Distributional Chaotic Generalized Shifts

Abstract

Suppose X is a finite discrete space with at least two elements, Γ is a nonempty countable set, and consider self–map φ: Γ → Γ. We prove that the generalized shift σφ : X Γ →X Γ with σφ((Xα) α ∈Γ) = (Xφ (α))α∈Γ (for (Xα ) α ∈Γ ∈ X Γ) is: distributional chaotic (uniform, type 1, type 2) if and only if φ : Γ → Γ has at least a non-quasi-periodic point, dense distributional chaotic if and only if φ : Γ → Γ does not have any periodic point, transitive distributional chaotic if and only if φ : Γ → Γ is one–to–one without any periodic point. We complete the text by counterexamples.