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

TL;DR

This paper investigates the topological pressures of preimages of epsilon-stable sets and certain closed subsets in systems with positive entropy, establishing their relation to measure-theoretic pressures.

Contribution

It introduces methods to compute topological pressure via preimages of epsilon-stable sets and relates pressures of specific stable and unstable subsets to measure-theoretic pressure.

Findings

Topological pressure can be calculated using preimages of epsilon-stable sets.

Pressures of certain stable and unstable subsets are at least the measure-theoretic pressure.

Results apply to systems with positive entropy and weakly mixing subsets.

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.