Ergodic Theory and Dynamics on Geometrically Infinite Spaces

传统上，动力学是在以某种方式有限的相空间（例如紧致或有限测度）上研究的。越来越多的研究人员将这一研究扩展到几何或测度论意义上的无限空间。这方面的例子是斜积扩张，其中紧基被无限群扩展，或覆盖紧流形的空间，甚至具有一些更一般的几何有限性性质的流形，如凸余紧性。最近的工作是A Gogolev（俄亥俄州）和F Rodriguez Hertz（宾夕法尼亚州），他们探索了无限阿贝尔覆盖上某些混沌流的拓扑传递性与R Sharp引入的一个称为“同调充满度”的性质之间的关系。而且，在一个稍微不同的方向，最近的工作A Fathi（格鲁吉亚技术）的递归性质的某些同胚slifted无限阿贝尔覆盖，这再次是有关的“同调丰满”的概念。一个项目是探索类似的问题时，封面是顺从的，一个更一般的类比阿贝尔。拟议中的研究将使用双曲流的符号编码，表面同胚理论和遍历理论的分支称为代数形式主义，涉及转移算子和动态zeta函数的分析技术。该研究是在几何和拓扑和数学分析的研究领域，并完全在数学科学的主题。

Traditionally, dynamics has been studied on phase spaces that are finite in some way (e.g. compact or or finite measure). Increasingly, researchers arebroadening this study to spaces which might be thought of as infinite in a geometric or measure-theoretic sense. Examples of this are skew-productexpenstions, where a compact base is extended by an infinite group, or covering spaces of compact manifolds, or even manifolds with some more generalgeometric finiteness property, such as convex co-compactness. There has been recent work of A Gogolev (Ohio State) and F Rodriguez Hertz (Penn State) that explores the relationship between topological transitivity of certain chaotic flows on infinite abelian covers and a property called "homological fullness" introduced by R Sharp. And, in a slightly different direction, recent work of A Fathi (Georgia Tech) on the recurrence properties of certain homeomorphismslifted to infinite abelian covers, which again is related to the "homological fullness" concept. A project is to explore similar questions when the covers areamenable, a more general class than abelian. The proposed research will use symbolic coding of hyperbolic flows, the theory of surface homeomorphisms andthe branch of ergodic theory known as thermodynamical formalism, involving the analytic techniques of transfer operators and dynamical zeta functions. The research is in the research areas of Geometry and Topology and Mathematical Analysis, and is wholly within the Mathematical Sciences theme.