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

Consider a compact metric space $X$, and let $\mathcal{F}=\{f_1,f_2,\dots ,f_k\}$ be a set of contracting and continuous self maps on $X$. Let $\mathrm{\Sigma }$ be a sub-shift on $k$ symbols, and let ${\mathrm{\Sigma }}_k$ be the full shift. Define ${\mathcal{L}}_n(\mathrm{\Sigma })$ as the set of words of length $n$ in $\mathrm{\Sigma }$. For $u=u_1\cdots u_n\in {\mathcal{L}}_n(\mathrm{\Sigma })$, set $f_u:=f_{u_n}\circ \cdots \circ f_{u_1}$ and $H^n(\cdot ):={\cup }_{u\in {\mathcal{L}}_n({\mathrm{\Sigma }}_k)}{f}_u(\cdot )$. When $\mathrm{\Sigma }={\mathrm{\Sigma }}_k$, $H^n(\cdot )$ is the $n$th iteration of the Hutchinson’s operator, and there exists a compact set $S={\lim }_{n\to \mathrm{\infty }}{H}^n(A)$ for any compact $A\subseteq X$ with $H^n(S)=S$ (self-similarity criteria) for $n\in \mathbb{N}$. For arbitrary $\mathrm{\Sigma }$, the above limit exists; but it is not necessarily true that $H^n(S)=S$. Sufficient conditions on $\mathrm{\Sigma }$ are provided to have $H^n(S)=S$ for all or some $n\in \mathbb{N}$, and then the dynamics of $S$ under the admissible iterations of $f_i$’s defined by $\mathrm{\Sigma }$ are investigated.