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Rigidity of Conformal Iterated Function Systems

Published online by Cambridge University Press:  04 December 2007

R. Daniel Mauldin
Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430 Denton, TX 76203-1430, U.S.A. E-mail: [email protected]
Feliks Przytycki
Affiliation:
Institute of Mathematics, Polish Academy of Science, ul. Śniadeckich 8, 00-950 Warsaw, Poland. E-mail: [email protected]
Mariusz Urbański
Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430 Denton, TX 76203-1430, U.S.A. E-mail: [email protected]
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Abstract

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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 conformal, satisfying the ‘Open Set Condition’, iterated function system of countably many holomorphic contractions of an open connected subset of a complex plane, the Radon–Nikodym derivative dμ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and μ is the unique probability invariant measure equivalent with m. Next, we introduce the concept of nonlinearity for iterated function systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the following rigidity result: If h, the topological conjugacy between two nonlinear systems F and G, transports the conformal measure mF to the equivalence class of the conformal measure mG, 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 countably many holomorphic contractions is nonlinear, which allows us to extend our rigidity result to the case of parabolic systems.

Type
Research Article
Copyright
© 2001 Kluwer Academic Publishers