Computational Methods and Function Theory 4 (2004), No. 1, 47--58
Copyright Heldermann Verlag 2004
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.
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.
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