Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385709000789How to cite this article: KEMAL ILGAR EROĞLU, STEFFEN ROHDE and BORIS SOLOMYAK (2010). 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' 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 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) = z 2 + 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.

Examples of coarse expanding conformal maps

“…al. establish Proposition 2.1 [ERS,Prop. 5.2] Suppose λ ∈ T and c ∈ C is a parameter such that J c is a dendrite.…”

Connectedness locus for pairs of affine maps and zeros of power series

“…The set M has been extensively studied as the Mandelbrot set for the pair of linear maps, see e.g. [4,5,1,13,12,6] and references therein.…”

Finite state automata and homeomorphism of self-similar sets

“…The study of homeomorphism of fractal sets dated back to Whyburn [20]. For studies of quasi-symmetric equivalence of fractal sets, see [4,19]. The study of Lipschitz equivalence of fractal sets derives from 1990's and it becomes a very active topic in recent years [5,6,9,11,14,15,18,22], where most of the studies focus on self-similar sets which are totally disconnected.…”

“…Whyburn [20] proved that all the Sierpinski curves are homeomorphic, which can be applied to a class of connected fractal squares. Solomyak [19] proved that a Julia set is always quasi-symmetric equivalent to a planar self-similar set with two branches. Bonk and Merenkov [4] proved that the quasi-symmetric map from Sierpinski carpet to itself must be an isometry.…”