Iteration of quasiregular tangent functions in three dimensions

Abstract

We define a new quasiregular mapping T : R 3 → R 3 ∪ { ∞ } 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 { λ T : λ > 0 } \{\lambda T:\lambda >0\} , establishing results analogous to those of Devaney and Keen for the meromorphic family { z ↦ λ tan ⁡ z : λ > 0 } \{z\mapsto \lambda \tan z:\lambda >0\} , although the methods used are necessarily original.