Non-linear iterated function systems with overlaps
type: Item
instance of: scholarly article
title: Non-linear iterated function systems with overlaps
zbMATH Open document ID: 0932.54036
DOI: 10.1023/A:1004738721853
author: B. Solomyak
published in: Periodica Mathematica Hungarica
publication date: 2000-02-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1404458
zbMATH Keywords: iterated function system
zbMATH Keywords: open set condition
zbMATH Keywords: attractor
reviewed by: Jean Ludwig
MaRDI profile type: Publication
full work available at URL: a:1004738721853
OpenAlex ID: W1699001311
description: scientific article; zbMATH DE number 1404458