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Uniformly quasiregular mappings of Lattès type. (English) Zbl 0897.30008
Summary: Using an analogy of the Lattès’ construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the \(n\)-sphere \(\overline{\mathbb{R}}^n \) whose Julia set is the whole sphere. Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is \({\mathbb{S}}^{n-1}\) and its complement in \({\mathbb{S}}^{n}\) consists of two superattracting basins. In the chaotic case we study the invariant conformal structures and show that Lattès type rational mappings are either rigid or form a 1-parameter family of quasiconformal deformations.

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
37F99 Dynamical systems over complex numbers
Full Text: DOI
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