Mathematical Sciences: Topics in Dynamical Systems and Smooth Erogodic Theory

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 study of the path of points under repeated iterations of a continuous mapping. Ergodic theory studies the long term behavior of most of the points under many iterations. Riemannian geometry describes the nature of the space in which the points reside; quantities such as curvature are used in the description. The principal investigator will analyze several problems which stem from attempts to apply theories from all three of these areas.