The basic idea is to define new semimetrics \({\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s\) on the hyperspace of closed subsets of a locally compact metric space \(X\) with the following properties:
1. \(0\leq {\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s(A, B)\leq h_s(A, B)\) for all non-empty closed sets \(A\), \(B\) and \(h_s\) the Hausdorff semimetric.
2. \({\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s(A, B)= 0\) iff \(A\subseteq B\),
3. \({\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s\) satisfies the triangle inequality.
4. \(\lim_{n\to \infty} {\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s(\text{cl} \cup^\infty_{k= 1} A_k, A_n)= 0\) for every increasing sequence of non-empty closed sets \(\{A_n\}\).
The authors give a simple construction of some \({\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s\). They consider so-called regular finite IFS \(\{S_i; i\in I\}\) on \(X\), i.e., the Barnsley operator \(F_{I_0}\) for some \(I_0\subset I\) has a fixpoint \(A_0\) and \(F_{I_0}(A)\to A_0\) in the Hausdorff metric \(h\) for every compact non-empty subset \(A\subset X\). If \(F_I\) denotes the full Barnsley operator then \(A_*= \text{cl} \bigcup^\infty_{n= 1} F^n(A_0)\) is said to be a semifractal associated to the full IFS. For example, \(F(A_*)= A_*\) and \(\lim_{n\to \infty} {\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}(F^n(A), A_*)= 0\) for \(A\subseteq A_*\), \(A\neq \emptyset\), moreover, the semifractal is independent of the nucleus \(A_0\). Note that \({\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}\) is the metric arising from \({\mathbf d}{\mathbf i}{\mathbf s}{\mathbf t}_s\). \(A_*\) is the support for certain distributions if probabilities \(p_i\) are associated to the \(S_i\). The main result is that the authors can give a rigorous proof why certain computer programs based on non-hyperbolic IFS must work.