Topological pressures for $\epsilon$-stable and stable sets

Abstract

Topological pressures of the preimages of $\epsilon$-stable sets and some certain closed subsets of stable sets in positive entropy systems are investigated. It is showed that the topological pressure of any topological system can be calculated in terms of the topological pressure of the preimages of $\epsilon$-stable sets. For the constructed closed subset of the stable set or the unstable set of any point in a measure-theoretic `rather big' set of a topological system with positive entropy, especially for the weakly mixing subset contained in the closure of stable set and unstable set, it is proved that topological pressures of these subsets can be no less than the measure-theoretic pressure.