TY - JOUR

T1 - Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

AU - Eroǧlu, Kemal Ilgar

AU - Rohde, Steffen

AU - Solomyak, Boris

PY - 2010/12

Y1 - 2010/12

N2 - 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 thatq 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 p c(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.

AB - 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 thatq 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 p c(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.

UR - http://www.scopus.com/inward/record.url?scp=79451469893&partnerID=8YFLogxK

U2 - 10.1017/S0143385709000789

DO - 10.1017/S0143385709000789

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AN - SCOPUS:79451469893

SN - 0143-3857

VL - 30

SP - 1665

EP - 1684

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 6

ER -