The principal investigator will analyze interrelated problems concerning hyperbolic dynamical systems and compact Riemannian manifolds of negative curvature. Specifically, he will investigate the classification, differentiability, and regularity of stable and unstable foliations. Other directions of research include investigations of rigidity and regularity of global invariants with respect to external parameters. This project involves three areas of mathematical research. Dynamical systems is the study of where points go after many repeated iterations of a continuous function. Ergodic theorists ask where most, but not all, points go under many such iterations. And Riemannian geometry is concerned with the curvature and other aspects of the base space from which these points are taken. The principal investigator will analyze several problems which stem from attempts to apply theories from all three of these areas at once.
