Computational Methods and Function Theory

, Volume 4, Issue 1, pp 47–58

# Iterated Function Systems, Capacity and Green’s Functions

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

## 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.

## Keywords

Iterated function system attractor capacity Green’s function Hölder continuous

31A15 28A80

## References

1. 1.
V. V. Andrievskii, The highest smoothness of the Green function implies the highest density of a set, preprint.Google Scholar
2. 2.
A. F. Beardon, Complex Analysis, Wiley, Chichester, 1979.
3. 3.
L. Bialas and A. Volberg, Markov’s property of the Cantor ternary set, Studia Math. 104 (1993), 259–268.
4. 4.
L. Carleson and T. W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.
5. 5.
K. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
6. 6.
T. J. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.
7. 7.
A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111–115.
8. 8.
R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s, Proc. Amer. Math. Soc. 128 (2000), 2569–2575.
9. 9.
R. Stankewitz, Uniformly perfect analytic and conformal attractor sets, Bull. London Math. Soc. 33 (2001), 320–330.
10. 10.
T. Sugawa, Uniformly perfect sets: analytic and geometric aspects, Sugaku Expositions 16 (2003), 225–242.
11. 11.
T. Sugawa, On boundary regularity of the Dirichlet problem for plane domains, preprint.Google Scholar
12. 12.
F. Xie, Y. Yin and Y. Sun, Uniform perfectness of self-affine sets, Proc. Amer. Math. Soc. 131 (2003), 3053–3057.

## Authors and Affiliations

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