This research will focus on the study of the dynamics of manifolds. Interest in this topic stems principally from Thurston's work on the classification of diffeomorphisms of surfaces. The questions to be addressed concern invariant sets of automorphisms of torii, flows on surfaces of negative curvature and diffeomorphisms of Lorenz manifolds. Albert Fathi will use his knowledge of pseudo-Anosov flows to construct and investigate compact invariant subsets of hyperbolic toral maps. In particular he will investigate estimates for the Hausdorff dimension of these sets. Fathi will also continue his investigations of the connections between the intersection of geodesic currents and metric entropy. The final part of the project involves the study of Anosov diffeomorphisms of Lorenz manifolds. The objective here is to try to extend established results about flows on Riemannian manifolds of negative curvature to time-like geodesic flow on certain Lorenz manifolds. Special attention will be devoted to those manifolds of importance in general relativity.
