On the rigidity of one-dimensional systems of contraction similitudes (Q2371283)

The main result reads as follows: Let \(S=\{S_1,\dots, S_m\}\), \(T=\{T_1,\dots, T_m\}\) be systems of contraction similitudes in \(\mathbb R\), the invariant set of each being the segment \([0, 1]\), and let \(\varphi: K(S)\to K(T)\) be a structure-preserving homeomorphism for these two systems, such that \(\varphi(0)=0\) and \(\varphi(1)=1\). If \(Id\) is a limit point of the associated family \(F(S)\) for the system \(S\), then \(\varphi(x)\equiv x\), and \(S=T\).