The authors study continuous iterated function systems (IFSs) defined on a compact metric space \(X\). A continuous IFS is called hyperbolic if all of its maps are uniformly contractive; otherwise non-hyperbolic. Amongst these non-hyperbolic IFSs are the weakly hyperbolic ones \(\mathrm{IFS}(f_1,\ldots,f_k)\) defined by \[ \lim_{n\to\infty} \text{diam}\, f_{\omega_0}\circ \cdots\circ f_{\omega_n}(X) = 0, \] for all \(\omega = \omega_0\ldots\omega_n\ldots\in \Sigma_k:= \{1,\ldots, k\}^\mathbb{N}\).
Define the subset \(S_{\text{wh}}\subset\Sigma_k\) of weakly hyperbolic sequences by \[ S_{\text{wh}} := \left\{\omega\in \Sigma_k : \lim_{n\to\infty}\text{diam}\, f_{\omega_0}\circ \cdots\circ f_{\omega_n}(X) = 0\right\}. \] With \(S_{\text{wh}}\) one can associate a coding map \(\pi:S_{\text{wh}}\to X\) given by \(\pi (\omega):= \lim\limits_{n\to\infty} f_{\omega_0}\circ \cdots\circ f_{\omega_n}(p)\), for arbitrary \(p\in X\).
The authors focus on the properties of so-called target sets which are defined by \[ A_{\text{tar}} := \pi (S_{\text{wh}}). \] It is shown that the closure of \(A_{\text{tar}}\) is a semi-fractal set and necessary and sufficient conditions are given for the closure of \(A_{\text{tar}}\) to be a local attractor of an IFS. In addition, random orbits of the IFSs are studied and it is proven that such orbits draw target set that are stable in a certain sense.