  The orbit of a point $x\in X$ in a classical iterated function system (IFS)
can be defined as $\{f_u(x)=f_{u_n}\circ\cdots \circ f_{u_1}(x):$ $u=u_1\cdots
u_n$ is a word of a full shift $\Sigma$ on finite symbols and $f_{u_i}$ is a
continuous self map on $X$ $\}$. One also can associate to
$\sigma=\sigma_1\sigma_2\cdots\in\Sigma$ a non-autonomous system
$(X,\,f_\sigma)$ where the trajectory of $x\in X$ is defined as
$x,\,f_{\sigma_1}(x),\,f_{\sigma_1\sigma_2}(x),\ldots$.Here instead of the full
shift, we consider an arbitrary shift space $\Sigma$. Then we investigate basic
properties related to this IFS and the associated non-autonomous systems. In
particular, we look for sufficient conditions that guarantees that in a
transitive IFS one may have a transitive $(X,\,f_\sigma)$ for some
$\sigma\in\Sigma$ and how abundance are such $\sigma$'s.
