Journal of the
Korean Mathematical Society JKMS

Recently, Chung and Lee \cite{CL} introduced the notion of topological stability for a finitely generated group action, and proved a group action version of the Walters's stability theorem. In this paper, we introduce the concepts of continuous shadowing and continuous inverse shadowing of a finitely generated group action on a compact metric space $X$ with respect to various classes of admissible pseudo orbits and study the relationships between topological stability and continuous shadowing and continuous inverse shadowing property of group actions. Moreover, we introduce the notion of structural stability for a finitely generated group action, and we prove that an expansive action on a compact manifold is structurally stable if and only if it is continuous inverse shadowing.

Keywords: continuous shadowing, expansiveness, group action, inverse shadowing, structural stability, topological stability

MSC numbers: 37C85, 37C50, 37C75

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(No. NRF-2017R1D1A1B03032148) and (No. NRF-2018R1A2B3001457).