Set-theoretical entropies of generalized shifts

In the following text for arbitrary X with at least two elements, nonempty set Γ and self-map φ: Γ → Γ, we prove the set-theoretical entropy of generalized shift σφ: XΓ → XΓ (σφ((xα)α ∈ Γ) = (xφ(α))α ∈ Γ (para (xα)α ∈ Γ ∈ XΓ) is either zero or infinity, moreover it is zero if and only if φ is quasi-...