Topological pressures for 𝜖-stable and stable sets

Xianfeng Ma and Ercai Chen

Vol. 263 (2013), No. 1, 117–142

DOI: 10.2140/pjm.2013.263.117

Abstract

In this paper, topological pressures of the preimages of 𝜖-stable sets and certain closed subsets of stable sets in positive entropy systems are investigated. It is shown that the topological pressure of any topological system can be calculated in terms of the topological pressure of the preimages of 𝜖-stable sets. For the constructed closed subset (W. Huang, Commun. Math. Phys. 279, 535–557 (2008)) 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 the stable and unstable sets, it is proved that topological pressures of these subsets can be no less than the measure-theoretic pressure.

Keywords

topological pressure, stable sets, 𝜖-stable sets, weak mixing sets

Mathematical Subject Classification 2010

Primary: 37A35

Milestones

Received: 27 April 2012

Revised: 27 October 2012

Accepted: 19 December 2012

Published: 14 May 2013