Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

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 A_λ 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 q_λ on A_λ. 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 J_c such that (A_λ,q_λ) and (J_c,p_c) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.