然的动力系统与这个几何图形相关联，叫做测地线流，负曲率使这个系统变得混乱。此外，这种混沌行为与宇宙覆盖空间中“无穷远处”的行为相似。同样的现象发生在更高的维度和几何结构“粗糙”而不是“光滑”的情况下。在这个理论中出现的群有各种与之相关的数值特征，特别是所谓的临界指数。这可以被描述为在测地线流的动态复杂性的群作用下，通用覆盖层的增长。它通常等于一个潜在的复杂集合的分形维数，它位于通用覆盖的边界内。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这个群体。特别是，从一个固定的组开始，然后考虑各种子组。我们期望与这些子群的纯代数性质建立关系。当子群产生几何上无限的空间时，这个理论就变得有趣了，因为许多标准理论在这种情况下并不适用。为了分析这些问题，我们将研究作为测地线流动模型的符号动力系统。这种方法允许用一种叫做热力学形式论的理论体系来描述临界指数之类的量。它起源于统计力学，但已成功地应用于理解混沌动力系统。我们的第二个目标是为符号动力系统的无限群扩展发展这一理论。在分析测地线流动和其他动力系统时，一个非常成功的工具就是所谓的系统的zeta函数。这些是由系统周期轨道给出的局部数据组合而成的复变量的函数。它们是由一个无穷积在复点的适当区域内的收敛来定义的，但如果能扩展它们的解析域，就能得到重要的信息，而这种扩展的获得与热力学的形式化密切相关。例如，已经有可能为这些系统建立非常精确的渐近结果。然而，即使从定义的角度来看，对于几何上无限的空间，这些函数也很难理解。我们的第三个目标是在这种情况下为这些函数开发和分析一个合适的理论。

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.