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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.

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 37F99 Dynamical systems over complex numbers
##### Keywords:
Julia set; Lattès type rational mappings
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##### References:
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