Iteration of quasiregular tangent functions in three dimensions

We define a new quasiregular mapping $ T:\mathbb{R}^3\to \mathbb{R}^3 \cup \{\infty \}$ that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family $ \{\lambda T:\lambda >0\}$, establishing results analogous to those of Devaney and Keen for the meromorphic family $ \{z\mapsto \lambda \tan z:\lambda >0\}$, although the methods used are necessarily original.