Analytic Multifunctions, Holomorphic Motions and Hausdorff Dimension in IFSs

We study iterated function systems of contractions which depend holomorphically on a complex parameter λ. We first restrict our attention to systems which consist of similarities that satisfy the OSC. In this setting, we prove that the Hausdorff dimension of the limit set J(λ) is a continuous, subharmonic function of λ. In the remainder of the paper, systems consisting of conformal contractions are considered. We give conditions under which J(λ) and A(λ) = describe a holomorphic motion, and construct an example that shows that this is not the case in general. We finally show that A(λ) is best described as an analytic multifunction of λ, a notion that generalizes that of holomorphic motion. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT). This research was supported by the FQRNT.