Rigidity of Multi - Dimensional Conformal Iterated Function

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, where m is the conformal measure and is the unique probability invariant measure equivalent with m. Next, we explore in this context the concept of essential aanity of iterated function systems providing its several necessary and suucient conditions. We prove the following rigidity result. If d 3 and h, a topological conjugacy between two not essentially aane 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 MPU] we extend our rigidity result to the case of parabolic systems.