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Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Published online by Cambridge University Press:  24 November 2009

KEMAL ILGAR EROĞLU ,
STEFFEN ROHDE  and
BORIS SOLOMYAK
KEMAL ILGAR EROĞLU
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD) FI-40014, Finland (email: [email protected])
STEFFEN ROHDE
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle WA 98195, USA (email: [email protected], [email protected])
BORIS SOLOMYAK
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle WA 98195, USA (email: [email protected], [email protected])
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Abstract

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ‘overlap set’ 𝒪 is finite, and which are ‘invertible’ on the attractor A, in the sense that there is a continuous surjection q:A→A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at 𝒪. We suppose also that there is a rational function p with the Julia set J such that (A,q) and (J,p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz,λz+1} where λ is a complex parameter in the unit disk, such that its attractor Aλ is a dendrite, which happens whenever 𝒪 is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map qλ on Aλ. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map pc (z)=z2 +c, with the Julia set Jc such that (Aλ,qλ) and (Jc,pc) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1665 - 1684
Copyright
Copyright © Cambridge University Press 2009

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