Specification properties on uniform spaces

Abstract

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space $X$ with at least two elements, a nonempty set $\Gamma$ and a self--map $\varphi:\Gamma\to\Gamma$ the generalized shift dynamical system $(X^\Gamma,\sigma_\varphi)$: \begin{itemize} \item has (almost) weak specification property if and only if $\varphi:\Gamma\to\Gamma$ does not have any periodic point, \item has (uniform) stroboscopical property if and only if $\varphi:\Gamma\to\Gamma$ is one-to-one. \end{itemize}