Iteration of quasiregular tangent functions in three dimensions

TL;DR

This paper introduces a new three-dimensional quasiregular mapping generalizing the tangent function, and studies its dynamics, revealing parallels with complex plane tangent function behavior through novel methods.

Contribution

It defines a new quasiregular map in three dimensions and analyzes its dynamics, extending known complex tangent function results to higher dimensions with original techniques.

Findings

Established dynamics of the family λT in three dimensions

Identified geometric properties analogous to complex tangent functions

Developed original methods for quasiregular map analysis

Abstract

We define a new quasiregular mapping T in three dimensions 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 for \lambda>0, establishing results analogous to those of Devaney and Keen for the meromorphic family \lambda tan z, \lambda>0, although the methods used are necessarily original.