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 (XΓ,σφ):
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has (almost) weak specification property if and only if φ : Γ →Γ does not have any periodic point,
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has (uniform) stroboscopical property if and only if φ : Γ →Γ is one-to-one.
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The authors would like to gratefully thank the anonymous referee for his(her) useful comments.
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Shirazi, F.A.Z., Ahmadabadi, Z.N., Taherkhani, B. et al. Specification Properties on Uniform Spaces. J Dyn Control Syst 27, 321–333 (2021). https://doi.org/10.1007/s10883-020-09499-x
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DOI: https://doi.org/10.1007/s10883-020-09499-x