Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Abstract

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection $q: A\to A$ 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 $\Ok$. 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 $A_\lam$ is a dendrite, which happens whenever $\Ok$ 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 $q_\lam$ on $A_\lam$. 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 $(A_\lam,q_\lam)$ 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.