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Let (X, F = {fn}n =0∞) be a non-autonomous discrete system by a compact metric space X and continuous maps fn : X → X, n = 0, 1, ....We introduce functional envelope (S(X), G = {Gn}n =0∞), of (X, F = {fn}n =0∞), where S(X) is the space of all continuous self maps of X and the map Gn : S(X) → S(X) is defined by Gn(ϕ) = Fn ∘ ϕ, Fn = fn ∘ fn-1 ∘ . . . ∘ f1 ∘ f0. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope.
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