Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Details Quasisymmetric conjugacy between quadratic dynamics and iterated function systems Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

2008-06-24

arxiv.org/abs/0806.3952v2

Ergod. Th. & Dynam. Sys. (2010), 30, 1665-1684

Mathematics

Dynamical Systems

24 pages, 4 figures; minor corrections in Section 2; accepted for publication in Ergodic Th. & Dynamical Systems

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Quasisymmetric conjugacy between quadratic dynamics and iterated function systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Quasisymmetric conjugacy between quadratic dynamics and iterated function systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quasisymmetric conjugacy between quadratic dynamics and iterated function systems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-98152