Shadowing and mixing on systems of countable group actions

Abstract

The theory of zero-dimensional extensions is a useful tool in the study of entropy theory. In the present paper, we aim to study which properties can be preserved by a suitable zero-dimensional extension. For a dynamical system (X,G,Φ), we prove that the shadowing property implies that it is a factor of the inverse limit of subshifts with the shadowing property. Moreover, if X is totally disconnected, the extension is a conjugate. Besides, we prove that a system with transitivity (or mixng) is a factor of the inverse limit of subshifts with transitivity (or mixng, respectively). Moreover, if X is totally disconnected, the extension is a conjugate.