Stable Sets and $$\epsilon$$ -Stable Sets in Positive-Entropy Systems

Abstract

In this paper, the chaoticity appearing in the stable and unstable sets of a dynamical system with positive entropy is investigated. It is shown that in any positive-entropy system, there is a measure-theoretically “rather big” set such that the closure of the stable or unstable set of any point from the set contains a weak mixing set. Moreover, the Bowen entropy of these weak mixing sets are also estimated. At the same time, it is proved that the topological entropy of any topological system can be calculated in terms of the dispersion of the pre-images of $$\epsilon$$ -stable sets, which answers an open question posed by D. Fiebig, U.R. Fiebig and Z.H. Nitecki (Ergod. Th & Dynam. Sys. 23, 1785-1806 (2003)).