The paper starts with an appropriate version of the bounded distortion theorem. We show that for a regular, satisfying the "Open Set Condition", iterated function system of countably many conformal contractions of an open connected subset of a Euclidean space IR d with d 3, the Radon-Nikodym derivative dd=dm has a real-analytic extension on an open neighbourhood of the limit set of this system,...

The following discussion is closely connected with Lebesgue's theorem that the derivative of an integral is equal to the integrand almost everywhere. I t is well known that in generalizing this theorem to higher dimensions, great care must be exercised in the choice of the systems of intervals or sets used for ^-dimensional differentiation. Lebesgue had already observed that arbitrary intervals...