Sliding of self-similar sets
Let \(\mathcal{R}=\{r_1,\dots,r_m\}\) with \(r_i\in (0,1)\) and \(r_1+\dots+ r_m<1\), and denote by \(S_{\mathcal{R}}\) the collection of self-similar sets satisfying (1) \(E=\bigcup_{i=1}^m S_i(E)\), where \(S_i(x)=r_i x+b_i\) and \(b_i=0\), \(b_m=1-r_m\); (2) \([0,1]\) contains intervals \(S_1([0,1]),\dots, S_m([0,1])\) with their interior pairwise disjoint; (3) it follows from \(S_i([0,1])\cap S_j([0,1])\neq \emptyset\) that \(r_i=r_j\). The authors prove that in case \(r_1=r_m\) any two self-similar sets in \(S_{\mathcal{R}}\) are Lipschitz equivalent. Further, they consider the case \(m=3\), \(r_1+r_2+r_3<1\), such that there exists \(s\in (0,1)\) such that \(r_1^s+r_2^s +r_3^s=1\). In this case two self-similar sets \(F_{\{r_1,r_2,r_3\}}\) and \(G_{\{r_1,r_2,r_3\}}\), which are invariant sets of certain contractions \((T_1,T_2,T_3)\) and \((T'_1,T'_2,T'_3)\), are Lipschitz equivalent, when \(r_1^s\) and \(r_3^s\) are algebraically independent, i.e. \(P(r_1^s,r_2^s)\neq 0\) for any non-zero polynomial \(P\) of two variables, with integer coefficients.
scientific article; zbMATH DE number 5164227