Uniformly quasiregular mappings of Lattès type

Using an analogy of the Lattès' construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the $n$-sphere $\overline{{\Bbb R}}^n$ whose Julia set is the whole sphere. Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is ${\Bbb S}^{n-1}$ and its complement in ${\Bbb S}^{n}$ 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.