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 Γ and a self-map ϕ : Γ → Γ the generalized shift dynamical system (X Γ , σϕ):• has (almost) weak specification property if and only if ϕ : Γ → Γ does not have any periodic point, • has (uniform) stroboscopical property if and only if ϕ : Γ → Γ is one-toone.