Non-linear iterated function systems with overlaps
Let \(\Phi=\{\varphi_1,\dots,\varphi_k\}\) be a collection of selfmaps on a closed interval \(X\subset \mathbb R\). The collection \(\Phi\) is called an iterated function system (i.f.s.) and it is assumed that all the functions \(\varphi_i\) belong to the space \(C^{1+\Theta}(X)\) for some \(\Theta\in(0,1]\). A nonempty compact subset \(I_\Phi\subset X\) satisfying \(I_\Phi=\bigcup^k_{i=1}\varphi_i(I_\Phi)\) is called the attractor of the system \(\Phi\). The i.f.s. is said to satisfy the Open Set Condition (OSC) if there is an open set \(\mathcal U\subset X\) such that \(\varphi_i(\mathcal U) \subset \mathcal U\) for all \(i\) and \(\varphi_i(\mathcal U)\cap\varphi_j(\mathcal U)\) is empty for \(i\neq j\). The author investigates the Hausdorff dimension of the attractor \(I_\Phi\) and its Lebesgue measure for systems \(\Phi\) which do not necessarily satisfy the condition (OSC). He also studies conditions guaranteeing the existence of invariant measure and its absolute continuity.
scientific article; zbMATH DE number 1404458