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

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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.