Georgia Institute of Technology College of Sciences

Stable sets and unstable sets of a dynamical system with positive entropy are investigated. It is shown that in any invertible system with positive entropy, there is a measure-theoretically ?rather big? set such that for any point from the set, the intersection of the closure of the stable set and the closure of the unstable set of the point has positive entropy. Moreover, for several kinds of specific systems, the lower bound of Hausdorff dimension of these sets is estimated. Particularly the lower bound of the Hausdorff dimension of such sets appearing in a positive entropy diffeomorphism on a smooth Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.