Iterated function systems, capacity and Green's functions
type: Item
instance of: scholarly article
title: Iterated function systems, capacity and Green's functions
zbMATH Open document ID: 1052.31001
DOI: 10.1007/BF03321055
author: Line Baribeau
author: Dominique Brunet
author: Thomas J. Ransford
author: Jérémie Rostand
published in: Computational Methods and Function Theory
publication date: 2004-09-28T00:00:00Z
review text: The iterated function system \(f_1, \ldots, f_m: \mathbb C\to\mathbb C\) with Lipschitz bounds \[ a_j | z-w| \leq | f_j(z)-f_j(w)| \leq b_j | z-w| , \] for \(0<a_j\leq b_j<1\) and \(1\leq i\leq m\), defines a unique nonvoid and compact attractor \(E\), a fractal. The authors consider the potential theoretic properties of \(E\subset \mathbb C\), provided it is not a singleton: Global and local lower bounds of the (logarithmic) capacity of \(E\), \(E\) is not thin at every point of itself. When the iterated function system in addition consists of similitudes, then \(E\) is uniformly perfect and the Green's function vanishing on \(E\) is Hölder continuous.
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2104114
zbMATH Keywords: logarithmic capacity
zbMATH Keywords: logarithmic Green's function
reviewed by: Volker Metz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03321055
OpenAlex ID: W2016156932
description: scientific article; zbMATH DE number 2104114