1. School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China;
2. School of Mathematics, South China Normal University, Guangzhou 510630, P. R. China

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}(?)=T?? for any ?∈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.