Specification properties on uniform spaces

2017/03/07 by Shirazi, Fatemah Ayatollah Zadeh, Ahmadabadi, Zahra Nili, Taherkhani, Bahman +1
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1703.02288

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,σ_φ): \beginitemize \item has (almost) weak specification property if and only if φ:Γ→Γ does not have any periodic point, \item has (uniform) stroboscopical property if and only if φ:Γ→Γ is one-to-one. \enditemize

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