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）,FT）, where FT is defined by FT（φ） = T o φ for any φ ∈ S（X）. We show that （1） If （∑, T） is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where ∑ is any closed subset of a Cantor set and T a selfmap of ∑ （2） If （S（∑）,Fσ） is transitive then it is Devaney chaos, where （∑, σ） is a subshift of finite type; （3） If （∑, T） has shadowing property, then （Su （∑）, FT） has shadowing property, where ∑ is any closed subset of a Cantor set and T a selfmap of ∑; （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 （Su（X）,FT） is sensitive; （5） If ∑, is a closed subset of a Cantor set with infinite points and T ：∑ →∑ is positively expansive then the entropy entv（FT） of the functional envelope of （∑, T） is infinity.