Ergodic Theory and Dynamics on Geometrically Infinite Spaces

传统上，动力学是在相空间上研究的，这些相空间在某种程度上是有限的(例如紧度量或有限度量)。研究人员越来越多地将这项研究扩大到在几何或测度论意义上可能被认为是无限的空间。这方面的例子有斜积扩张，其中紧基被无限群扩张，或覆盖紧流形的空间，甚至具有一些更一般的几何有限性质的流形，如凸余紧性。最近，俄亥俄州立大学的A Gogolv和宾夕法尼亚州立大学的F Rodriguez Hertz研究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“同调满度”性质之间的关系。还有，在一个略有不同的方向上，A Fathi(乔治亚理工学院)最近关于某些同胚的常返性的工作被滑动到无限阿贝尔覆盖，这也与“同调满度”的概念有关。一个项目是探索类似的问题，当封面听从的时候，这是一个比阿贝尔语更一般的课程。拟议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热力学形式主义，涉及转移算子和动态Zeta函数的分析技术。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。

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.