Semi-distality and related topics in generalized shift dynamical systems

Document Type : Regular Paper


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


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, $Per(varphi)=Gamma$).


Main Subjects