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 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 {lambda z, lambda z + 1} where lambda is a complex parameter in the unit disk, such that its attractor A(lambda) 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(lambda) on A(lambda). If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map p(c)(z) = z(2) + c, with the Julia set J(c) such that (A(lambda), q(lambda)) 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.