Geometric Structures on 3-manifolds

Thurston's hyperbolization of fibered 3-manifolds is based on his classification theorem for isotopy classes of surface homeomorphisms. This classification has also been extremely important to the study of dynamical systems on surfaces. The classification theorem holds for surfaces of finite topological type, but in dynamics one is always interested in infinities: infinite time, infinite orbits, etc. In this talk we discuss some specific instances of this contrast. We introduce generalized pseudo-Anosov maps, which are pA maps which are allowed to have infinitely many singularities. We then discuss the possibility of hyperbolizing the associated mapping tori and some similarities with the Otal proof of hyperbolization. We also discuss families of mapping tori associated to families of unimodal maps, limits of certain sequences of hyperbolic 3-manifolds within these families and relate this to the previous hyperbolization discussion.