Semi-distality and Related Topics in Generalized Shift Dynamical Systems

Abstract

In the following text, we prove that the generalized shift dynamical system $(X^{\varGamma } ,\sigma_{\varphi } )$ with nonempty $\varGamma$, finite discrete $X$ with at least two elements, and arbitrary map $\varphi \colon \varGamma \to \varGamma$ is semi-distal if and only if all the points of $\varGamma$ are quasi-periodic points of $\varphi ;$ moreover, for countable $\varGamma$, semi-distality and almost distality of $(X^{\varGamma } ,\sigma_{\varphi } )$ are equivalent. Also, we prove that the following statements are equivalent: The generalized shift $(X^{\varGamma } ,\sigma_{\varphi } )$ is pointwise minimal; The generalized shift dynamical system $(X^{\varGamma } ,\sigma_{\varphi } )$ is distal; the map $\varphi \colon \varGamma \to \varGamma$ is pointwise periodic (i.e., $Per(\varphi ) = \varGamma$).