Dimension of Stable Sets and Scrambled Sets in Positive Finite Entropy Systems

We study dimension of stable sets and scrambled sets of a dynamical system with positive finite entropy. We show that there is a measure-theoretically “large” set containing points whose sets of “hyperbolic points” (i.e., points lying in the intersections of the closures of the stable and unstable sets) admit positive Bowen dimension entropies, which, under a continuum hypothesis, also contains a scrambled set with positive Bowen dimension entropy. For several kinds of specific invertible dynamical systems, lower bounds of Hausdorff dimension of these sets are estimated. In particular, for a diffeomorphism on a smooth Riemannian manifold with positive entropy, such a lower bound is given in terms of the metric entropy and Lyapunov exponent.