a2 School of Mathematics, Jilin University, Changchun, 130012, PR China (email: firstname.lastname@example.org)
a3 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
We study the dimensions 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; under the continuum hypothesis, this set also contains a scrambled set with positive Bowen dimension entropies. For several kinds of specific invertible dynamical systems, the lower bounds of the 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.
(Received September 11 2010)
(Revised December 13 2010)
(Online publication April 28 2011)
In memory of Dan Rudolph