Mathematical Sciences: Limit Sets of Foliations in Hyperbolic 3-Manifolds and Anosov Flows

双曲三维流形中的余维一叶状结构F提升到泛覆盖中的余维一叶状结构F‘，它是双曲的三维空间。研究者将研究F‘的叶的渐近行为以及它们逼近极限集的速度。他还将分析F的拓扑结构在多大程度上决定了流形的几何。最后，一个非奇异流是Anosov的，如果它诱导出与流动方向相反的双曲行为，从而在维度3中它确定了两个横向余维一叶系。研究人员计划检查万能盖子中诱导出的叶理的结构。这与流的闭合轨道的同伦性质密切相关。他还计划研究流线的度量性质，例如，是否所有的流线都可以是准线。流形的叶状结构是用较低维度的部件填充流形的一种方式。在余维一叶理的情况下，这些片断的尺寸比给定流形的尺寸小一。想想洋葱或洋蓟。流形的拓扑与它所支持的叶层的种类密切相关，在熟练的人手中，这种关系已经被锻造成研究流形拓扑的有力工具。裂缝也与流形上可能的流动类型密切相关。特别是在低维领域，标准的代数拓扑学技术不那么有效，使用叶面来研究流形和流形上的可能流动是一个非常受欢迎的补充。

A codimension-one foliation F in a hyperbolic 3-manifold lifts to a foliation F' in the universal cover, which is hyperbolic 3- space. The investigator will study the asymptotic behavior of leaves of F' and how fast they approach their limit sets. He will also analyze to what extent the topological structure of F determines the geometry of the manifold. Finally, a non-singular flow is Anosov if it induces hyperbolic behavior transversally to the flow direction, so that in dimension 3 it determines two transverse codimension-one foliations. The investigator plans to examine the structure of the induced foliations in the universal cover. This is strongly related to homotopic properties of closed orbits of the flow. He also plans to study metric properties of flow lines, for example, whether all flow lines can be quasigeodesic. A foliation of a manifold is a way of filling the manifold with lower dimensional pieces. In the case of a codimension-one foliation, these pieces are of dimension one less than that of the given manifold. Think of an onion or an artichoke. The topology of a manifold is strongly related to the kind of foliation which it will support, and in skillful hands this relation has been forged into a powerful tool for investigating the topology of manifolds. Foliations are also intimately related to the types of flows possible on a manifold. Particularly in low dimensions, where the standard algebraic techniques of topology work less well, using foliations to investigate manifolds and the possible flows on them is a very welcome addition to the arsenal.