Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Abstract

AbstractWe 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.