Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components

We construct examples of uniformly quasiregular mappings (uqr) acting on a sphere and having Fatou set consisting of infinitely many components. In particular we construct a uqr mapping providing a higher dimensional counterpart for the polynomial $z\mapsto z^2-1$ acting on the extended complex plane $\widehat{\mathbb{C}}$.