Ergodic Theory and Dynamics on Geometrically Infinite Spaces

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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.