Rigidity of multi-dimensional conformal iterated function systems

This paper starts with an appropriate version of the bounded distortion theorem. We show that for a regular iterated function system of countably many conformal contractions of an open connected subset of a Euclidean space R-d with d greater than or equal to 3, satisfying the 'open set condition', the Radon-Nikodym derivative d mu /dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and mu is the unique probability invariant measure equivalent with tn. Next, within this context we explore the concept of the essential affinity of iterated function systems providing the several necessary and sufficient conditions. We prove the following rigidity result. If d greater than or equal to 3 and h, a topological. conjugacy between two not essentially affine systems F and G sends the conformal measure m(F) to a measure equivalent with the conformal measure m(G), then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, in exactly the same way as in Mauldin et al (2001 Compos. Math. to appear) we extend our rigidity result to the case of parabolic systems.