Rigidity of infinite one-dimensional iterated function systems (Q1978962)

Let \(X \subset \mathbb R\) be a compact interval and \(I\) be a countable set containing at least two elements. Let \(S = \{\varphi _i:X \to X;\;i \in I\}\) be a collection of injective contractions of \(X\) into \(X\) such that there exists \(s \in (0,1)\) with \(|\varphi _i(x) - \varphi _i(y)|\leq s|x-y|\) for every \(i \in I,\;x,y \in X\). Then \(S\) is called an iterated function system. The authors study a special class of iterated function systems, so-called conformal iterated function systems, which were introduced in the paper by \textit{R. D. Mauldin} and \textit{M. Urbański} [Proc. Lond. Math. Soc., III. Ser. 73, No. 1, 105-154 (1996; Zbl 0852.28005)]. The sufficient and necessary conditions, under which two conformal systems of bounded geometry are bi-Lipschitz equivalent, are proved. The authors deals also with real analytic systems and scaling functions.