Let C λ and Cγ be two affine Cantor sets in R with similarity dimensions d λ and dγ , respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having H d λ +dγ (C λ + Cγ ) > 0 where C λ + Cγ denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form C λ × Cγ . We prove that for Lebesgue almost all directions θ for which the projection is not one-to-one, the projection has zero (d λ + dγ )-dimensional Hausdorff measure. We demonstrate the results on the case when C λ and Cγ are the middle-(1 − 2λ) and middle-(1 − 2γ) sets.

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.

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