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Attractor sets and Julia sets in low dimensions. (English) Zbl 1445.30014
Let $$j \in \{1,2\}$$, and let $$X$$ be a conformal iterated function system generated by contractions $$\varphi_1, \ldots, \varphi_m$$ acting on the closed unit ball of $$\mathbb{R}^j$$ and so that at least two of the maps generating $$X$$ have distinct fixed points. The main theorem states that if the images of the open unit ball $$\mathbb{B}^j$$ under $$\varphi_1, \ldots, \varphi_m$$ are quasiballs contained relatively compact in $$\mathbb{B}^j$$, then there exists a quasiregular semigroup $$G$$ acting on $$\mathbb{R}^j$$ so that the Julia set of $$G$$ equals the attractor set of $$X$$. Moreover, it is shown that in $$2D$$ every $$X$$ which satisfies the strong open set condition (that is, the closures of $$\varphi_i(\mathbb{B}^2)$$, $$i=1, \ldots, m$$, are pairwise disjoint and contained in $$\mathbb{B}^2$$), there exists a uniformly quasiregular map $$f :\mathbb{C} \to \mathbb{C}$$ with Julia set equal to the attractor set of $$X$$. As a consequence, in this case the attractor set of $$X$$ is quasiconformally equivalent to the Julia set of a rational map.
##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 30C62 Quasiconformal mappings in the complex plane 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 37F05 Dynamical systems involving relations and correspondences in one complex variable
##### Keywords:
iterated function system; Julia set; quasiregular semigroup
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##### References:
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