Computational Methods and Function Theory

, Volume 4, Issue 1, pp 47–58 | Cite as

Iterated Function Systems, Capacity and Green’s Functions

  • Line Baribeau
  • Dominique Brunet
  • Thomas Ransford
  • Jérémie Rostand


Let \(f_{1},\dots.,f_{m}\colon {\rm C}\rightarrow{\rm C}\) be maps satisfying
$$a_{j}|z- w|\leq |f_{j}(z)- f_{j}(w)|\leq b_{j}|z- w|,\ z,w\in {C},j=1,\dots,m,$$
where \(0< a_{j}\leq b_{j}< 1, j=1,\dots,m\). Let E be the attractor of this iterated function system, namely the unique compact subset of ℂ satisfying \(E=\bigcup_{1}^{m}f_{j}(E)\). 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
$$c(E\cap \overline{D}(w,r))\geq Cr^{\alpha},\qquad w\in E,0< r\leq {\rm diam}(E),$$
, 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.


Iterated function system attractor capacity Green’s function Hölder continuous 

2000 MSC

31A15 28A80 


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Copyright information

© Heldermann  Verlag 2004

Authors and Affiliations

  • Line Baribeau
    • 1
  • Dominique Brunet
    • 1
  • Thomas Ransford
    • 1
  • Jérémie Rostand
    • 1
  1. 1.Département de mathématiques et de statistiqueUniversité LavalQuébecCanada

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