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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
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