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