Exploring the asymptotic average shadowing property

Abstract

Abstract The main aim of this paper is to relate a recently introduced notion of the asymptotic average shadowing property (AASP) with other notions known from topological dynamics. For a compact metric space X and a continuous map we prove the following facts: If f has the AASP and the minimal points of f are dense in X then f is totally transitive. If f is a surjection and has the specification property then f has the AASP. If f is c-expansive and has the shadowing property then f is mixing if and only if f has the AASP. Keywords: shadowingspecification propertyhyperbolic homeomorphismtopological mixingAMS Subject Classification:: 54H20 (primary)34D05, 37D45 (secondary) Acknowledgements Piotr Oprocha was supported in part by the Project No. NN201272333 for years 2007–2009 from Polish Ministry of Science and Higher Education, an annual national scholarship for young scientists from the Foundation for Polish Science and AGH local Grant No. 10.420.03.