Pointwise almost periodicity in a generalized shift dynamical system
In the following text we prove that in a generalized shift dynamical system (X Γ , σϕ) for discrete X with at least two elements, arbitrary nonempty Γ and bijection ϕ : Γ → Γ, the following statements are equivalent: • (X Γ , σϕ) is pointwise recurrent; • (X Γ , σϕ) is pointwise almost periodic; • (X Γ , σϕ) is pointwise regularly almost periodic; • (X Γ , σϕ) is compactly almost periodic; • Per(ϕ) = Γ (ϕ : Γ → Γ is pointwise periodic). Keywords: Almost periodic, Compactly almost periodic, Compactly recurrent, Dynamical system, Generalized shift, Periodic, Pointwise almost periodic, Pointwise periodic, Pointwise regularly almost periodic, Pointwise recurrent, Recurrent. 2000 Mathematics subject classification: 37B05, 54H20
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