Functional envelope of Cantor spaces

Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of (X, T) is the system (S(X), F _T ), where F _T is defined by F _T (φ) = T ? φ for any φ ∈ S(X). We show that (1) If (Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (2) If (S(Σ), F _σ ) is transitive then it is Devaney chaos, where (Σ, σ) is a subshift of finite type; (3) If (Σ, T) has shadowing property, then (S _U (Σ), F _T ) has shadowing property, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (4) If (X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T: X → X is continuous, then (S _U (X), F _T ) is sensitive; (5) If Σ is a closed subset of a Cantor set with infinite points and T: Σ → Σ is positively expansive then the entropy ent _U (F _T ) of the functional envelope of (Σ, T) is infinity.