Semi-distality and related topics in generalized shift dynamical systems

Ayatollah Zadeh Shirazi, Fatemah; Ebrahimifar, Fatemeh; Taherkhani, Bahman

doi:10.22099/ijsts.2016.3575

Semi-distality and related topics in generalized shift dynamical systems

Document Type : Regular Paper

Authors

¹ Faculty of Mathematics, Statistics and Computer Sience, College of Science, University of Tehran, Tehran, Iran

² Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran

10.22099/ijsts.2016.3575

Abstract

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

Keywords