Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 4 (2004), No. 1, 47--58 Copyright Heldermann Verlag 2004 Iterated Function Systems, Capacity and Green's Functions Line Baribeau lbarib@mat.ulaval.ca , Département de mathématiques et de statistique, Université Laval, Québec (QC), G1K 7P4, Canada. Dominique Brunet dbrunet@mat.ulaval.ca , Département de mathématiques et de statistique, Université Laval, Québec (QC), G1K 7P4, Canada. Thomas Ransford ransford@mat.ulaval.ca , Département de mathématiques et de statistique, Université Laval, Québec (QC), G1K 7P4, Canada. Jérémie Rostand jrostand@mat.ulaval.ca , Département de mathématiques et de statistique, Université Laval, Québec (QC), G1K 7P4, Canada. [Abstract-pdf] [Abstract-ps] Let $f_1,\dots,f_m\colon \mathbb{C}\to\mathbb{C}$ be maps satisfying $$a_j|z-w|\le |f_j(z)-f_j(w)|\le b_j|z-w|, \qquad z,w\in\mathbb{C},\,j=1,\dots,m,$$ where $0< a_j\le b_j< 1$, $j=1,\dots,m$. Let $E$ be the attractor of this iterated function system, namely the unique compact subset of $\mathbb{C}$ satisfying $E=\bigcup_1^mf_j(E)$. Assume that $E$ does not reduce to a singleton (i.e.\ that the maps $f_j$ have no common fixed point). We give a lower bound for the logarithmic capacity $c(E)$ of $E$ in terms of the diameter $\diam(E)$ and the constants $a_1,\dots,a_m,b_1,\dots,b_m$. We further prove that $$c(E\cap \overline{D}(w,r))\ge Cr^\alpha, \qquad w\in E,\,0< r\le\diam(E),$$ where $C>0$ and $\alpha=\max_j(\log a_j/\log b_j)$, and deduce that $E$ is non-thin at every point of itself. Finally, in the case where $a_j=b_j$ for each $j$ (so all the~$f_j$ are similarities), we give a simple proof that the Green's function of $E$ is Hölder continuous, and obtain estimates for the exponent of Hölder continuity. Keywords: Iterated function system, attractor, capacity, Green's function, Hölder continuous. MSC 2000: Primary 31A15, Secondary 28A80. [FullText-pdf (340 K)] [FullText-ps (272 K)]