Iterated function systems, capacity and Green's functions
The iterated function system \(f_1, \ldots, f_m: \mathbb C\to\mathbb C\) with Lipschitz bounds \[ a_j | z-w| \leq | f_j(z)-f_j(w)| \leq b_j | z-w| , \] for \(0<a_j\leq b_j<1\) and \(1\leq i\leq m\), defines a unique nonvoid and compact attractor \(E\), a fractal. The authors consider the potential theoretic properties of \(E\subset \mathbb C\), provided it is not a singleton: Global and local lower bounds of the (logarithmic) capacity of \(E\), \(E\) is not thin at every point of itself. When the iterated function system in addition consists of similitudes, then \(E\) is uniformly perfect and the Green's function vanishing on \(E\) is Hölder continuous.
scientific article; zbMATH DE number 2104114