Recurrence of Transitive Points in Dynamical Systems with the Specification Property

Let T: X → X be a continuous map of a compact metric space X. A point x ∈ X is called Banach recurrent point if for all neighborhood V of x, {n ∈ ℕ: Tⁿ(x) ∈ V} has positive upper Banach density. Denote by Tr(T), W(T), QW(T) and BR(T) the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of (X, T). If (X, T) has the specification property, then we show that every transitive point is Banach recurrent and ∅ ≠ W(T) ∩ Tr(T) ⫋ W * (T) ∩ Tr(T) ⫋ QW(T) ∩ Tr(T) ⫋ BR(T) ∩ Tr(T), in which W * (T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W * (T) / W(T) is residual in X. Moreover, we construct a point x ∈ BR / QW in symbol dynamical system, and demonstrate that the sets W(T),QW(T) and BR(T) of a dynamical system are all Borel sets.