Attractor and its self-similarities for an IFS over arbitrary sub-shift

Consider a compact metric space X, and let F=₁,\,f₂,…,\, fₖ\ be a set of contracting and continuous self maps on X. Let Σ be a sub-shift on k symbols, and let Σₖ be the full shift. Define Lₙ(Σ) as the set of words of length n in Σ. For u=u₁⋯ uₙ∈ Lₙ(Σ), set fᵤ:=fuₙ∘⋯ ∘ fu₁ and Hⁿ(·):=∪_u ∈ Lₙ(Σₖ) fᵤ(·). When Σ=Σₖ, Hⁿ(·) is the nth iteration of the Hutchinson's operator, and there exists a compact set S= limn → ∞ Hⁿ(A) for any compact A⊆ X with Hⁿ(S)=S (self-similarity criteria) for n∈. For arbitrary Σ, the above limit exists; but it is not necessarily true that Hⁿ(S)=S. Sufficient conditions on Σ are provided to have Hⁿ(S)=S for all or some n∈, and then the dynamics of S under the admissible iterations of fᵢ's defined by Σ are investigated.