Rigidity of conformal iterated function systems

The paper extends the rigidity of the mixing expanding repellers theorem of D. Sullivan announced at the 1986 IMC. We show that, for a regular conform al, satisfying the 'Open Set Condition', iterated function system of counta bly many holomorphic contractions of an open connected subset of a complex plane, 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 co nformal measure and mu is the unique probability invariant measure equivale nt with m. Next, we introduce the concept of nonlinearity for iterated func tion systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the fo llowing rigidity result: If h, the topological conjugacy between two nonlin ear systems F and G, transports the conformal measure m(F) to the equivalen ce class of the conformal measure m(G), then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, we prove that the hyperbolic system associated to a given parabolic system of counta bly many holomorphic contractions is nonlinear, which allows us to extend o ur rigidity result to the case of parabolic systems.