Non-linear iterated function systems with overlaps (Q1964617)

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.