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We construct a quasiregular analogue of the function $z\exp(z)$ in dimension3, which gives the first explicit example of a quasiregular mapping oftranscendental type that has exactly one zero. We then modify the constructionto create a family of such quasiregular mappings and study their dynamics. Fromthis, we also construct the first quasimeromorphic mappings with an essentialsingularity at infinity where the backward orbit of infinity is non-empty andfinite.

Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3