Iterated Function Systems, Capacity and Green’s Functions

Abstract

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.